# ctrsylv - Man Page

## Name

ctrsylv — C-Interface

— C-Interface for triangular standard Sylvester equations.

## Synopsis

### Functions

void **mepack_double_trsylv_dag** (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation with DAG parallelization.

void **mepack_single_trsylv_dag** (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation with DAG parallelization.

void **mepack_double_trsylv2_dag** (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation with DAG parallelization.

void **mepack_single_trsylv2_dag** (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation with DAG parallelization.

void **mepack_double_trsylv_level2** (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation.

void **mepack_double_trsylv_level2_reorder** (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, 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_trsylv_level2_unopt** (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Unoptimized variant)

void **mepack_double_trsylv_level2_local_copy** (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant)

void **mepack_double_trsylv_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 *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 32)

void **mepack_double_trsylv_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 *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 64)

void **mepack_double_trsylv_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 *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N<=96)

void **mepack_double_trsylv_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 *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 128)

void **mepack_single_trsylv_level2** (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation.

void **mepack_single_trsylv_level2_reorder** (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, 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_trsylv_level2_unopt** (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Unoptimized variant)

void **mepack_single_trsylv_level2_local_copy** (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant)

void **mepack_single_trsylv_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 *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 32 )

void **mepack_single_trsylv_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 *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 64)

void **mepack_single_trsylv_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 *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 96)

void **mepack_single_trsylv_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 *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 128)

void **mepack_double_trsylv2_level2** (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation.

void **mepack_double_trsylv2_level2_reorder** (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, 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_trsylv2_level2_unopt** (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Unoptimized variant)

void **mepack_double_trsylv2_level2_local_copy** (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant)

void **mepack_double_trsylv2_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 *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 32)

void **mepack_double_trsylv2_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 *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 64)

void **mepack_double_trsylv2_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 *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N<=96)

void **mepack_double_trsylv2_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 *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 128)

void **mepack_single_trsylv2_level2** (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation.

void **mepack_single_trsylv2_level2_reorder** (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, 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_trsylv2_level2_unopt** (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Unoptimized variant)

void **mepack_single_trsylv2_level2_local_copy** (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant)

void **mepack_single_trsylv2_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 *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 32 )

void **mepack_single_trsylv2_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 *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 64)

void **mepack_single_trsylv2_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 *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 96)

void **mepack_single_trsylv2_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 *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 128)

void **mepack_double_trsylv_level3** (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation.

void **mepack_double_trsylv_level3_unopt** (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Unoptimized variant)

void **mepack_double_trsylv_level3_2stage** (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, 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_trsylv_level3** (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation.

void **mepack_single_trsylv_level3_unopt** (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Unoptimized variant)

void **mepack_single_trsylv_level3_2stage** (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm with blocking for the generalized Sylvester equation.

void **mepack_double_trsylv2_level3** (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation.

void **mepack_double_trsylv2_level3_unopt** (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Unoptimized variant)

void **mepack_double_trsylv2_level3_2stage** (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm with blocking for the generalized Sylvester equation.

void **mepack_single_trsylv2_level3** (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation.

void **mepack_single_trsylv2_level3_unopt** (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Unoptimized variant)

void **mepack_single_trsylv2_level3_2stage** (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm with sub-blocking for the generalized Sylvester equation.

void **mepack_double_trsylv_recursive** (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Recursive blocking Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation.

void **mepack_single_trsylv_recursive** (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Recursive blocking Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation.

void **mepack_double_trsylv2_recursive** (const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Recursive blocking Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation.

void **mepack_single_trsylv2_recursive** (const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Recursive blocking Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation.

## Detailed Description

C-Interface for triangular standard Sylvester equations.

The Fortran routines to solve the standard 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_trsylv2_dag (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation with DAG parallelization.

**Purpose:**

mepack_double_trsylv2_dag solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv2_dag**.**See also****dla_trsylv2_dag****Parameters***TRANSA*TRANSA is string Specifies the form of the system of equations with respect to A: == 'N': op1(A) = A == 'T': op1(A) = A**T

*TRANSB*TRANSB is string Specifies the form of the system of equations with respect to B: == 'N': op2(B) = B, == 'T': op2(B) = B**T

*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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= 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. LDB >= 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 **166** of file **trsylv2.c**.

### void mepack_double_trsylv2_level2 (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation.

**Purpose:**

mepack_double_trsylv2_level2 solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv2_l2**.**See also****dla_trsylv2_l2****Parameters***TRANSA*TRANSA is string Specifies the form of the system of equations with respect to A: == 'N': op1(A) = A == 'T': op1(A) = A**T

*TRANSB*TRANSB is string Specifies the form of the system of equations with respect to B: == 'N': op2(B) = B, == 'T': op2(B) = B**T

*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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= 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. LDB >= 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 **165** of file **trsylv2.c**.

### void mepack_double_trsylv2_level2_local_copy (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant)

**Purpose:**

mepack_double_trsylv2_level2_local_copy solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv2_l2_local_copy**.**See also****dla_trsylv2_l2_local_copy****Parameters***TRANSA*TRANSA is string Specifies the form of the system of equations with respect to A: == 'N': op1(A) = A == 'T': op1(A) = A**T

*TRANSB*TRANSB is string Specifies the form of the system of equations with respect to B: == 'N': op2(B) = B, == 'T': op2(B) = B**T

*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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= 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. LDB >= 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 **637** of file **trsylv2.c**.

### void mepack_double_trsylv2_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 * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 128)

**Purpose:**

mepack_double_trsylv2_level2_local_copy_128 solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv2_l2_local_copy_128**.**See also****dla_trsylv2_l2_local_copy_128****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **1269** of file **trsylv2.c**.

### void mepack_double_trsylv2_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 * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 32)

**Purpose:**

mepack_double_trsylv2_level2_local_copy_32 solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv2_l2_local_copy_32**.**See also****dla_trsylv2_l2_local_copy_32****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **795** of file **trsylv2.c**.

### void mepack_double_trsylv2_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 * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 64)

**Purpose:**

mepack_double_trsylv2_level2_local_copy_64 solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv2_l2_local_copy_64**.**See also****dla_trsylv2_l2_local_copy_64****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **953** of file **trsylv2.c**.

### void mepack_double_trsylv2_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 * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N<=96)

**Purpose:**

mepack_double_trsylv2_level2_local_copy_96 solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv2_l2_local_copy_96**.**See also****dla_trsylv2_l2_local_copy_96****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **1111** of file **trsylv2.c**.

### void mepack_double_trsylv2_level2_reorder (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, 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_trsylv2_level2_reorder solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv2_l2_reorder**.**See also****dla_trsylv2_l2_reorder****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **323** of file **trsylv2.c**.

### void mepack_double_trsylv2_level2_unopt (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Unoptimized variant)

**Purpose:**

mepack_double_trsylv2_level2_unopt solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv2_l2_unopt**.**See also****dla_trsylv2_l2_unopt****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **480** of file **trsylv2.c**.

### void mepack_double_trsylv2_level3 (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation.

**Purpose:**

mepack_double_trsylv2_level3 solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv2_l3**.**See also****dla_trsylv2_l3****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **168** of file **trsylv2.c**.

### void mepack_double_trsylv2_level3_2stage (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm with blocking for the generalized Sylvester equation.

**Purpose:**

mepack_double_trsylv2_level3_2stage solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv2_l3_2s**.**See also****dla_trsylv2_l3_2s****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **483** of file **trsylv2.c**.

### void mepack_double_trsylv2_level3_unopt (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Unoptimized variant)

**Purpose:**

mepack_double_trsylv2_level3_unopt solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv2_l3_unopt**.**See also****dla_trsylv2_l3_unopt****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **326** of file **trsylv2.c**.

### void mepack_double_trsylv2_recursive (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Recursive blocking Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation.

**Purpose:**

mepack_double_trsylv2_recursive solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv2_recursive**.**See also****dla_trsylv2_recursive****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **168** of file **trsylv2.c**.

### void mepack_double_trsylv_dag (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation with DAG parallelization.

**Purpose:**

mepack_double_trsylv_dag solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv_dag**.**See also****dla_trsylv_dag****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **166** of file **trsylv.c**.

### void mepack_double_trsylv_level2 (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation.

**Purpose:**

mepack_double_trsylv_level2 solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv_l2**.**See also****dla_trsylv_l2****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **165** of file **trsylv.c**.

### void mepack_double_trsylv_level2_local_copy (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant)

**Purpose:**

mepack_double_trsylv_level2_local_copy solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv_l2_local_copy**.**See also****dla_trsylv_l2_local_copy****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **637** of file **trsylv.c**.

### void mepack_double_trsylv_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 * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 128)

**Purpose:**

mepack_double_trsylv_level2_local_copy_128 solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv_l2_local_copy_128**.**See also****dla_trsylv_l2_local_copy_128****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **1269** of file **trsylv.c**.

### void mepack_double_trsylv_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 * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 32)

**Purpose:**

mepack_double_trsylv_level2_local_copy_32 solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv_l2_local_copy_32**.**See also****dla_trsylv_l2_local_copy_32****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **795** of file **trsylv.c**.

### void mepack_double_trsylv_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 * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 64)

**Purpose:**

mepack_double_trsylv_level2_local_copy_64 solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv_l2_local_copy_64**.**See also****dla_trsylv_l2_local_copy_64****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **953** of file **trsylv.c**.

### void mepack_double_trsylv_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 * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N<=96)

**Purpose:**

mepack_double_trsylv_level2_local_copy_96 solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv_l2_local_copy_96**.**See also****dla_trsylv_l2_local_copy_96****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **1111** of file **trsylv.c**.

### void mepack_double_trsylv_level2_reorder (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, 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_trsylv_level2_reorder solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv_l2_reorder**.**See also****dla_trsylv_l2_reorder****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **323** of file **trsylv.c**.

### void mepack_double_trsylv_level2_unopt (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Unoptimized variant)

**Purpose:**

mepack_double_trsylv_level2_unopt solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv_l2_unopt**.**See also****dla_trsylv_l2_unopt****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **480** of file **trsylv.c**.

### void mepack_double_trsylv_level3 (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation.

**Purpose:**

mepack_double_trsylv_level3 solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv_l3**.**See also****dla_trsylv_l3****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **168** of file **trsylv.c**.

### void mepack_double_trsylv_level3_2stage (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, 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_trsylv_level3_2stage solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv_l3_2s**.**See also****dla_trsylv_l3_2s****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **483** of file **trsylv.c**.

### void mepack_double_trsylv_level3_unopt (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Unoptimized variant)

**Purpose:**

mepack_double_trsylv_level3_unopt solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv_l3_unopt**.**See also****dla_trsylv_l3_unopt****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **326** of file **trsylv.c**.

### void mepack_double_trsylv_recursive (const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Recursive blocking Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation.

**Purpose:**

mepack_double_trsylv_recursive solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv_recursive**.**See also****dla_trsylv_recursive****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **168** of file **trsylv.c**.

### void mepack_single_trsylv2_dag (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation with DAG parallelization.

**Purpose:**

mepack_single_trsylv2_dag solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv2_dag**.**See also****sla_trsylv2_dag****Parameters***TRANSA**TRANSB**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= 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. LDB >= 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***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **324** of file **trsylv2.c**.

### void mepack_single_trsylv2_level2 (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation.

**Purpose:**

mepack_single_trsylv2_level2 solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv2_l2**.**See also****sla_trsylv2_l2****Parameters***TRANSA**TRANSB**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= 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. LDB >= 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***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **1428** of file **trsylv2.c**.

### void mepack_single_trsylv2_level2_local_copy (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant)

**Purpose:**

mepack_single_trsylv2_level2_local_copy solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv2_l2_local_copy**.**See also****sla_trsylv2_l2_local_copy****Parameters***TRANSA**TRANSB**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= 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. LDB >= 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***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **1903** of file **trsylv2.c**.

### void mepack_single_trsylv2_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 * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 128)

**Purpose:**

mepack_single_trsylv2_level2_local_copy_128 solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv2_l2_local_copy_128**.**See also****sla_trsylv2_l2_local_copy_128****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **2535** of file **trsylv2.c**.

### void mepack_single_trsylv2_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 * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 32 )

**Purpose:**

mepack_single_trsylv2_level2_local_copy_32 solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv2_l2_local_copy_32**.**See also****sla_trsylv2_l2_local_copy_32****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **2060** of file **trsylv2.c**.

### void mepack_single_trsylv2_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 * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 64)

**Purpose:**

mepack_double_trsylv2_level2_local_copy_64 solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv2_l2_local_copy_64**.**See also****dla_trsylv2_l2_local_copy_64****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **2217** of file **trsylv2.c**.

### void mepack_single_trsylv2_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 * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 96)

**Purpose:**

mepack_single_trsylv2_level2_local_copy_96 solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv2_l2_local_copy_96**.**See also****sla_trsylv2_l2_local_copy_96****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **2376** of file **trsylv2.c**.

### void mepack_single_trsylv2_level2_reorder (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, 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_trsylv2_level2_reorder solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv2_l2_reorder**.**See also****sla_trsylv2_l2_reorder****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **1587** of file **trsylv2.c**.

### void mepack_single_trsylv2_level2_unopt (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Unoptimized variant)

**Purpose:**

mepack_single_trsylv2_level2_unopt solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv2_l2_unopt**.**See also****sla_trsylv2_l2_unopt****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **1744** of file **trsylv2.c**.

### void mepack_single_trsylv2_level3 (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation.

**Purpose:**

mepack_single_trsylv2_level3 solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv2_l3**.**See also****sla_trsylv2_l3****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **640** of file **trsylv2.c**.

### void mepack_single_trsylv2_level3_2stage (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm with sub-blocking for the generalized Sylvester equation.

**Purpose:**

mepack_single_trsylv2_level3_2stage solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv2_l3_2s**.**See also****sla_trsylv2_l3_2s****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **956** of file **trsylv2.c**.

### void mepack_single_trsylv2_level3_unopt (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Unoptimized variant)

**Purpose:**

mepack_single_trsylv2_level3_unopt solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv2_l3_unopt**.**See also****sla_trsylv2_l3_unopt****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **799** of file **trsylv2.c**.

### void mepack_single_trsylv2_recursive (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Recursive blocking Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation.

**Purpose:**

mepack_single_trsylv2_recursive solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv2_recursive**.**See also****sla_trsylv2_recursive****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **327** of file **trsylv2.c**.

### void mepack_single_trsylv_dag (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation with DAG parallelization.

**Purpose:**

mepack_single_trsylv_dag solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv_dag**.**See also****sla_trsylv_dag****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **324** of file **trsylv.c**.

### void mepack_single_trsylv_level2 (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation.

**Purpose:**

mepack_single_trsylv_level2 solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv_l2**.**See also****sla_trsylv_l2****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **1428** of file **trsylv.c**.

### void mepack_single_trsylv_level2_local_copy (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant)

**Purpose:**

mepack_single_trsylv_level2_local_copy solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv_l2_local_copy**.**See also****sla_trsylv_l2_local_copy****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **1903** of file **trsylv.c**.

### void mepack_single_trsylv_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 * X, int LDX, float * SCALE, float * WORK, int * INFO)

**Purpose:**

mepack_single_trsylv_level2_local_copy_128 solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv_l2_local_copy_128**.**See also****sla_trsylv_l2_local_copy_128****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **2535** of file **trsylv.c**.

### void mepack_single_trsylv_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 * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 32 )

**Purpose:**

mepack_single_trsylv_level2_local_copy_32 solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv_l2_local_copy_32**.**See also****sla_trsylv_l2_local_copy_32****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **2060** of file **trsylv.c**.

### void mepack_single_trsylv_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 * X, int LDX, float * SCALE, float * WORK, int * INFO)

**Purpose:**

mepack_double_trsylv_level2_local_copy_64 solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via DGEES form LAPACK.

**Remarks**This function is a wrapper around

**dla_trsylv_l2_local_copy_64**.**See also****dla_trsylv_l2_local_copy_64****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **2217** of file **trsylv.c**.

### void mepack_single_trsylv_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 * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Local copy variant, M,N <= 96)

**Purpose:**

mepack_single_trsylv_level2_local_copy_96 solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv_l2_local_copy_96**.**See also****sla_trsylv_l2_local_copy_96****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **2376** of file **trsylv.c**.

### void mepack_single_trsylv_level2_reorder (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, 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_trsylv_level2_reorder solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv_l2_reorder**.**See also****sla_trsylv_l2_reorder****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **1587** of file **trsylv.c**.

### void mepack_single_trsylv_level2_unopt (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Unoptimized variant)

**Purpose:**

mepack_single_trsylv_level2_unopt solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv_l2_unopt**.**See also****sla_trsylv_l2_unopt****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **1744** of file **trsylv.c**.

### void mepack_single_trsylv_level3 (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation.

**Purpose:**

mepack_single_trsylv_level3 solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv_l3**.**See also****sla_trsylv_l3****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **640** of file **trsylv.c**.

### void mepack_single_trsylv_level3_2stage (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm with blocking for the generalized Sylvester equation.

**Purpose:**

mepack_single_trsylv_level3_2stage solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv_l3_2s**.**See also****sla_trsylv_l3_2s****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **956** of file **trsylv.c**.

### void mepack_single_trsylv_level3_unopt (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation. (Unoptimized variant)

**Purpose:**

mepack_single_trsylv_level3_unopt solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv_l3_unopt**.**See also****sla_trsylv_l3_unopt****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **799** of file **trsylv.c**.

### void mepack_single_trsylv_recursive (const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Recursive blocking Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation.

**Purpose:**

mepack_single_trsylv_recursive solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices A and B are generated via SGEES form LAPACK.

**Remarks**This function is a wrapper around

**sla_trsylv_recursive**.**See also****sla_trsylv_recursive****Parameters***TRANSA**TRANSB**SGN**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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*X**LDX*LDX is INTEGER The leading dimension of the array X. LDB >= max(1,M).

*SCALE**WORK**INFO***Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **327** of file **trsylv.c**.

## Author

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