ctgsylv - Man Page
Name
ctgsylv — C-Interface
— C-Interface for triangular generalized Sylvester equations.
Synopsis
Functions
void mepack_double_tgcsylv_dag (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
Level-3 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation with DAG scheduling.
void mepack_single_tgcsylv_dag (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
Level-3 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation with DAG scheduling.
void mepack_double_tgcsylv_dual_dag (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
OpenMP4 - DAG Algorithm for the dual generalized coupled Sylvester equation.
void mepack_single_tgcsylv_dual_dag (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
OpenMP4 - DAG Algorithm for the dual generalized coupled Sylvester equation.
void mepack_double_tgsylv_dag (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *X, int LDX, double *SCALE, double *WORK, int *INFO)
Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation with DAG based parallelization.
void mepack_single_tgsylv_dag (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *X, int LDX, float *SCALE, float *WORK, int *INFO)
Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation with DAG based parallelization.
void mepack_double_tgcsylv_level2 (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation.
void mepack_double_tgcsylv_level2_unopt (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Unoptimized variant)
void mepack_double_tgcsylv_level2_reorder (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Reordered variant)
void mepack_double_tgcsylv_level2_local_copy (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Local copies, unaligned)
void mepack_double_tgcsylv_level2_local_copy_32 (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Local copies, M,N<=32)
void mepack_double_tgcsylv_level2_local_copy_64 (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Local copies, M,N<=64)
void mepack_double_tgcsylv_level2_local_copy_96 (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Local copies, M,N<=96)
void mepack_double_tgcsylv_level2_local_copy_128 (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Local copies, M,N<=128)
void mepack_single_tgcsylv_level2 (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation.
void mepack_single_tgcsylv_level2_unopt (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Unoptimized variant)
void mepack_single_tgcsylv_level2_reorder (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Reordered variant)
void mepack_single_tgcsylv_level2_local_copy (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Local copies, unaligned)
void mepack_single_tgcsylv_level2_local_copy_32 (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Local copies, M,N<=32)
void mepack_single_tgcsylv_level2_local_copy_64 (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Local copies, M,N<=64)
void mepack_single_tgcsylv_level2_local_copy_96 (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Local copies, M,N<=96)
void mepack_single_tgcsylv_level2_local_copy_128 (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Local copies, M,N<=128)
void mepack_double_tgcsylv_dual_level2 (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation.
void mepack_double_tgcsylv_dual_level2_local_copy (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation (Local copy variant)
void mepack_double_tgcsylv_dual_level2_local_copy_32 (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation (Local copy variant, M,N <= 32)
void mepack_double_tgcsylv_dual_level2_local_copy_64 (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation (Local copy variant, M,N <= 64)
void mepack_double_tgcsylv_dual_level2_local_copy_96 (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation (Local copy variant, M,N <= 96)
void mepack_double_tgcsylv_dual_level2_local_copy_128 (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation (Local copy variant, M,N <= 128)
void mepack_single_tgcsylv_dual_level2 (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation.
void mepack_single_tgcsylv_dual_level2_local_copy (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation ( Local Copies )
void mepack_single_tgcsylv_dual_level2_local_copy_32 (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation ( Local Copies, M,N <= 32 )
void mepack_single_tgcsylv_dual_level2_local_copy_64 (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation ( Local Copies, M,N <= 64 )
void mepack_single_tgcsylv_dual_level2_local_copy_96 (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation ( Local Copies, M,N <= 96 )
void mepack_single_tgcsylv_dual_level2_local_copy_128 (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation ( Local Copies, M,N <= 128 )
void mepack_double_tgsylv_level2_local_copy_32 (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *X, int LDX, double *SCALE, double *WORK, int *INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Local copies, M,N<=32)
void mepack_double_tgsylv_level2_local_copy_64 (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *X, int LDX, double *SCALE, double *WORK, int *INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Local copies, M,N<=64)
void mepack_double_tgsylv_level2_local_copy_96 (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *X, int LDX, double *SCALE, double *WORK, int *INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Local copies, M,N<=96)
void mepack_double_tgsylv_level2_local_copy_128 (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *X, int LDX, double *SCALE, double *WORK, int *INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Local copies, M,N<=128)
void mepack_double_tgsylv_level2_local_copy (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *X, int LDX, double *SCALE, double *WORK, int *INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Local copies, unaligned)
void mepack_double_tgsylv_level2_reorder (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *X, int LDX, double *SCALE, double *WORK, int *INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Reordered variant)
void mepack_double_tgsylv_level2 (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *X, int LDX, double *SCALE, double *WORK, int *INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Classical level 2 variant)
void mepack_double_tgsylv_level2_rowwise (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *X, int LDX, double *SCALE, double *WORK, int *INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Row-by-Row variant)
void mepack_double_tgsylv_level2_colwise (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *X, int LDX, double *SCALE, double *WORK, int *INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Column-by-column variant)
void mepack_double_tgsylv_gardiner_laub (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *X, int LDX, double *SCALE, double *WORK, int *INFO)
GARDINER-Laub Algorithm for the generalized Sylvester equation.
void mepack_single_tgsylv_level2_local_copy_32 (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *X, int LDX, float *SCALE, float *WORK, int *INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Local copies, M,N<=32)
void mepack_single_tgsylv_level2_local_copy_64 (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *X, int LDX, float *SCALE, float *WORK, int *INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Local copies, M,N<=64)
void mepack_single_tgsylv_level2_local_copy_96 (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *X, int LDX, float *SCALE, float *WORK, int *INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Local copies, M,N<=96)
void mepack_single_tgsylv_level2_local_copy_128 (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *X, int LDX, float *SCALE, float *WORK, int *INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Local copies, M,N<=128)
void mepack_single_tgsylv_level2_local_copy (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *X, int LDX, float *SCALE, float *WORK, int *INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Local copies, unaligned)
void mepack_single_tgsylv_level2_reorder (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *X, int LDX, float *SCALE, float *WORK, int *INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Reordered variant)
void mepack_single_tgsylv_level2 (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *X, int LDX, float *SCALE, float *WORK, int *INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Classic level 2variant)
void mepack_single_tgsylv_level2_rowwise (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *X, int LDX, float *SCALE, float *WORK, int *INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Row-by-row variant)
void mepack_single_tgsylv_level2_colwise (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *X, int LDX, float *SCALE, float *WORK, int *INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Column-by-column variant)
void mepack_single_tgsylv_gardiner_laub (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *X, int LDX, float *SCALE, float *WORK, int *INFO)
Gardiner-Laub Algorithm for the generalized Sylvester equation.
void mepack_double_tgcsylv_level3 (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
Level-3 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation.
void mepack_double_tgcsylv_level3_unopt (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
Level-3 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Unoptimized variant)
void mepack_double_tgcsylv_level3_2stage (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
Level-3 Kagstroem-Westin/Bartels-Stewart Algorithm with sub-blocking for the generalized coupled Sylvester equation.
void mepack_single_tgcsylv_level3 (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
Level-3 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation.
void mepack_single_tgcsylv_level3_unopt (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
Level-3 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation.
void mepack_single_tgcsylv_level3_2stage (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
Level-3 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation.
void mepack_double_tgcsylv_dual_level3 (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
Level-3 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation.
void mepack_single_tgcsylv_dual_level3 (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
Level-3 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation.
void mepack_double_tgcsylv_dual_level3_2stage (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
Level-3 with sub blocking Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation.
void mepack_single_tgcsylv_dual_level3_2stage (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
Level-3 Kagstroem-Westin/Bartels-Stewart Algorithm with sub blocking for the dual generalized coupled Sylvester equation.
void mepack_double_tgsylv_level3 (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *X, int LDX, double *SCALE, double *WORK, int *INFO)
Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation.
void mepack_double_tgsylv_level3_rowwise (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *X, int LDX, double *SCALE, double *WORK, int *INFO)
Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation (Row-by-row variant)
void mepack_double_tgsylv_level3_colwise (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *X, int LDX, double *SCALE, double *WORK, int *INFO)
Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation (Column-by-Column variant)
void mepack_double_tgsylv_level3_2stage (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *X, int LDX, double *SCALE, double *WORK, int *INFO)
Level-3 Bartels-Stewart Algorithm with sub-blocking for the generalized Sylvester equation.
void mepack_single_tgsylv_level3 (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *X, int LDX, float *SCALE, float *WORK, int *INFO)
Level-3 Bartels-Stewart Algorithm for the generalized Sylvester.
void mepack_single_tgsylv_level3_rowwise (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *X, int LDX, float *SCALE, float *WORK, int *INFO)
Level-3 Bartels-Stewart Algorithm for the generalized Sylvester (Row-by-row variant)
void mepack_single_tgsylv_level3_colwise (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *X, int LDX, float *SCALE, float *WORK, int *INFO)
Level-3 Bartels-Stewart Algorithm for the generalized Sylvester (Column-by-column variant)
void mepack_single_tgsylv_level3_2stage (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *X, int LDX, float *SCALE, float *WORK, int *INFO)
Level-3 Bartels-Stewart Algorithm with sub-blocking for the generalized Sylvester.
void mepack_double_tgcsylv_recursive (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
Recursive blocking Algorithm for the generalized coupled Sylvester equation.
void mepack_single_tgcsylv_recursive (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
Recursive blocking Algorithm for the generalized coupled Sylvester equation.
void mepack_double_tgcsylv_dual_recursive (const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, int *INFO)
Recursive blocking algorithm for the dual generalized coupled Sylvester equation.
void mepack_single_tgcsylv_dual_recursive (const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, int *INFO)
Recursive blocking algorithm for the dual generalized coupled Sylvester equation.
void mepack_double_tgsylv_recursive (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *X, int LDX, double *SCALE, double *WORK, int *INFO)
Recursive blocking Algorithm for the generalized Sylvester equation.
void mepack_single_tgsylv_recursive (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *X, int LDX, float *SCALE, float *WORK, int *INFO)
Recursive blocking algorithm for the generalized Sylvester equation.
Detailed Description
C-Interface for triangular generalized Sylvester equations.
The Fortran routines to solve the generalized Sylvester equation with triangular coefficients are wrapped in C to provide an easier access to them. All wrapper routines are direct wrappers to the corresponding Fortran subroutines without sanity checks. These are performed by the Fortran routines. Since the routines are using int values to pass sizes the work_space query will fail for large scale problems. For this reason the function mepack_memory should be used to query the required work_space from a C code. This function is aware of 64 bit integers if MEPACK is compiled with it.
Function Documentation
void mepack_double_tgcsylv_dag (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
Level-3 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation with DAG scheduling.
Purpose:
mepack_double_tgcsylv_dag solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
The function is a wrapper for dla_tgcsylv_dag
- See also
dla_tgcsylv_dag
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 214 of file tgcsylv.c.
void mepack_double_tgcsylv_dual_dag (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
OpenMP4 - DAG Algorithm for the dual generalized coupled Sylvester equation.
Purpose:
mepack_double_tgcsylv_dual_dag solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * L = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular
matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or
D can be quasi upper triangular as well. The right hand side Y and the solution X
M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are
created by DGGES from LAPACK.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
| kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for dla_tgcsylv_dual_dag.
See also
dla_tgcsylv_dual_dag
Parameters
TRANSA
TRANSA is CHARACTER(1)
Specifies the form of the system of equations with respect to A and C:
== 'N': op1(A) = A, op1(C) = C (No transpose for A and C)
== 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is CHARACTER(1)
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B, op2(D) = D (No transpose for B and D)
== 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N)
The matrix B must be (quasi-) upper triangular. If the matrix D is already
quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M)
The matrix C must be upper triangular.LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N)
The matrix D must be (quasi-) upper triangular. If the matrix B is already
quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER
The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory.INFO
INFO is INTEGER
On input:
== -1 : Perform a workspace query ( not recommended in the C interface)
<> -1 : normal operation
On exit, workspace query:
< 0 : if INFO == -i, the i-Th argument had an illegal value
>= 0: The value of INFO is the required number of elements in the workspace.
On exit, normal operation:
== 0: successful exit
< 0: if INFO == -i, the i-Th argument had an illegal value
> 0: The equation is not solved correctly. One of the arising inner
system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 232 of file tgcsylv_dual.c.
void mepack_double_tgcsylv_dual_level2 (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation.
Purpose:
mepack_double_tgcsylv_dual_level2 solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * L = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular
matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or
D can be quasi upper triangular as well. The right hand side Y and the solution X
M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are
created by DGGES from LAPACK.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
| kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for dla_tgcsylv_dual_l2.
See also
dla_tgcsylv_dual_l2
Parameters
TRANSA
TRANSA is CHARACTER(1)
Specifies the form of the system of equations with respect to A and C:
== 'N': op1(A) = A, op1(C) = C (No transpose for A and C)
== 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is CHARACTER(1)
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B, op2(D) = D (No transpose for B and D)
== 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N)
The matrix B must be (quasi-) upper triangular. If the matrix D is already
quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M)
The matrix C must be upper triangular.LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N)
The matrix D must be (quasi-) upper triangular. If the matrix B is already
quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER
The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory.INFO
INFO is INTEGER
On input:
== -1 : Perform a workspace query ( not recommended in the C interface)
<> -1 : normal operation
On exit, workspace query:
< 0 : if INFO == -i, the i-Th argument had an illegal value
>= 0: The value of INFO is the required number of elements in the workspace.
On exit, normal operation:
== 0: successful exit
< 0: if INFO == -i, the i-Th argument had an illegal value
> 0: The equation is not solved correctly. One of the arising inner
system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 231 of file tgcsylv_dual.c.
void mepack_double_tgcsylv_dual_level2_local_copy (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation (Local copy variant)
Purpose:
mepack_double_tgcsylv_dual_level2_local_copy solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * R = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular
matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or
D can be quasi upper triangular as well. The right hand side Y and the solution X
M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are
created by DGGES from LAPACK.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
| kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for dla_tgcsylv_dual_l2_local_copy.
See also
dla_tgcsylv_dual_l2_local_copy
Parameters
TRANSA
TRANSA is CHARACTER(1)
Specifies the form of the system of equations with respect to A and C:
== 'N': op1(A) = A, op1(C) = C (No transpose for A and C)
== 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is CHARACTER(1)
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B, op2(D) = D (No transpose for B and D)
== 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N)
The matrix B must be (quasi-) upper triangular. If the matrix D is already
quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M)
The matrix C must be upper triangular.LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N)
The matrix D must be (quasi-) upper triangular. If the matrix B is already
quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER
The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory.INFO
INFO is INTEGER
On input:
== -1 : Perform a workspace query ( not recommended in the C interface)
<> -1 : normal operation
On exit, workspace query:
< 0 : if INFO == -i, the i-Th argument had an illegal value
>= 0: The value of INFO is the required number of elements in the workspace.
On exit, normal operation:
== 0: successful exit
< 0: if INFO == -i, the i-Th argument had an illegal value
> 0: The equation is not solved correctly. One of the arising inner
system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 460 of file tgcsylv_dual.c.
void mepack_double_tgcsylv_dual_level2_local_copy_128 (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation (Local copy variant, M,N <= 128)
Purpose:
mepack_double_tgcsylv_dual_level2_local_copy_128 solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * R = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular
matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or
D can be quasi upper triangular as well. The right hand side Y and the solution X
M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are
created by DGGES from LAPACK.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
| kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for dla_tgcsylv_dual_l2_local_copy_128.
See also
dla_tgcsylv_dual_l2_local_copy_128
Parameters
TRANSA
TRANSA is CHARACTER(1)
Specifies the form of the system of equations with respect to A and C:
== 'N': op1(A) = A, op1(C) = C (No transpose for A and C)
== 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is CHARACTER(1)
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B, op2(D) = D (No transpose for B and D)
== 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N)
The matrix B must be (quasi-) upper triangular. If the matrix D is already
quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M)
The matrix C must be upper triangular.LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N)
The matrix D must be (quasi-) upper triangular. If the matrix B is already
quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER
The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory.INFO
INFO is INTEGER
On input:
== -1 : Perform a workspace query ( not recommended in the C interface)
<> -1 : normal operation
On exit, workspace query:
< 0 : if INFO == -i, the i-Th argument had an illegal value
>= 0: The value of INFO is the required number of elements in the workspace.
On exit, normal operation:
== 0: successful exit
< 0: if INFO == -i, the i-Th argument had an illegal value
> 0: The equation is not solved correctly. One of the arising inner
system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 1376 of file tgcsylv_dual.c.
void mepack_double_tgcsylv_dual_level2_local_copy_32 (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation (Local copy variant, M,N <= 32)
Purpose:
mepack_double_tgcsylv_dual_level2_local_copy_32 solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * R = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular
matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or
D can be quasi upper triangular as well. The right hand side Y and the solution X
M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are
created by DGGES from LAPACK.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
| kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for dla_tgcsylv_dual_l2_local_copy_32.
See also
dla_tgcsylv_dual_l2_local_copy_32
Parameters
TRANSA
TRANSA is CHARACTER(1)
Specifies the form of the system of equations with respect to A and C:
== 'N': op1(A) = A, op1(C) = C (No transpose for A and C)
== 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is CHARACTER(1)
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B, op2(D) = D (No transpose for B and D)
== 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N)
The matrix B must be (quasi-) upper triangular. If the matrix D is already
quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M)
The matrix C must be upper triangular.LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N)
The matrix D must be (quasi-) upper triangular. If the matrix B is already
quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER
The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory.INFO
INFO is INTEGER
On input:
== -1 : Perform a workspace query ( not recommended in the C interface)
<> -1 : normal operation
On exit, workspace query:
< 0 : if INFO == -i, the i-Th argument had an illegal value
>= 0: The value of INFO is the required number of elements in the workspace.
On exit, normal operation:
== 0: successful exit
< 0: if INFO == -i, the i-Th argument had an illegal value
> 0: The equation is not solved correctly. One of the arising inner
system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 689 of file tgcsylv_dual.c.
void mepack_double_tgcsylv_dual_level2_local_copy_64 (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation (Local copy variant, M,N <= 64)
Purpose:
mepack_double_tgcsylv_dual_level2_local_copy_64 solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * R = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular
matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or
D can be quasi upper triangular as well. The right hand side Y and the solution X
M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are
created by DGGES from LAPACK.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
| kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for dla_tgcsylv_dual_l2_local_copy_64.
See also
dla_tgcsylv_dual_l2_local_copy_64
Parameters
TRANSA
TRANSA is CHARACTER(1)
Specifies the form of the system of equations with respect to A and C:
== 'N': op1(A) = A, op1(C) = C (No transpose for A and C)
== 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is CHARACTER(1)
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B, op2(D) = D (No transpose for B and D)
== 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N)
The matrix B must be (quasi-) upper triangular. If the matrix D is already
quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M)
The matrix C must be upper triangular.LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N)
The matrix D must be (quasi-) upper triangular. If the matrix B is already
quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER
The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory.INFO
INFO is INTEGER
On input:
== -1 : Perform a workspace query ( not recommended in the C interface)
<> -1 : normal operation
On exit, workspace query:
< 0 : if INFO == -i, the i-Th argument had an illegal value
>= 0: The value of INFO is the required number of elements in the workspace.
On exit, normal operation:
== 0: successful exit
< 0: if INFO == -i, the i-Th argument had an illegal value
> 0: The equation is not solved correctly. One of the arising inner
system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 917 of file tgcsylv_dual.c.
void mepack_double_tgcsylv_dual_level2_local_copy_96 (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation (Local copy variant, M,N <= 96)
Purpose:
mepack_double_tgcsylv_dual_level2_local_copy_96 solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * R = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular
matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or
D can be quasi upper triangular as well. The right hand side Y and the solution X
M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are
created by DGGES from LAPACK.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
| kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for dla_tgcsylv_dual_l2_local_copy_96.
See also
dla_tgcsylv_dual_l2_local_copy_96
Parameters
TRANSA
TRANSA is CHARACTER(1)
Specifies the form of the system of equations with respect to A and C:
== 'N': op1(A) = A, op1(C) = C (No transpose for A and C)
== 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is CHARACTER(1)
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B, op2(D) = D (No transpose for B and D)
== 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N)
The matrix B must be (quasi-) upper triangular. If the matrix D is already
quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M)
The matrix C must be upper triangular.LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N)
The matrix D must be (quasi-) upper triangular. If the matrix B is already
quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER
The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory.INFO
INFO is INTEGER
On input:
== -1 : Perform a workspace query ( not recommended in the C interface)
<> -1 : normal operation
On exit, workspace query:
< 0 : if INFO == -i, the i-Th argument had an illegal value
>= 0: The value of INFO is the required number of elements in the workspace.
On exit, normal operation:
== 0: successful exit
< 0: if INFO == -i, the i-Th argument had an illegal value
> 0: The equation is not solved correctly. One of the arising inner
system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 1147 of file tgcsylv_dual.c.
void mepack_double_tgcsylv_dual_level3 (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
Level-3 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation.
Purpose:
mepack_double_tgcsylv_dual_level3 solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * L = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular
matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or
D can be quasi upper triangular as well. The right hand side Y and the solution X
M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are
created by DGGES from LAPACK.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
| kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for dla_tgcsylv_dual_l3.
See also
dla_tgcsylv_dual_l3
Parameters
TRANSA
TRANSA is CHARACTER(1)
Specifies the form of the system of equations with respect to A and C:
== 'N': op1(A) = A, op1(C) = C (No transpose for A and C)
== 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is CHARACTER(1)
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B, op2(D) = D (No transpose for B and D)
== 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N)
The matrix B must be (quasi-) upper triangular. If the matrix D is already
quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M)
The matrix C must be upper triangular.LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N)
The matrix D must be (quasi-) upper triangular. If the matrix B is already
quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER
The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory.INFO
INFO is INTEGER
On input:
== -1 : Perform a workspace query ( not recommended in the C interface)
<> -1 : normal operation
On exit, workspace query:
< 0 : if INFO == -i, the i-Th argument had an illegal value
>= 0: The value of INFO is the required number of elements in the workspace.
On exit, normal operation:
== 0: successful exit
< 0: if INFO == -i, the i-Th argument had an illegal value
> 0: The equation is not solved correctly. One of the arising inner
system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 231 of file tgcsylv_dual.c.
void mepack_double_tgcsylv_dual_level3_2stage (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
Level-3 with sub blocking Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation.
Purpose:
mepack_double_tgcsylv_dual_level3_2stage solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * R = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular
matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or
D can be quasi upper triangular as well. The right hand side Y and the solution X
M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are
created by DGGES from LAPACK.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
| kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for dla_tgcsylv_dual_l3_2s.
See also
dla_tgcsylv_dual_l3_2s
Parameters
TRANSA
TRANSA is CHARACTER(1)
Specifies the form of the system of equations with respect to A and C:
== 'N': op1(A) = A, op1(C) = C (No transpose for A and C)
== 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is CHARACTER(1)
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B, op2(D) = D (No transpose for B and D)
== 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N)
The matrix B must be (quasi-) upper triangular. If the matrix D is already
quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M)
The matrix C must be upper triangular.LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N)
The matrix D must be (quasi-) upper triangular. If the matrix B is already
quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER
The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory.INFO
INFO is INTEGER
On input:
== -1 : Perform a workspace query ( not recommended in the C interface)
<> -1 : normal operation
On exit, workspace query:
< 0 : if INFO == -i, the i-Th argument had an illegal value
>= 0: The value of INFO is the required number of elements in the workspace.
On exit, normal operation:
== 0: successful exit
< 0: if INFO == -i, the i-Th argument had an illegal value
> 0: The equation is not solved correctly. One of the arising inner
system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 687 of file tgcsylv_dual.c.
void mepack_double_tgcsylv_dual_recursive (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
Recursive blocking algorithm for the dual generalized coupled Sylvester equation.
Purpose:
mepack_double_tgcsylv_dual_recursive solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * L = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular
matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or
D can be quasi upper triangular as well. The right hand side Y and the solution X
M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are
created by DGGES from LAPACK.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
| kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for dla_tgcsylv_dual_recursive.
See also
dla_tgcsylv_dual_recursive
Parameters
TRANSA
TRANSA is CHARACTER(1)
Specifies the form of the system of equations with respect to A and C:
== 'N': op1(A) = A, op1(C) = C (No transpose for A and C)
== 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is CHARACTER(1)
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B, op2(D) = D (No transpose for B and D)
== 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N)
The matrix B must be (quasi-) upper triangular. If the matrix D is already
quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M)
The matrix C must be upper triangular.LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N)
The matrix D must be (quasi-) upper triangular. If the matrix B is already
quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER
The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory.INFO
INFO is INTEGER
On input:
== -1 : Perform a workspace query ( not recommended in the C interface)
<> -1 : normal operation
On exit, workspace query:
< 0 : if INFO == -i, the i-Th argument had an illegal value
>= 0: The value of INFO is the required number of elements in the workspace.
On exit, normal operation:
== 0: successful exit
< 0: if INFO == -i, the i-Th argument had an illegal value
> 0: The equation is not solved correctly. One of the arising inner
system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 231 of file tgcsylv_dual.c.
void mepack_double_tgcsylv_level2 (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation.
Purpose:
mepack_double_tgcsylv_level2 solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
The function is a wrapper for dla_tgcsylv_l2
- See also
dla_tgcsylv_l2
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 214 of file tgcsylv.c.
void mepack_double_tgcsylv_level2_local_copy (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Local copies, unaligned)
Purpose:
mepack_double_tgcsylv_level2_local_copy solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
The function is a wrapper for dla_tgcsylv_l2_local_copy
- See also
dla_tgcsylv_l2_local_copy
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 847 of file tgcsylv.c.
void mepack_double_tgcsylv_level2_local_copy_128 (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Local copies, M,N<=128)
Purpose:
mepack_double_tgcsylv_level2_local_copy_128 solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
The function is a wrapper for dla_tgcsylv_l2_local_copy_128
- See also
dla_tgcsylv_l2_local_copy_128
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 1688 of file tgcsylv.c.
void mepack_double_tgcsylv_level2_local_copy_32 (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Local copies, M,N<=32)
Purpose:
mepack_double_tgcsylv_level2_local_copy_32 solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
The function is a wrapper for dla_tgcsylv_l2_local_copy_32
- See also
dla_tgcsylv_l2_local_copy_32
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 1058 of file tgcsylv.c.
void mepack_double_tgcsylv_level2_local_copy_64 (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Local copies, M,N<=64)
Purpose:
mepack_double_tgcsylv_level2_local_copy_64 solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
The function is a wrapper for dla_tgcsylv_l2_local_copy_64
- See also
dla_tgcsylv_l2_local_copy_64
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 1268 of file tgcsylv.c.
void mepack_double_tgcsylv_level2_local_copy_96 (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Local copies, M,N<=96)
Purpose:
mepack_double_tgcsylv_level2_local_copy_96 solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
The function is a wrapper for dla_tgcsylv_l2_local_copy_96
- See also
dla_tgcsylv_l2_local_copy_96
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 1478 of file tgcsylv.c.
void mepack_double_tgcsylv_level2_reorder (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Reordered variant)
Purpose:
mepack_double_tgcsylv_level2_reorder solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
The function is a wrapper for dla_tgcsylv_l2_reorder
- See also
dla_tgcsylv_l2_reorder
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 636 of file tgcsylv.c.
void mepack_double_tgcsylv_level2_unopt (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Unoptimized variant)
Purpose:
mepack_double_tgcsylv_level2_unopt solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
The function is a wrapper for dla_tgcsylv_l2_unopt
- See also
dla_tgcsylv_l2_unopt
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 425 of file tgcsylv.c.
void mepack_double_tgcsylv_level3 (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
Level-3 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation.
Purpose:
mepack_double_tgcsylv_level3 solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
The function is a wrapper for dla_tgcsylv_l3
- See also
dla_tgcsylv_l3
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 214 of file tgcsylv.c.
void mepack_double_tgcsylv_level3_2stage (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
Level-3 Kagstroem-Westin/Bartels-Stewart Algorithm with sub-blocking for the generalized coupled Sylvester equation.
Purpose:
mepack_double_tgcsylv_level3_2stage solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
The function is a wrapper for dla_tgcsylv_l3_2s
- See also
dla_tgcsylv_l3_2s
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 636 of file tgcsylv.c.
void mepack_double_tgcsylv_level3_unopt (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
Level-3 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Unoptimized variant)
Purpose:
mepack_double_tgcsylv_level3_unopt solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
The function is a wrapper for dla_tgcsylv_l3_unopt
- See also
dla_tgcsylv_l3_unopt
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 425 of file tgcsylv.c.
void mepack_double_tgcsylv_recursive (const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, int * INFO)
Recursive blocking Algorithm for the generalized coupled Sylvester equation.
Purpose:
mepack_double_tgcsylv_recursive solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
The function is a wrapper for dla_tgcsylv_recursive
- See also
dla_tgcsylv_recursive
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 214 of file tgcsylv.c.
void mepack_double_tgsylv_dag (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * X, int LDX, double * SCALE, double * WORK, int * INFO)
Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation with DAG based parallelization.
Purpose:
mepack_double_tgsylv_dag solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
This function is a wrapper for dla_tgsylv_dag.
- See also
dla_tgsylv_dag
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is DOUBLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 195 of file tgsylv.c.
void mepack_double_tgsylv_gardiner_laub (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * X, int LDX, double * SCALE, double * WORK, int * INFO)
GARDINER-Laub Algorithm for the generalized Sylvester equation.
Purpose:
mepack_double_tgsylv_gardiner_laub solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
This function is a wrapper for dla_tgsylv_gardiner_laub.
- See also
dla_tgsylv_gardiner_laub
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is DOUBLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 1941 of file tgsylv.c.
void mepack_double_tgsylv_level2 (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * X, int LDX, double * SCALE, double * WORK, int * INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Classical level 2 variant)
Purpose:
mepack_double_tgsylv_level2 solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
This function is a wrapper for dla_tgsylv_l2.
- See also
dla_tgsylv_l2
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is DOUBLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 1359 of file tgsylv.c.
void mepack_double_tgsylv_level2_colwise (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * X, int LDX, double * SCALE, double * WORK, int * INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Column-by-column variant)
Purpose:
mepack_double_tgsylv_level2_colwise solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
This function is a wrapper for dla_tgsylv_l2_colwise.
- See also
dla_tgsylv_l2_colwise
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is DOUBLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 1747 of file tgsylv.c.
void mepack_double_tgsylv_level2_local_copy (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * X, int LDX, double * SCALE, double * WORK, int * INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Local copies, unaligned)
Purpose:
mepack_double_tgsylv_level2_local_copy solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
This function is a wrapper for dla_tgsylv_l2_local_copy.
- See also
dla_tgsylv_l2_local_copy
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is DOUBLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 971 of file tgsylv.c.
void mepack_double_tgsylv_level2_local_copy_128 (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * X, int LDX, double * SCALE, double * WORK, int * INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Local copies, M,N<=128)
Purpose:
mepack_double_tgsylv_level2_local_copy_128 solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
This function is a wrapper for dla_tgsylv_l2_local_copy_128.
- See also
dla_tgsylv_l2_local_copy_128
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is DOUBLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 777 of file tgsylv.c.
void mepack_double_tgsylv_level2_local_copy_32 (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * X, int LDX, double * SCALE, double * WORK, int * INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Local copies, M,N<=32)
Purpose:
mepack_double_tgsylv_level2_local_copy_32 solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
This function is a wrapper for dla_tgsylv_l2_local_copy_32.
- See also
dla_tgsylv_l2_local_copy_32
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is DOUBLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 195 of file tgsylv.c.
void mepack_double_tgsylv_level2_local_copy_64 (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * X, int LDX, double * SCALE, double * WORK, int * INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Local copies, M,N<=64)
Purpose:
mepack_double_tgsylv_level2_local_copy_32 solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
This function is a wrapper for dla_tgsylv_l2_local_copy_64.
- See also
dla_tgsylv_l2_local_copy_64
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is DOUBLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 389 of file tgsylv.c.
void mepack_double_tgsylv_level2_local_copy_96 (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * X, int LDX, double * SCALE, double * WORK, int * INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Local copies, M,N<=96)
Purpose:
mepack_double_tgsylv_level2_local_copy_96 solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
This function is a wrapper for dla_tgsylv_l2_local_copy_96.
- See also
dla_tgsylv_l2_local_copy_96
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is DOUBLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 583 of file tgsylv.c.
void mepack_double_tgsylv_level2_reorder (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * X, int LDX, double * SCALE, double * WORK, int * INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Reordered variant)
Purpose:
mepack_double_tgsylv_level2_reorder solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
This function is a wrapper for dla_tgsylv_l2_reorder.
- See also
dla_tgsylv_l2_reorder
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is DOUBLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 1165 of file tgsylv.c.
void mepack_double_tgsylv_level2_rowwise (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * X, int LDX, double * SCALE, double * WORK, int * INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Row-by-Row variant)
Purpose:
mepack_double_tgsylv_level2_rowwise solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
This function is a wrapper for dla_tgsylv_l2_rowwise.
- See also
dla_tgsylv_l2_rowwise
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is DOUBLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 1553 of file tgsylv.c.
void mepack_double_tgsylv_level3 (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * X, int LDX, double * SCALE, double * WORK, int * INFO)
Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation.
Purpose:
mepack_double_tgsylv_level3 solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
This function is a wrapper for dla_tgsylv_l3.
- See also
dla_tgsylv_l3
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is DOUBLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 195 of file tgsylv.c.
void mepack_double_tgsylv_level3_2stage (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * X, int LDX, double * SCALE, double * WORK, int * INFO)
Level-3 Bartels-Stewart Algorithm with sub-blocking for the generalized Sylvester equation.
Purpose:
mepack_double_tgsylv_level3_2stage solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
This function is a wrapper for dla_tgsylv_l3_2s.
- See also
dla_tgsylv_l3_2s
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is DOUBLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 777 of file tgsylv.c.
void mepack_double_tgsylv_level3_colwise (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * X, int LDX, double * SCALE, double * WORK, int * INFO)
Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation (Column-by-Column variant)
Purpose:
mepack_double_tgsylv_level3_colwise solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
This function is a wrapper for dla_tgsylv_l3_colwise.
- See also
dla_tgsylv_l3_colwise
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is DOUBLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 583 of file tgsylv.c.
void mepack_double_tgsylv_level3_rowwise (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * X, int LDX, double * SCALE, double * WORK, int * INFO)
Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation (Row-by-row variant)
Purpose:
mepack_double_tgsylv_level3_rowwise solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
This function is a wrapper for dla_tgsylv_l3_rowwise.
- See also
dla_tgsylv_l3_rowwise
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is DOUBLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 389 of file tgsylv.c.
void mepack_double_tgsylv_recursive (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * X, int LDX, double * SCALE, double * WORK, int * INFO)
Recursive blocking Algorithm for the generalized Sylvester equation.
Purpose:
mepack_double_tgsylv_recursive solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by DGGES from LAPACK.
- Remarks
This function is a wrapper for dla_tgsylv_recursive.
- See also
dla_tgsylv_recursive
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is DOUBLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 195 of file tgsylv.c.
void mepack_single_tgcsylv_dag (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
Level-3 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation with DAG scheduling.
Purpose:
mepack_single_tgcsylv_dag solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
The function is a wrapper for sla_tgcsylv_dag
- See also
sla_tgcsylv_dag
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 425 of file tgcsylv.c.
void mepack_single_tgcsylv_dual_dag (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
OpenMP4 - DAG Algorithm for the dual generalized coupled Sylvester equation.
Purpose:
mepack_single_tgcsylv_dual_dag solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * R = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular
matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or
D can be quasi upper triangular as well. The right hand side Y and the solution X
M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are
created by DGGES from LAPACK.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
| kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for sla_tgcsylv_dual_dag.
See also
sla_tgcsylv_dual_dag
Parameters
TRANSA
TRANSA is CHARACTER(1)
Specifies the form of the system of equations with respect to A and C:
== 'N': op1(A) = A, op1(C) = C (No transpose for A and C)
== 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is CHARACTER(1)
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B, op2(D) = D (No transpose for B and D)
== 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N)
The matrix B must be (quasi-) upper triangular. If the matrix D is already
quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M)
The matrix C must be upper triangular.LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N)
The matrix D must be (quasi-) upper triangular. If the matrix B is already
quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER
The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory.INFO
INFO is INTEGER
On input:
== -1 : Perform a workspace query ( not recommended in the C interface)
<> -1 : normal operation
On exit, workspace query:
< 0 : if INFO == -i, the i-Th argument had an illegal value
>= 0: The value of INFO is the required number of elements in the workspace.
On exit, normal operation:
== 0: successful exit
< 0: if INFO == -i, the i-Th argument had an illegal value
> 0: The equation is not solved correctly. One of the arising inner
system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 459 of file tgcsylv_dual.c.
void mepack_single_tgcsylv_dual_level2 (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation.
Purpose:
mepack_single_tgcsylv_dual_level2 solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * R = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular
matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or
D can be quasi upper triangular as well. The right hand side Y and the solution X
M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are
created by DGGES from LAPACK.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for sla_tgcsylv_dual_l2.
See also
sla_tgcsylv_dual_l2
Parameters
TRANSA
TRANSA is CHARACTER(1)
Specifies the form of the system of equations with respect to A and C:
== 'N': op1(A) = A, op1(C) = C (No transpose for A and C)
== 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is CHARACTER(1)
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B, op2(D) = D (No transpose for B and D)
== 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N)
The matrix B must be (quasi-) upper triangular. If the matrix D is already
quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M)
The matrix C must be upper triangular.LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N)
The matrix D must be (quasi-) upper triangular. If the matrix B is already
quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER
The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory.INFO
INFO is INTEGER
On input:
== -1 : Perform a workspace query ( not recommended in the C interface)
<> -1 : normal operation
On exit, workspace query:
< 0 : if INFO == -i, the i-Th argument had an illegal value
>= 0: The value of INFO is the required number of elements in the workspace.
On exit, normal operation:
== 0: successful exit
< 0: if INFO == -i, the i-Th argument had an illegal value
> 0: The equation is not solved correctly. One of the arising inner
system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 1603 of file tgcsylv_dual.c.
void mepack_single_tgcsylv_dual_level2_local_copy (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation ( Local Copies )
Purpose:
mepack_single_tgcsylv_dual_level2 solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * R = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular
matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or
D can be quasi upper triangular as well. The right hand side Y and the solution X
M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are
created by DGGES from LAPACK.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
| kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for sla_tgcsylv_dual_l2_local_copy.
See also
sla_tgcsylv_dual_l2_local_copy
Parameters
TRANSA
TRANSA is CHARACTER(1)
Specifies the form of the system of equations with respect to A and C:
== 'N': op1(A) = A, op1(C) = C (No transpose for A and C)
== 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is CHARACTER(1)
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B, op2(D) = D (No transpose for B and D)
== 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N)
The matrix B must be (quasi-) upper triangular. If the matrix D is already
quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M)
The matrix C must be upper triangular.LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N)
The matrix D must be (quasi-) upper triangular. If the matrix B is already
quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER
The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory.INFO
INFO is INTEGER
On input:
== -1 : Perform a workspace query ( not recommended in the C interface)
<> -1 : normal operation
On exit, workspace query:
< 0 : if INFO == -i, the i-Th argument had an illegal value
>= 0: The value of INFO is the required number of elements in the workspace.
On exit, normal operation:
== 0: successful exit
< 0: if INFO == -i, the i-Th argument had an illegal value
> 0: The equation is not solved correctly. One of the arising inner
system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 1832 of file tgcsylv_dual.c.
void mepack_single_tgcsylv_dual_level2_local_copy_128 (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation ( Local Copies, M,N <= 128 )
Purpose:
mepack_single_tgcsylv_dual_level2_local_copy_128 solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * R = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular
matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or
D can be quasi upper triangular as well. The right hand side Y and the solution X
M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are
created by DGGES from LAPACK.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
| kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for sla_tgcsylv_dual_l2_local_copy_128.
See also
sla_tgcsylv_dual_l2_local_copy_128
Parameters
TRANSA
TRANSA is CHARACTER(1)
Specifies the form of the system of equations with respect to A and C:
== 'N': op1(A) = A, op1(C) = C (No transpose for A and C)
== 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is CHARACTER(1)
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B, op2(D) = D (No transpose for B and D)
== 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N)
The matrix B must be (quasi-) upper triangular. If the matrix D is already
quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M)
The matrix C must be upper triangular.LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N)
The matrix D must be (quasi-) upper triangular. If the matrix B is already
quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER
The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory.INFO
INFO is INTEGER
On input:
== -1 : Perform a workspace query ( not recommended in the C interface)
<> -1 : normal operation
On exit, workspace query:
< 0 : if INFO == -i, the i-Th argument had an illegal value
>= 0: The value of INFO is the required number of elements in the workspace.
On exit, normal operation:
== 0: successful exit
< 0: if INFO == -i, the i-Th argument had an illegal value
> 0: The equation is not solved correctly. One of the arising inner
system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 2752 of file tgcsylv_dual.c.
void mepack_single_tgcsylv_dual_level2_local_copy_32 (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation ( Local Copies, M,N <= 32 )
Purpose:
mepack_single_tgcsylv_dual_level2_local_copy_32 solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * R = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular
matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or
D can be quasi upper triangular as well. The right hand side Y and the solution X
M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are
created by DGGES from LAPACK.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
| kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for sla_tgcsylv_dual_l2_local_copy_32.
See also
sla_tgcsylv_dual_l2_local_copy_32
Parameters
TRANSA
TRANSA is CHARACTER(1)
Specifies the form of the system of equations with respect to A and C:
== 'N': op1(A) = A, op1(C) = C (No transpose for A and C)
== 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is CHARACTER(1)
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B, op2(D) = D (No transpose for B and D)
== 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N)
The matrix B must be (quasi-) upper triangular. If the matrix D is already
quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M)
The matrix C must be upper triangular.LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N)
The matrix D must be (quasi-) upper triangular. If the matrix B is already
quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER
The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory.INFO
INFO is INTEGER
On input:
== -1 : Perform a workspace query ( not recommended in the C interface)
<> -1 : normal operation
On exit, workspace query:
< 0 : if INFO == -i, the i-Th argument had an illegal value
>= 0: The value of INFO is the required number of elements in the workspace.
On exit, normal operation:
== 0: successful exit
< 0: if INFO == -i, the i-Th argument had an illegal value
> 0: The equation is not solved correctly. One of the arising inner
system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 2062 of file tgcsylv_dual.c.
void mepack_single_tgcsylv_dual_level2_local_copy_64 (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation ( Local Copies, M,N <= 64 )
Purpose:
mepack_single_tgcsylv_dual_level2_local_copy_64 solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * R = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular
matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or
D can be quasi upper triangular as well. The right hand side Y and the solution X
M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are
created by DGGES from LAPACK.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
| kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for sla_tgcsylv_dual_l2_local_copy_64.
See also
sla_tgcsylv_dual_l2_local_copy_64
Parameters
TRANSA
TRANSA is CHARACTER(1)
Specifies the form of the system of equations with respect to A and C:
== 'N': op1(A) = A, op1(C) = C (No transpose for A and C)
== 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is CHARACTER(1)
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B, op2(D) = D (No transpose for B and D)
== 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N)
The matrix B must be (quasi-) upper triangular. If the matrix D is already
quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M)
The matrix C must be upper triangular.LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N)
The matrix D must be (quasi-) upper triangular. If the matrix B is already
quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER
The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory.INFO
INFO is INTEGER
On input:
== -1 : Perform a workspace query ( not recommended in the C interface)
<> -1 : normal operation
On exit, workspace query:
< 0 : if INFO == -i, the i-Th argument had an illegal value
>= 0: The value of INFO is the required number of elements in the workspace.
On exit, normal operation:
== 0: successful exit
< 0: if INFO == -i, the i-Th argument had an illegal value
> 0: The equation is not solved correctly. One of the arising inner
system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 2291 of file tgcsylv_dual.c.
void mepack_single_tgcsylv_dual_level2_local_copy_96 (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation ( Local Copies, M,N <= 96 )
Purpose:
mepack_single_tgcsylv_dual_level2_local_copy_96 solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * R = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular
matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or
D can be quasi upper triangular as well. The right hand side Y and the solution X
M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are
created by DGGES from LAPACK.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
| kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for sla_tgcsylv_dual_l2_local_copy_96.
See also
sla_tgcsylv_dual_l2_local_copy_96
Parameters
TRANSA
TRANSA is CHARACTER(1)
Specifies the form of the system of equations with respect to A and C:
== 'N': op1(A) = A, op1(C) = C (No transpose for A and C)
== 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is CHARACTER(1)
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B, op2(D) = D (No transpose for B and D)
== 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N)
The matrix B must be (quasi-) upper triangular. If the matrix D is already
quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M)
The matrix C must be upper triangular.LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N)
The matrix D must be (quasi-) upper triangular. If the matrix B is already
quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER
The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory.INFO
INFO is INTEGER
On input:
== -1 : Perform a workspace query ( not recommended in the C interface)
<> -1 : normal operation
On exit, workspace query:
< 0 : if INFO == -i, the i-Th argument had an illegal value
>= 0: The value of INFO is the required number of elements in the workspace.
On exit, normal operation:
== 0: successful exit
< 0: if INFO == -i, the i-Th argument had an illegal value
> 0: The equation is not solved correctly. One of the arising inner
system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 2521 of file tgcsylv_dual.c.
void mepack_single_tgcsylv_dual_level3 (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
Level-3 Kagstroem-Westin/Bartels-Stewart Algorithm for the dual generalized coupled Sylvester equation.
Purpose:
mepack_single_tgcsylv_dual_level3 solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * R = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular
matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or
D can be quasi upper triangular as well. The right hand side Y and the solution X
M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are
created by DGGES from LAPACK.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for sla_tgcsylv_dual_l3.
See also
sla_tgcsylv_dual_l3
Parameters
TRANSA
TRANSA is CHARACTER(1)
Specifies the form of the system of equations with respect to A and C:
== 'N': op1(A) = A, op1(C) = C (No transpose for A and C)
== 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is CHARACTER(1)
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B, op2(D) = D (No transpose for B and D)
== 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N)
The matrix B must be (quasi-) upper triangular. If the matrix D is already
quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M)
The matrix C must be upper triangular.LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N)
The matrix D must be (quasi-) upper triangular. If the matrix B is already
quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER
The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory.INFO
INFO is INTEGER
On input:
== -1 : Perform a workspace query ( not recommended in the C interface)
<> -1 : normal operation
On exit, workspace query:
< 0 : if INFO == -i, the i-Th argument had an illegal value
>= 0: The value of INFO is the required number of elements in the workspace.
On exit, normal operation:
== 0: successful exit
< 0: if INFO == -i, the i-Th argument had an illegal value
> 0: The equation is not solved correctly. One of the arising inner
system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 459 of file tgcsylv_dual.c.
void mepack_single_tgcsylv_dual_level3_2stage (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
Level-3 Kagstroem-Westin/Bartels-Stewart Algorithm with sub blocking for the dual generalized coupled Sylvester equation.
Purpose:
mepack_single_tgcsylv_dual_level3_2stage solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * R = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular
matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or
D can be quasi upper triangular as well. The right hand side Y and the solution X
M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are
created by DGGES from LAPACK.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for sla_tgcsylv_dual_l3_2s.
See also
sla_tgcsylv_dual_l3_2s
Parameters
TRANSA
TRANSA is CHARACTER(1)
Specifies the form of the system of equations with respect to A and C:
== 'N': op1(A) = A, op1(C) = C (No transpose for A and C)
== 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is CHARACTER(1)
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B, op2(D) = D (No transpose for B and D)
== 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N)
The matrix B must be (quasi-) upper triangular. If the matrix D is already
quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M)
The matrix C must be upper triangular.LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N)
The matrix D must be (quasi-) upper triangular. If the matrix B is already
quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER
The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory.INFO
INFO is INTEGER
On input:
== -1 : Perform a workspace query ( not recommended in the C interface)
<> -1 : normal operation
On exit, workspace query:
< 0 : if INFO == -i, the i-Th argument had an illegal value
>= 0: The value of INFO is the required number of elements in the workspace.
On exit, normal operation:
== 0: successful exit
< 0: if INFO == -i, the i-Th argument had an illegal value
> 0: The equation is not solved correctly. One of the arising inner
system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 915 of file tgcsylv_dual.c.
void mepack_single_tgcsylv_dual_recursive (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
Recursive blocking algorithm for the dual generalized coupled Sylvester equation.
Purpose:
mepack_single_tgcsylv_dual_recursive solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * R = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular
matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or
D can be quasi upper triangular as well. The right hand side Y and the solution X
M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are
created by DGGES from LAPACK.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for sla_tgcsylv_dual_recursive.
See also
sla_tgcsylv_dual_recursive
Parameters
TRANSA
TRANSA is CHARACTER(1)
Specifies the form of the system of equations with respect to A and C:
== 'N': op1(A) = A, op1(C) = C (No transpose for A and C)
== 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is CHARACTER(1)
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B, op2(D) = D (No transpose for B and D)
== 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N)
The matrix B must be (quasi-) upper triangular. If the matrix D is already
quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M)
The matrix C must be upper triangular.LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N)
The matrix D must be (quasi-) upper triangular. If the matrix B is already
quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L of Equation (1)
as selected by TRANSA, TRANSB, and SGN1/SGN2.
Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER
The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory.INFO
INFO is INTEGER
On input:
== -1 : Perform a workspace query ( not recommended in the C interface)
<> -1 : normal operation
On exit, workspace query:
< 0 : if INFO == -i, the i-Th argument had an illegal value
>= 0: The value of INFO is the required number of elements in the workspace.
On exit, normal operation:
== 0: successful exit
< 0: if INFO == -i, the i-Th argument had an illegal value
> 0: The equation is not solved correctly. One of the arising inner
system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 459 of file tgcsylv_dual.c.
void mepack_single_tgcsylv_level2 (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation.
Purpose:
mepack_single_tgcsylv_level2 solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
The function is a wrapper for sla_tgcsylv_l2
- See also
sla_tgcsylv_l2
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 1900 of file tgcsylv.c.
void mepack_single_tgcsylv_level2_local_copy (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Local copies, unaligned)
Purpose:
mepack_single_tgcsylv_level2_local_copy solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
The function is a wrapper for sla_tgcsylv_l2_local_copy
- See also
sla_tgcsylv_l2_local_copy
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 2533 of file tgcsylv.c.
void mepack_single_tgcsylv_level2_local_copy_128 (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Local copies, M,N<=128)
Purpose:
mepack_single_tgcsylv_level2_local_copy_128 solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
The function is a wrapper for sla_tgcsylv_l2_local_copy_128
- See also
sla_tgcsylv_l2_local_copy_128
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 3374 of file tgcsylv.c.
void mepack_single_tgcsylv_level2_local_copy_32 (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Local copies, M,N<=32)
Purpose:
mepack_single_tgcsylv_level2_local_copy_32 solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
The function is a wrapper for sla_tgcsylv_l2_local_copy_32
- See also
sla_tgcsylv_l2_local_copy_32
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 2744 of file tgcsylv.c.
void mepack_single_tgcsylv_level2_local_copy_64 (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Local copies, M,N<=64)
Purpose:
mepack_single_tgcsylv_level2_local_copy_64 solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
The function is a wrapper for sla_tgcsylv_l2_local_copy_64
- See also
sla_tgcsylv_l2_local_copy_64
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 2954 of file tgcsylv.c.
void mepack_single_tgcsylv_level2_local_copy_96 (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Local copies, M,N<=96)
Purpose:
mepack_single_tgcsylv_level2_local_copy_96 solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
The function is a wrapper for sla_tgcsylv_l2_local_copy_96
- See also
sla_tgcsylv_l2_local_copy_96
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 3164 of file tgcsylv.c.
void mepack_single_tgcsylv_level2_reorder (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Reordered variant)
Purpose:
mepack_single_tgcsylv_level2_reorder solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
The function is a wrapper for sla_tgcsylv_l2_reorder
- See also
sla_tgcsylv_l2_reorder
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 2322 of file tgcsylv.c.
void mepack_single_tgcsylv_level2_unopt (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
Level-2 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation (Unoptimized variant)
Purpose:
mepack_single_tgcsylv_level2_unopt solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
The function is a wrapper for sla_tgcsylv_l2_unopt
- See also
sla_tgcsylv_l2_unopt
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 2111 of file tgcsylv.c.
void mepack_single_tgcsylv_level3 (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
Level-3 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation.
Purpose:
mepack_single_tgcsylv_level3 solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
The function is a wrapper for sla_tgcsylv_l3
- See also
sla_tgcsylv_l3
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 847 of file tgcsylv.c.
void mepack_single_tgcsylv_level3_2stage (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
Level-3 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation.
Purpose:
mepack_single_tgcsylv_level3_2stage solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
The function is a wrapper for sla_tgcsylv_l3_2s
- See also
sla_tgcsylv_l3_2s
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 1269 of file tgcsylv.c.
void mepack_single_tgcsylv_level3_unopt (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
Level-3 Kagstroem-Westin/Bartels-Stewart Algorithm for the generalized coupled Sylvester equation.
Purpose:
mepack_single_tgcsylv_level3_unopt solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
The function is a wrapper for sla_tgcsylv_l3_unopt
- See also
sla_tgcsylv_l3_unopt
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 1058 of file tgcsylv.c.
void mepack_single_tgcsylv_recursive (const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, int * INFO)
Recursive blocking Algorithm for the generalized coupled Sylvester equation.
Purpose:
mepack_single_tgcsylv_recursive solves a generalized coupled Sylvester equation of the following form op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
The function is a wrapper for sla_tgcsylv_recursive
- See also
sla_tgcsylv_recursive
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).E
E is SINGLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side E and the solution R are M-by-N matrices.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L of Equation (1) as selected by TRANSA, TRANSB, and SGN1/SGN2. Right hand side F and the solution L are M-by-N matrices.LDF
LDF is INTEGER The leading dimension of the array F. LDE >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 423 of file tgcsylv.c.
void mepack_single_tgsylv_dag (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * X, int LDX, float * SCALE, float * WORK, int * INFO)
Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation with DAG based parallelization.
Purpose:
mepack_single_tgsylv_dag solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
This function is a wrapper for sla_tgsylv_dag.
- See also
sla_tgsylv_dag
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is SINGLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 389 of file tgsylv.c.
void mepack_single_tgsylv_gardiner_laub (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * X, int LDX, float * SCALE, float * WORK, int * INFO)
Gardiner-Laub Algorithm for the generalized Sylvester equation.
Purpose:
mepack_single_tgsylv_gardiner_laub solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
This function is a wrapper for sla_tgsylv_gardiner_laub.
- See also
sla_tgsylv_gardiner_laub
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is SINGLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 3894 of file tgsylv.c.
void mepack_single_tgsylv_level2 (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * X, int LDX, float * SCALE, float * WORK, int * INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Classic level 2variant)
Purpose:
mepack_single_tgsylv_level2 solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
This function is a wrapper for sla_tgsylv_l2.
- See also
sla_tgsylv_l2
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is SINGLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 3311 of file tgsylv.c.
void mepack_single_tgsylv_level2_colwise (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * X, int LDX, float * SCALE, float * WORK, int * INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Column-by-column variant)
Purpose:
mepack_single_tgsylv_level2_colwise solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
This function is a wrapper for sla_tgsylv_l2_colwise.
- See also
sla_tgsylv_l2_colwise
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is SINGLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 3699 of file tgsylv.c.
void mepack_single_tgsylv_level2_local_copy (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * X, int LDX, float * SCALE, float * WORK, int * INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Local copies, unaligned)
Purpose:
mepack_single_tgsylv_level2_local_copy solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
This function is a wrapper for sla_tgsylv_l2_local_copy.
- See also
sla_tgsylv_l2_local_copy
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is SINGLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 2921 of file tgsylv.c.
void mepack_single_tgsylv_level2_local_copy_128 (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * X, int LDX, float * SCALE, float * WORK, int * INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Local copies, M,N<=128)
Purpose:
mepack_single_tgsylv_level2_local_copy_128 solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
This function is a wrapper for sla_tgsylv_l2_local_copy_128.
- See also
sla_tgsylv_l2_local_copy_128
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is SINGLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 2725 of file tgsylv.c.
void mepack_single_tgsylv_level2_local_copy_32 (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * X, int LDX, float * SCALE, float * WORK, int * INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Local copies, M,N<=32)
Purpose:
mepack_single_tgsylv_level2_local_copy_32 solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
This function is a wrapper for sla_tgsylv_l2_local_copy_32.
- See also
sla_tgsylv_l2_local_copy_32
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is SINGLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 2137 of file tgsylv.c.
void mepack_single_tgsylv_level2_local_copy_64 (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * X, int LDX, float * SCALE, float * WORK, int * INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Local copies, M,N<=64)
Purpose:
mepack_single_tgsylv_level2_local_copy_64 solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
This function is a wrapper for sla_tgsylv_l2_local_copy_64.
- See also
sla_tgsylv_l2_local_copy_64
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is SINGLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 2333 of file tgsylv.c.
void mepack_single_tgsylv_level2_local_copy_96 (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * X, int LDX, float * SCALE, float * WORK, int * INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Local copies, M,N<=96)
Purpose:
mepack_single_tgsylv_level2_local_copy_96 solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
This function is a wrapper for sla_tgsylv_l2_local_copy_96.
- See also
sla_tgsylv_l2_local_copy_96
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is SINGLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 2529 of file tgsylv.c.
void mepack_single_tgsylv_level2_reorder (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * X, int LDX, float * SCALE, float * WORK, int * INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Reordered variant)
Purpose:
mepack_single_tgsylv_level2_reorder solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
This function is a wrapper for sla_tgsylv_l2_reorder.
- See also
sla_tgsylv_l2_reorder
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is SINGLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 3115 of file tgsylv.c.
void mepack_single_tgsylv_level2_rowwise (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * X, int LDX, float * SCALE, float * WORK, int * INFO)
Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Row-by-row variant)
Purpose:
mepack_single_tgsylv_level2_reorder solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
This function is a wrapper for sla_tgsylv_l2_rowwise.
- See also
sla_tgsylv_l2_rowwise
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is SINGLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 3505 of file tgsylv.c.
void mepack_single_tgsylv_level3 (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * X, int LDX, float * SCALE, float * WORK, int * INFO)
Level-3 Bartels-Stewart Algorithm for the generalized Sylvester.
Purpose:
mepack_single_tgsylv_level3 solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
This function is a wrapper for sla_tgsylv_l3.
- See also
sla_tgsylv_l3
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is SINGLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 971 of file tgsylv.c.
void mepack_single_tgsylv_level3_2stage (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * X, int LDX, float * SCALE, float * WORK, int * INFO)
Level-3 Bartels-Stewart Algorithm with sub-blocking for the generalized Sylvester.
Purpose:
mepack_single_tgsylv_level3_2stage solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
This function is a wrapper for sla_tgsylv_l3_2s.
- See also
sla_tgsylv_l3_2s
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is SINGLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 1557 of file tgsylv.c.
void mepack_single_tgsylv_level3_colwise (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * X, int LDX, float * SCALE, float * WORK, int * INFO)
Level-3 Bartels-Stewart Algorithm for the generalized Sylvester (Column-by-column variant)
Purpose:
mepack_single_tgsylv_level3_colwise solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
This function is a wrapper for sla_tgsylv_l3_colwise.
- See also
sla_tgsylv_l3_colwise
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is SINGLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 1362 of file tgsylv.c.
void mepack_single_tgsylv_level3_rowwise (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * X, int LDX, float * SCALE, float * WORK, int * INFO)
Level-3 Bartels-Stewart Algorithm for the generalized Sylvester (Row-by-row variant)
Purpose:
mepack_single_tgsylv_level3_rowwise solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
This function is a wrapper for sla_tgsylv_l3_rowwise.
- See also
sla_tgsylv_l3_rowwise
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is SINGLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 1166 of file tgsylv.c.
void mepack_single_tgsylv_recursive (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * X, int LDX, float * SCALE, float * WORK, int * INFO)
Recursive blocking algorithm for the generalized Sylvester equation.
Purpose:
mepack_single_tgsylv_recursive solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, C is a M-by-M upper triangular matrix and B and D are N-by-N upper triangular matrices. Furthermore either B or D can be quasi upper triangular as well. The right hand side Y and the solution X M-by-N matrices. Typically the matrix pencils (A,C) and (B,D) ( or (D,B) ) are created by SGGES from LAPACK.
- Remarks
This function is a wrapper for sla_tgsylv_recursive.
- See also
sla_tgsylv_recursive
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C: == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C)TRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D)SGN
SGN is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N) The matrix B must be (quasi-) upper triangular. If the matrix D is already quasi-upper triangular the matrix B must be upper triangular.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The matrix C must be upper triangular.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N) The matrix D must be (quasi-) upper triangular. If the matrix B is already quasi-upper triangular the matrix D must be upper triangular.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is SINGLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is SINGLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. If SCALE .NE. 1 the problem is no solved correctly in this case one have to use an other solver.WORK
WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.INFO
INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 389 of file tgsylv.c.
Author
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