cggsylv - Man Page
Name
cggsylv — C-Interface
— C-Interface for generalized Sylvester equations.
Synopsis
Functions
void mepack_double_ggcsylv (const char *FACTA, const char *FACTB, const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *QA, int LDQA, double *ZA, int LDZA, double *QB, int LDQB, double *ZB, int LDZB, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, size_t LDWORK, int *INFO)
Frontend for the solution of Coupled Generalized Sylvester Equations.
void mepack_single_ggcsylv (const char *FACTA, const char *FACTB, const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *QA, int LDQA, float *ZA, int LDZA, float *QB, int LDQB, float *ZB, int LDZB, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, size_t LDWORK, int *INFO)
Frontend for the solution of Coupled Generalized Sylvester Equations.
void mepack_double_ggcsylv_dual (const char *FACTA, const char *FACTB, const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *QA, int LDQA, double *ZA, int LDZA, double *QB, int LDQB, double *ZB, int LDZB, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, size_t LDWORK, int *INFO)
Frontend for the solution of the dual Coupled Generalized Sylvester Equations.
void mepack_single_ggcsylv_dual (const char *FACTA, const char *FACTB, const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *QA, int LDQA, float *ZA, int LDZA, float *QB, int LDQB, float *ZB, int LDZB, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, size_t LDWORK, int *INFO)
Frontend for the solution of the dual Coupled Generalized Sylvester Equations.
void mepack_double_ggsylv (const char *FACTA, const char *FACTB, const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *QA, int LDQA, double *ZA, int LDZA, double *QB, int LDQB, double *ZB, int LDZB, double *X, int LDX, double *SCALE, double *WORK, size_t LDWORK, int *INFO)
Frontend for the solution of Generalized Sylvester Equations.
void mepack_single_ggsylv (const char *FACTA, const char *FACTB, const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *QA, int LDQA, float *ZA, int LDZA, float *QB, int LDQB, float *ZB, int LDZB, float *X, int LDX, float *SCALE, float *WORK, size_t LDWORK, int *INFO)
Frontend for the solution of Generalized Sylvester Equations.
void mepack_double_ggcsylv_refine (const char *TRANSA, const char *TRANSB, const char *GUESS, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *R, int LDR, double *L, int LDL, double *E, int LDE, double *F, int LDF, double *AS, int LDAS, double *BS, int LDBS, double *CS, int LDCS, double *DS, int LDDS, double *Q, int LDQ, double *Z, int LDZ, double *U, int LDU, double *V, int LDV, int *MAXIT, double *TAU, double *CONVLOG, double *WORK, size_t LDWORK, int *INFO)
Iterative Refinement for the Coupled Generalized Sylvester Equations.
void mepack_single_ggcsylv_refine (const char *TRANSA, const char *TRANSB, const char *GUESS, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *R, int LDR, float *L, int LDL, float *E, int LDE, float *F, int LDF, float *AS, int LDAS, float *BS, int LDBS, float *CS, int LDCS, float *DS, int LDDS, float *Q, int LDQ, float *Z, int LDZ, float *U, int LDU, float *V, int LDV, int *MAXIT, float *TAU, float *CONVLOG, float *WORK, size_t LDWORK, int *INFO)
Iterative Refinement for the Coupled Generalized Sylvester Equations.
void mepack_double_ggcsylv_dual_refine (const char *TRANSA, const char *TRANSB, const char *GUESS, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *R, int LDR, double *L, int LDL, double *E, int LDE, double *F, int LDF, double *AS, int LDAS, double *BS, int LDBS, double *CS, int LDCS, double *DS, int LDDS, double *Q, int LDQ, double *Z, int LDZ, double *U, int LDU, double *V, int LDV, int *MAXIT, double *TAU, double *CONVLOG, double *WORK, size_t LDWORK, int *INFO)
Iterative Refinement for the dual Coupled Generalized Sylvester Equations.
void mepack_single_ggcsylv_dual_refine (const char *TRANSA, const char *TRANSB, const char *GUESS, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *R, int LDR, float *L, int LDL, float *E, int LDE, float *F, int LDF, float *AS, int LDAS, float *BS, int LDBS, float *CS, int LDCS, float *DS, int LDDS, float *Q, int LDQ, float *Z, int LDZ, float *U, int LDU, float *V, int LDV, int *MAXIT, float *TAU, float *CONVLOG, float *WORK, size_t LDWORK, int *INFO)
Iterative Refinement for the dual Coupled Generalized Sylvester Equations.
void mepack_double_ggsylv_refine (const char *TRANSA, const char *TRANSB, const char *GUESS, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *X, int LDX, double *Y, int LDY, double *AS, int LDAS, double *BS, int LDBS, double *CS, int LDCS, double *DS, int LDDS, double *Q, int LDQ, double *Z, int LDZ, double *U, int LDU, double *V, int LDV, int *MAXIT, double *TAU, double *CONVLOG, double *WORK, size_t LDWORK, int *INFO)
Iterative Refinement for the Generalized Sylvester Equations.
void mepack_single_ggsylv_refine (const char *TRANSA, const char *TRANSB, const char *GUESS, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *X, int LDX, float *Y, int LDY, float *AS, int LDAS, float *BS, int LDBS, float *CS, int LDCS, float *DS, int LDDS, float *Q, int LDQ, float *Z, int LDZ, float *U, int LDU, float *V, int LDV, int *MAXIT, float *TAU, float *CONVLOG, float *WORK, size_t LDWORK, int *INFO)
Iterative Refinement for the Generalized Sylvester Equations.
Detailed Description
C-Interface for generalized Sylvester equations.
The Fortran routines to solve the generalized Sylvester equation with arbitrary 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. The only difference is that the C interface does not allow LAPACK-like work_space queries. For this purpose the mepack_memory_frontend function needs to be used.
Function Documentation
void mepack_double_ggcsylv (const char * FACTA, const char * FACTB, const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * QA, int LDQA, double * ZA, int LDZA, double * QB, int LDQB, double * ZB, int LDZB, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, size_t LDWORK, int * INFO)
Frontend for the solution of Coupled Generalized Sylvester Equations.
Purpose:
mepack_double_ggcsylv solves a generalized Sylvester equation of the following forms
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
where (A,C) is a M-by-M matrix pencil and (B,D) is a N-by-N matrix pencil.
The right hand side (E,F) and the solution (R,L) are M-by-N matrix pencils. The matrix pencils (A,C)
and (B,D) can be either given as general unreduced matrices, as generalized
Hessenberg form, or in terms of their generalized Schur decomposition.
If they are given as general matrices or as a generalized Hessenberg form
their generalized Schur decomposition will be computed..fi
Remarks
This function is a wrapper for dla_ggcsylv.
See also
dla_ggcsylv
Parameters
FACTA
FACTA is String
Specifies how the matrix pencil (A,C) is given.
== 'N': The matrix pencil (A,C) is given as a general matrices and its Schur decomposition
A = QA*S*ZA**T, C = QA*R*ZA**T will be computed.
== 'F': The matrix pencil (A,C) is already in generalized Schur form and S, R, QA, and ZA
are given.
== 'H': The matrix pencil (A,C) is given in generalized Hessenberg form and its Schur decomposition
A = QA*S*ZA**T, C = QA*R*ZA**T will be computed.FACTB
FACTB is String
Specifies how the matrix pencil (B,D) is given.
== 'N': The matrix pencil (B,D) is given as a general matrices and its Schur decomposition
B = QB*U*ZB**T, D = QB*V*ZB**T will be computed.
== 'F': The matrix pencil (B,D) is already in generalized Schur form and U, V, QB, and ZB
are given.
== 'H': The matrix pencil (B,D) is given in generalized Hessenberg form and its Schur decomposition
B = QB*U*ZB**T, D = QB*V*ZB**T will be computed.TRANSA
TRANSA is String
Specifies the form of the system of equations with respect to A and C :
== 'N': op1(A) = A
== 'T': op1(A) = A**TTRANSB
TRANSB is String
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B,
== 'T': op2(B) = B**TSGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M)
If FACT == 'N', the matrix A is a general matrix and it is overwritten with the
(quasi-) upper triangular factor S of the Schur decomposition of (A,C).
If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix S of
the Schur decomposition of (A,C).
If FACT == 'H', the matrix A is an upper Hessenberg matrix of the generalized
Hessenberg form (A,C) and it is overwritten with the (quasi-) upper triangular
factor S of the Schur decomposition of (A,C).LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N)
If FACT == 'N', the matrix B is a general matrix and it is overwritten with the
(quasi-) upper triangular factor U of the Schur decomposition of (B,D).
If FACT == 'F', the matrix B contains its (quasi-) upper triangular matrix U of
the Schur decomposition of (B,D).
If FACT == 'H', the matrix B is an upper Hessenberg matrix of the generalized
Hessenberg form (B,D) and it is overwritten with the (quasi-) upper triangular
factor U of the Schur decomposition of (B,D).LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M)
If FACT == 'N', the matrix C is a general matrix and it is overwritten with the
upper triangular factor R of the Schur decomposition of (A,C).
If FACT == 'F', the matrix C contains its upper triangular matrix R of
the Schur decomposition of (A,C).
If FACT == 'H', the matrix C is the upper triangular matrix of the generalized Hessenberg form
(A,C) and it is overwritten with the upper triangular factor R of the Schur decomposition of (A,C).LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N)
If FACT == 'N', the matrix D is a general matrix and it is overwritten with the
upper triangular factor V of the Schur decomposition of (B,D).
If FACT == 'F', the matrix D contains its upper triangular matrix V of
the Schur decomposition of (B,D).
If FACT == 'H', the matrix D is the upper triangular matrix of the generalized Hessenberg form
(B,D) and it is overwritten with the upper triangular factor V of the Schur decomposition of (B,D).LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).QA
QA is DOUBLE PRECISION array, dimension (LDQA,M)
If FACT == 'N', the matrix QA is an empty M-by-M matrix on input and contains the
left Schur vectors of (A,C) on output.
If FACT == 'F', the matrix QA contains the left Schur vectors of (A,C).
If FACT == 'H', the matrix QA is an empty M-by-M matrix on input and contains the
left Schur vectors of (A,C) on output.LDQA
LDQA is INTEGER
The leading dimension of the array QA. LDQA >= max(1,M).ZA
ZA is DOUBLE PRECISION array, dimension (LDZA,M)
If FACT == 'N', the matrix ZA is an empty M-by-M matrix on input and contains the
right Schur vectors of (A,C) on output.
If FACT == 'F', the matrix ZA contains the right Schur vectors of (A,C).
If FACT == 'H', the matrix ZA is an empty M-by-M matrix on input and contains the
right Schur vectors of (A,C) on output.LDZA
LDZA is INTEGER
The leading dimension of the array ZA. LDZA >= max(1,M).QB
QB is DOUBLE PRECISION array, dimension (LDQB,N)
If FACT == 'N', the matrix QB is an empty N-by-N matrix on input and contains the
left Schur vectors of (B,D) on output.
If FACT == 'F', the matrix QB contains the left Schur vectors of (B,D).
If FACT == 'H', the matrix QB is an empty M-by-M matrix on input and contains the
left Schur vectors of (B,D) on output.LDQB
LDQB is INTEGER
The leading dimension of the array QB. LDQB >= max(1,N).ZB
ZB is DOUBLE PRECISION array, dimension (LDZB,N)
If FACT == 'N', the matrix ZB is an empty N-by-N matrix on input and contains the
right Schur vectors of (B,D) on output.
If FACT == 'F', the matrix ZB contains the right Schur vectors of (B,D).
If FACT == 'H', the matrix ZB is an empty M-by-M matrix on input and contains the
right Schur vectors of (B,D) on output.LDZB
LDZB is INTEGER
The leading dimension of the array ZB. LDZB >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L.LDF
LDF is INTEGER
The leading dimension of the array F. LDF >= max(1,M).SCALE
SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory_frontend.LDWORK
LDWORK is INTEGER
Size of the workspace for the algorithm counted in floating point numbers of the actual precision.
The C interface does not support the workspace query by setting LDWORK == -1 on input. In this case,
the \ref mepack_memory_frontend function have to be used.INFO
INFO is INTEGER
== 0: successful exit
= 1: DHGGES failed
= 2: DLA_SORT_GEV failed
= 3: Inner solver failed
< 0: if INFO == -i, the i-Th argument had an illegal value- Attention
The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.
- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 318 of file cgsylv.c.
void mepack_double_ggcsylv_dual (const char * FACTA, const char * FACTB, const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * QA, int LDQA, double * ZA, int LDZA, double * QB, int LDQB, double * ZB, int LDZB, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, size_t LDWORK, int * INFO)
Frontend for the solution of the dual Coupled Generalized Sylvester Equations.
Purpose:
mepack_double_ggcsylv_dual solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * L = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A and C are M-by-M matrices and B and D are N-by-N matrices.
The right hand sides E, F and the solutions R, L are M-by-N matrices.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
| kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for dla_ggcsylv_dual.
See also
dla_ggcsylv_dual
Parameters
FACTA
FACTA is String
Specifies how the matrix pencil (A,C) is given.
== 'N': The matrix pencil (A,C) is given as a general matrices and its Schur decomposition
A = QA*S*ZA**T, C = QA*R*ZA**T will be computed.
== 'F': The matrix pencil (A,C) is already in generalized Schur form and S, R, QA, and ZA
are given.
== 'H': The matrix pencil (A,C) is given in generalized Hessenberg form and its Schur decomposition
A = QA*S*ZA**T, C = QA*R*ZA**T will be computed.FACTB
FACTB is String
Specifies how the matrix pencil (B,D) is given.
== 'N': The matrix pencil (B,D) is given as a general matrices and its Schur decomposition
B = QB*U*ZB**T, D = QB*V*ZB**T will be computed.
== 'F': The matrix pencil (B,D) is already in generalized Schur form and U, V, QB, and ZB
are given.
== 'H': The matrix pencil (B,D) is given in generalized Hessenberg form and its Schur decomposition
B = QB*U*ZB**T, D = QB*V*ZB**T will be computed.TRANSA
TRANSA is String
Specifies the form of the system of equations with respect to A and C :
== 'N': op1(A) = A
== 'T': op1(A) = A**TTRANSB
TRANSB is String
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B,
== 'T': op2(B) = B**TSGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M)
If FACT == 'N', the matrix A is a general matrix and it is overwritten with the
(quasi-) upper triangular factor S of the Schur decomposition of (A,C).
If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix S of
the Schur decomposition of (A,C).
If FACT == 'H', the matrix A is an upper Hessenberg matrix of the generalized
Hessenberg form (A,C) and it is overwritten with the (quasi-) upper triangular
factor S of the Schur decomposition of (A,C).LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N)
If FACT == 'N', the matrix B is a general matrix and it is overwritten with the
(quasi-) upper triangular factor U of the Schur decomposition of (B,D).
If FACT == 'F', the matrix B contains its (quasi-) upper triangular matrix U of
the Schur decomposition of (B,D).
If FACT == 'H', the matrix B is an upper Hessenberg matrix of the generalized
Hessenberg form (B,D) and it is overwritten with the (quasi-) upper triangular
factor U of the Schur decomposition of (B,D).LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M)
If FACT == 'N', the matrix C is a general matrix and it is overwritten with the
upper triangular factor R of the Schur decomposition of (A,C).
If FACT == 'F', the matrix C contains its upper triangular matrix R of
the Schur decomposition of (A,C).
If FACT == 'H', the matrix C is the upper triangular matrix of the generalized Hessenberg form
(A,C) and it is overwritten with the upper triangular factor R of the Schur decomposition of (A,C).LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N)
If FACT == 'N', the matrix D is a general matrix and it is overwritten with the
upper triangular factor V of the Schur decomposition of (B,D).
If FACT == 'F', the matrix D contains its upper triangular matrix V of
the Schur decomposition of (B,D).
If FACT == 'H', the matrix D is the upper triangular matrix of the generalized Hessenberg form
(B,D) and it is overwritten with the upper triangular factor V of the Schur decomposition of (B,D).LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).QA
QA is DOUBLE PRECISION array, dimension (LDQA,M)
If FACT == 'N', the matrix QA is an empty M-by-M matrix on input and contains the
left Schur vectors of (A,C) on output.
If FACT == 'F', the matrix QA contains the left Schur vectors of (A,C).
If FACT == 'H', the matrix QA is an empty M-by-M matrix on input and contains the
left Schur vectors of (A,C) on output.LDQA
LDQA is INTEGER
The leading dimension of the array QA. LDQA >= max(1,M).ZA
ZA is DOUBLE PRECISION array, dimension (LDZA,M)
If FACT == 'N', the matrix ZA is an empty M-by-M matrix on input and contains the
right Schur vectors of (A,C) on output.
If FACT == 'F', the matrix ZA contains the right Schur vectors of (A,C).
If FACT == 'H', the matrix ZA is an empty M-by-M matrix on input and contains the
right Schur vectors of (A,C) on output.LDZA
LDZA is INTEGER
The leading dimension of the array ZA. LDZA >= max(1,M).QB
QB is DOUBLE PRECISION array, dimension (LDQB,N)
If FACT == 'N', the matrix QB is an empty N-by-N matrix on input and contains the
left Schur vectors of (B,D) on output.
If FACT == 'F', the matrix QB contains the left Schur vectors of (B,D).
If FACT == 'H', the matrix QB is an empty M-by-M matrix on input and contains the
left Schur vectors of (B,D) on output.LDQB
LDQB is INTEGER
The leading dimension of the array QB. LDQB >= max(1,N).ZB
ZB is DOUBLE PRECISION array, dimension (LDZB,N)
If FACT == 'N', the matrix ZB is an empty N-by-N matrix on input and contains the
right Schur vectors of (B,D) on output.
If FACT == 'F', the matrix ZB contains the right Schur vectors of (B,D).
If FACT == 'H', the matrix ZB is an empty M-by-M matrix on input and contains the
right Schur vectors of (B,D) on output.LDZB
LDZB is INTEGER
The leading dimension of the array ZB. LDZB >= max(1,N).E
E is DOUBLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L.LDF
LDF is INTEGER
The leading dimension of the array F. LDF >= max(1,M).SCALE
SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory_frontend.LDWORK
LDWORK is INTEGER
Size of the workspace for the algorithm counted in floating point numbers of the actual precision.
The C interface does not support the workspace query by setting LDWORK == -1 on input. In this case,
the \ref mepack_memory_frontend function have to be used.INFO
INFO is INTEGER
== 0: successful exit
= 1: DHGGES failed
= 2: DLA_SORT_GEV failed
= 3: Inner solver failed
< 0: if INFO == -i, the i-Th argument had an illegal value- Attention
The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.
- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 335 of file cgsylv_dual.c.
void mepack_double_ggcsylv_dual_refine (const char * TRANSA, const char * TRANSB, const char * GUESS, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * R, int LDR, double * L, int LDL, double * E, int LDE, double * F, int LDF, double * AS, int LDAS, double * BS, int LDBS, double * CS, int LDCS, double * DS, int LDDS, double * Q, int LDQ, double * Z, int LDZ, double * U, int LDU, double * V, int LDV, int * MAXIT, double * TAU, double * CONVLOG, double * WORK, size_t LDWORK, int * INFO)
Iterative Refinement for the dual Coupled Generalized Sylvester Equations.
Purpose:
mepack_double_ggcsylv_dual_refine solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * L = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A and C are M-by-M matrices and B and D are N-by-N matrices.
The right hand sides E, F and the solutions R, L are M-by-N matrices.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
| kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for dla_ggcsylv_dual_refine.
See also
dla_ggcsylv_dual_refine
Parameters
TRANSA
TRANSA is String
Specifies the form of the system of equations with respect to A and C :
== 'N': op1(A) = A
== 'T': op1(A) = A**TTRANSB
TRANSB is String
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B,
== 'T': op2(B) = B**TGUESS
GUESS is String
Specifies whether (R,L) contains an initial guess on input or not.
= 'I': (R,L) contains an initial guess for the solution
== 'N': No initial guess is provided. (R,L) are set to zero.SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M)
The array A contains the original matrix A defining the equation.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 array B contains the original matrix B defining the equation.LDB
LDB is INTEGER
The leading dimension of the array A. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M)
The array C contains the original matrix C defining the equation.LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N)
The array D contains the original matrix D defining the equation.LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).R
R is DOUBLE PRECISION array, dimension (LDR,N)
On input, the array R contains the initial guess for the first solution.
On output, the array R contains the solution R.LDR
LDR is INTEGER
The leading dimension of the array R. LDR >= max(1,M).L
L is DOUBLE PRECISION array, dimension (LDL,N)
On input, the array L contains the initial guess for the second solution.
On output, the array L contains the second solution.LDL
LDL is INTEGER
The leading dimension of the array L. LDL >= max(1,M).E
E is DOUBLE PRECISION array, dimension (LDE,N)
On input, the array E contains the right hand side E.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N)
On input, the array F contains the right hand side F.LDF
LDF is INTEGER
The leading dimension of the array F. LDF >= max(1,M).AS
AS is DOUBLE PRECISION array, dimension (LDAS,M)
The array AS contains the generalized Schur decomposition of the
A.LDAS
LDAS is INTEGER
The leading dimension of the array AS. LDAS >= max(1,M).BS
BS is DOUBLE PRECISION array, dimension (LDBS,N)
The array BS contains the generalized Schur decomposition of the
B.LDBS
LDBS is INTEGER
The leading dimension of the array BS. LDBS >= max(1,N).CS
CS is DOUBLE PRECISION array, dimension (LDC,M)
The array CS contains the generalized Schur decomposition of the
C.LDCS
LDCS is INTEGER
The leading dimension of the array CS. LDCS >= max(1,M).DS
DS is DOUBLE PRECISION array, dimension (LDDS,N)
The array DS contains the generalized Schur decomposition of the
D.LDDS
LDDS is INTEGER
The leading dimension of the array DS. LDDS >= max(1,N).Q
Q is DOUBLE PRECISION array, dimension (LDQ,M)
The array Q contains the left generalized Schur vectors for (A,C) as returned by DGGES.LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,M).Z
Z is DOUBLE PRECISION array, dimension (LDZ,M)
The array Z contains the right generalized Schur vectors for (A,C) as returned by DGGES.LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= max(1,M).U
U is DOUBLE PRECISION array, dimension (LDU,N)
The array U contains the left generalized Schur vectors for (B,D) as returned by DGGES.LDU
LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,N).V
V is DOUBLE PRECISION array, dimension (LDV,N)
The array V contains the right generalized Schur vectors for (B,D) as returned by DGGES.LDV
LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,N).MAXIT
MAXIT is INTEGER
On input, MAXIT contains the maximum number of iteration that are performed, MAXIT <= 100
On exit, MAXIT contains the number of iteration steps taken by the algorithm.TAU
TAU is DOUBLE PRECISION
On input, TAU contains the additional security factor for the stopping criterion, typical values are 0.1
On exit, TAU contains the last relative residual when the stopping criterion got valid.CONVLOG
CONVLOG is DOUBLE PRECISION array, dimension (MAXIT)
The CONVLOG array contains the convergence history of the iterative refinement. CONVLOG(I) contains the maximum
relative residual of both equations before it is solved for the I-Th time.WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm.LDWORK
LDWORK is INTEGER
Size of the workspace for the algorithm. This can be determined by a call \ref mepack_memory_frontend.INFO
INFO is INTEGER
== 0: Success
> 0: Iteration failed in step INFO
< 0: if INFO == -i, the i-Th argument had an illegal value- Attention
The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.
- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 374 of file cgsylv_dual.c.
void mepack_double_ggcsylv_refine (const char * TRANSA, const char * TRANSB, const char * GUESS, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * R, int LDR, double * L, int LDL, double * E, int LDE, double * F, int LDF, double * AS, int LDAS, double * BS, int LDBS, double * CS, int LDCS, double * DS, int LDDS, double * Q, int LDQ, double * Z, int LDZ, double * U, int LDU, double * V, int LDV, int * MAXIT, double * TAU, double * CONVLOG, double * WORK, size_t LDWORK, int * INFO)
Iterative Refinement for the Coupled Generalized Sylvester Equations.
Purpose:
mepack_double_ggcsylv_refine solves a coupled generalized Sylvester equation of the following forms op1(A) * R + SGN1 * L * op2(B) = E (1) op1(C) * R + SGN2 * L * op2(D) = F with iterative refinement, Thereby (A,C) is a M-by-M matrix pencil and (B,D) is a N-by-N matrix pencil. The right hand side (E,F) and the solution (R,L) are M-by-N matrices. The matrix pencils (A,C) and (B,D) need to be given in the original form as well as in their generalized Schur decomposition since both are required in the iterative refinement procedure.
- Remarks
This function is a wrapper for dla_ggcsylv_refine
- See also
dla_ggcsylv_refine
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C : == 'N': op1(A) = A == 'T': op1(A) = A**TTRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, == 'T': op2(B) = B**TGUESS
GUESS is String Specifies whether (R,L) contains an initial guess on input or not. = 'I': (R,L) contains an initial guess for the solution == 'N': No initial guess is provided. (R,L) are set to zero.SGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The array A contains the original matrix A defining the equation.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 array B contains the original matrix B defining the equation.LDB
LDB is INTEGER The leading dimension of the array A. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The array C contains the original matrix C defining the equation.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The array D contains the original matrix D defining the equation.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).R
R is DOUBLE PRECISION array, dimension (LDR,N) On input, the array R contains the initial guess for the first solution. On output, the array R contains the solution R.LDR
LDR is INTEGER The leading dimension of the array R. LDR >= max(1,M).L
L is DOUBLE PRECISION array, dimension (LDL,N) On input, the array L contains the initial guess for the second solution. On output, the array L contains the second solution.LDL
LDL is INTEGER The leading dimension of the array L. LDL >= max(1,M).E
E is DOUBLE PRECISION array, dimension (LDE,N) On input, the array E contains the right hand side E.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N) On input, the array F contains the right hand side F.LDF
LDF is INTEGER The leading dimension of the array F. LDF >= max(1,M).AS
AS is DOUBLE PRECISION array, dimension (LDAS,M) The array AS contains the generalized Schur decomposition of the A.LDAS
LDAS is INTEGER The leading dimension of the array AS. LDAS >= max(1,M).BS
BS is DOUBLE PRECISION array, dimension (LDBS,N) The array BS contains the generalized Schur decomposition of the B.LDBS
LDBS is INTEGER The leading dimension of the array BS. LDBS >= max(1,N).CS
CS is DOUBLE PRECISION array, dimension (LDCS,M) The array CS contains the generalized Schur decomposition of the C.LDCS
LDCS is INTEGER The leading dimension of the array CS. LDCS >= max(1,M).DS
DS is DOUBLE PRECISION array, dimension (LDDS,N) The array DS contains the generalized Schur decomposition of the D.LDDS
LDDS is INTEGER The leading dimension of the array DS. LDDS >= max(1,N).Q
Q is DOUBLE PRECISION array, dimension (LDQ,M) The array Q contains the left generalized Schur vectors for (A,C) as returned by DGGES.LDQ
LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,M).Z
Z is DOUBLE PRECISION array, dimension (LDZ,M) The array Z contains the right generalized Schur vectors for (A,C) as returned by DGGES.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= max(1,M).U
U is DOUBLE PRECISION array, dimension (LDU,N) The array U contains the left generalized Schur vectors for (B,D) as returned by DGGES.LDU
LDU is INTEGER The leading dimension of the array U. LDU >= max(1,N).V
V is DOUBLE PRECISION array, dimension (LDV,N) The array V contains the right generalized Schur vectors for (B,D) as returned by DGGES.LDV
LDV is INTEGER The leading dimension of the array V. LDV >= max(1,N).MAXIT
MAXIT is INTEGER On input, MAXIT contains the maximum number of iteration that are performed, MAXIT <= 100 On exit, MAXIT contains the number of iteration steps taken by the algorithm.TAU
TAU is DOUBLE PRECISION On input, TAU contains the additional security factor for the stopping criterion, typical values are 0.1 On exit, TAU contains the last relative residual when the stopping criterion got valid.CONVLOG
CONVLOG is DOUBLE PRECISION array, dimension (MAXIT) The CONVLOG array contains the convergence history of the iterative refinement. CONVLOG(I) contains the maximum relative residual of both equations before it is solved for the I-Th time.WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LDWORK)) Workspace for the algorithm.LDWORK
LDWORK is INTEGER Size of the workspace for the algorithm. This can be determined by a call \ref mepack_memory_frontend.INFO
INFO is INTEGER == 0: Success > 0: Iteration failed in step INFO < 0: if INFO == -i, the i-Th argument had an illegal value- Attention
The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.
- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 353 of file cgsylv.c.
void mepack_double_ggsylv (const char * FACTA, const char * FACTB, const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * QA, int LDQA, double * ZA, int LDZA, double * QB, int LDQB, double * ZB, int LDZB, double * X, int LDX, double * SCALE, double * WORK, size_t LDWORK, int * INFO)
Frontend for the solution of Generalized Sylvester Equations.
Purpose:
mepack_double_ggsylv solves a generalized Sylvester equation of the following forms
op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1)
or
op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2)
where (A,C) is a M-by-M matrix pencil and (B,D) is a N-by-N matrix pencil.
The right hand side Y and the solution X M-by-N matrices. The matrix pencils (A,C)
and (B,D) can be either given as general unreduced matrices, as generalized
Hessenberg form, or in terms of their generalized Schur decomposition.
If they are given as general matrices or as a generalized Hessenberg form
their generalized Schur decomposition will be computed..fi
Remarks
This function is a wrapper for dla_ggsylv.
See also
dla_ggsylv
Parameters
FACTA
FACTA is String
Specifies how the matrix pencil (A,C) is given.
== 'N': The matrix pencil (A,C) is given as a general matrices and its Schur decomposition
A = QA*S*ZA**T, C = QA*R*ZA**T will be computed.
== 'F': The matrix pencil (A,C) is already in generalized Schur form and S, R, QA, and ZA
are given.
== 'H': The matrix pencil (A,C) is given in generalized Hessenberg form and its Schur decomposition
A = QA*S*ZA**T, C = QA*R*ZA**T will be computed.FACTB
FACTB is String
Specifies how the matrix pencil (B,D) is given.
== 'N': The matrix pencil (B,D) is given as a general matrices and its Schur decomposition
B = QB*U*ZB**T, D = QB*V*ZB**T will be computed.
== 'F': The matrix pencil (B,D) is already in generalized Schur form and U, V, QB, and ZB
are given.
== 'H': The matrix pencil (B,D) is given in generalized Hessenberg form and its Schur decomposition
B = QB*U*ZB**T, D = QB*V*ZB**T will be computed.TRANSA
TRANSA is String
Specifies the form of the system of equations with respect to A and C :
== 'N': op1(A) = A
== 'T': op1(A) = A**TTRANSB
TRANSB is String
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B,
== 'T': op2(B) = B**TSGN
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)
If FACT == 'N', the matrix A is a general matrix and it is overwritten with the
(quasi-) upper triangular factor S of the Schur decomposition of (A,C).
If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix S of
the Schur decomposition of (A,C).
If FACT == 'H', the matrix A is an upper Hessenberg matrix of the generalized
Hessenberg form (A,C) and it is overwritten with the (quasi-) upper triangular
factor S of the Schur decomposition of (A,C).LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N)
If FACT == 'N', the matrix B is a general matrix and it is overwritten with the
(quasi-) upper triangular factor U of the Schur decomposition of (B,D).
If FACT == 'F', the matrix B contains its (quasi-) upper triangular matrix U of
the Schur decomposition of (B,D).
If FACT == 'H', the matrix B is an upper Hessenberg matrix of the generalized
Hessenberg form (B,D) and it is overwritten with the (quasi-) upper triangular
factor U of the Schur decomposition of (B,D).LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M)
If FACT == 'N', the matrix C is a general matrix and it is overwritten with the
upper triangular factor R of the Schur decomposition of (A,C).
If FACT == 'F', the matrix C contains its upper triangular matrix R of
the Schur decomposition of (A,C).
If FACT == 'H', the matrix C is the upper triangular matrix of the generalized Hessenberg form
(A,C) and it is overwritten with the upper triangular factor R of the Schur decomposition of (A,C).LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N)
If FACT == 'N', the matrix D is a general matrix and it is overwritten with the
upper triangular factor V of the Schur decomposition of (B,D).
If FACT == 'F', the matrix D contains its upper triangular matrix V of
the Schur decomposition of (B,D).
If FACT == 'H', the matrix D is the upper triangular matrix of the generalized Hessenberg form
(B,D) and it is overwritten with the upper triangular factor V of the Schur decomposition of (B,D).LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).QA
QA is DOUBLE PRECISION array, dimension (LDQA,M)
If FACT == 'N', the matrix QA is an empty M-by-M matrix on input and contains the
left Schur vectors of (A,C) on output.
If FACT == 'F', the matrix QA contains the left Schur vectors of (A,C).
If FACT == 'H', the matrix QA is an empty M-by-M matrix on input and contains the
left Schur vectors of (A,C) on output.LDQA
LDQA is INTEGER
The leading dimension of the array QA. LDQA >= max(1,M).ZA
ZA is DOUBLE PRECISION array, dimension (LDZA,M)
If FACT == 'N', the matrix ZA is an empty M-by-M matrix on input and contains the
right Schur vectors of (A,C) on output.
If FACT == 'F', the matrix ZA contains the right Schur vectors of (A,C).
If FACT == 'H', the matrix ZA is an empty M-by-M matrix on input and contains the
right Schur vectors of (A,C) on output.LDZA
LDZA is INTEGER
The leading dimension of the array ZA. LDZA >= max(1,M).QB
QB is DOUBLE PRECISION array, dimension (LDQB,N)
If FACT == 'N', the matrix QB is an empty N-by-N matrix on input and contains the
left Schur vectors of (B,D) on output.
If FACT == 'F', the matrix QB contains the left Schur vectors of (B,D).
If FACT == 'H', the matrix QB is an empty M-by-M matrix on input and contains the
left Schur vectors of (B,D) on output.LDQB
LDQB is INTEGER
The leading dimension of the array QB. LDQB >= max(1,N).ZB
ZB is DOUBLE PRECISION array, dimension (LDZB,N)
If FACT == 'N', the matrix ZB is an empty N-by-N matrix on input and contains the
right Schur vectors of (B,D) on output.
If FACT == 'F', the matrix ZB contains the right Schur vectors of (B,D).
If FACT == 'H', the matrix ZB is an empty M-by-M matrix on input and contains the
right Schur vectors of (B,D) on output.LDZB
LDZB is INTEGER
The leading dimension of the array ZB. LDZB >= 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)
Right hand side Y and the solution X are M-by-N matrices.LDX
LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,M).SCALE
SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory_frontend.LDWORK
LDWORK is INTEGER
Size of the workspace for the algorithm counted in floating point numbers of the actual precision.
The C interface does not support the workspace query by setting LDWORK == -1 on input. In this case,
the \ref mepack_memory_frontend function have to be used.INFO
INFO is INTEGER
== 0: successful exit
= 1: DHGGES failed
= 2: DLA_SORT_GEV failed
= 3: Inner solver failed
< 0: if INFO == -i, the i-Th argument had an illegal value- Attention
The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.
- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 303 of file ggsylv.c.
void mepack_double_ggsylv_refine (const char * TRANSA, const char * TRANSB, const char * GUESS, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * X, int LDX, double * Y, int LDY, double * AS, int LDAS, double * BS, int LDBS, double * CS, int LDCS, double * DS, int LDDS, double * Q, int LDQ, double * Z, int LDZ, double * U, int LDU, double * V, int LDV, int * MAXIT, double * TAU, double * CONVLOG, double * WORK, size_t LDWORK, int * INFO)
Iterative Refinement for the Generalized Sylvester Equations.
Purpose:
mepack_double_ggsylv_refine solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + SGN * op1(C) * X * op2(D) = Y (1) with iterative refinement, Thereby (A,C) is a M-by-M matrix pencil and (B,D) is a N-by-N matrix pencil. The right hand side Y and the solution X are M-by-N matrices. The matrix pencils (A,C) and (B,D) need to be given in the original form as well as in their generalized Schur decomposition since both are required in the iterative refinement procedure.
- Remarks
This function is a wrapper for dla_ggsylv_refine
- See also
dla_ggsylv_refine
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C : == 'N': op1(A) = A == 'T': op1(A) = A**TTRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, == 'T': op2(B) = B**TGUESS
GUESS is String Specifies whether X contains an initial guess on input or not. = 'I': X contains an initial guess for the solution == 'N': No initial guess is provided. X is set to zero.SGN
SGN is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M) The array A contains the original matrix A defining the equation.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 array B contains the original matrix B defining the equation.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M) The array C contains the original matrix C defining the equation.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N) The array D contains the original matrix D defining the equation.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is DOUBLE PRECISION array, dimension (LDX,N) On input, the array X contains the initial guess. On output, the array X contains the solution X.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).Y
Y is DOUBLE PRECISION array, dimension (LDY,N) On input, the array Y contains the right hand side.LDY
LDY is INTEGER The leading dimension of the array Y. LDY >= max(1,M).AS
AS is DOUBLE PRECISION array, dimension (LDAS,M) The array AS contains the generalized Schur decomposition of the A.LDAS
LDAS is INTEGER The leading dimension of the array AS. LDAS >= max(1,M).BS
BS is DOUBLE PRECISION array, dimension (LDBS,N) The array BS contains the generalized Schur decomposition of the B.LDBS
LDBS is INTEGER The leading dimension of the array BS. LDBS >= max(1,N).CS
CS is DOUBLE PRECISION array, dimension (LDCS,M) The array CS contains the generalized Schur decomposition of the C.LDCS
LDCS is INTEGER The leading dimension of the array CS. LDCS >= max(1,M).DS
DS is DOUBLE PRECISION array, dimension (LDDS,N) The array DS contains the generalized Schur decomposition of the D.LDDS
LDDS is INTEGER The leading dimension of the array DS. LDDS >= max(1,N).Q
Q is DOUBLE PRECISION array, dimension (LDQ,M) The array Q contains the left generalized Schur vectors for (A,C) as returned by DGGES.LDQ
LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,M).Z
Z is DOUBLE PRECISION array, dimension (LDZ,M) The array Z contains the right generalized Schur vectors for (A,C) as returned by DGGES.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= max(1,M).U
U is DOUBLE PRECISION array, dimension (LDU,N) The array U contains the left generalized Schur vectors for (B,D) as returned by DGGES.LDU
LDU is INTEGER The leading dimension of the array U. LDU >= max(1,N).V
V is DOUBLE PRECISION array, dimension (LDV,N) The array V contains the right generalized Schur vectors for (B,D) as returned by DGGES.LDV
LDV is INTEGER The leading dimension of the array V. LDV >= max(1,N).MAXIT
MAXIT is INTEGER On input, MAXIT contains the maximum number of iteration that are performed, MAXIT <= 100 On exit, MAXIT contains the number of iteration steps taken by the algorithm.TAU
TAU is DOUBLE PRECISION On input, TAU contains the additional security factor for the stopping criterion, typical values are 0.1 On exit, TAU contains the last relative residual when the stopping criterion got valid.CONVLOG
CONVLOG is DOUBLE PRECISION array, dimension (MAXIT) The CONVLOG array contains the convergence history of the iterative refinement. CONVLOG(I) contains the maximum relative residual before it is solved for the I-Th time.WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LDWORK)) Workspace for the algorithm.LDWORK
LDWORK is INTEGER Size of the workspace for the algorithm. This can be determined by a call \ref mepack_memory_frontend.INFO
INFO is INTEGER == 0: Success > 0: Iteration failed in step INFO < 0: if INFO == -i, the i-Th argument had an illegal value- Attention
The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.
- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 324 of file ggsylv.c.
void mepack_single_ggcsylv (const char * FACTA, const char * FACTB, const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * QA, int LDQA, float * ZA, int LDZA, float * QB, int LDQB, float * ZB, int LDZB, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, size_t LDWORK, int * INFO)
Frontend for the solution of Coupled Generalized Sylvester Equations.
Purpose:
mepack_single_ggcsylv solves a generalized Sylvester equation of the following forms
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
where (A,C) is a M-by-M matrix pencil and (B,D) is a N-by-N matrix pencil.
The right hand side (E,F) and the solution (R,L) M-by-N matrices. The matrix pencils (A,C)
and (B,D) can be either given as general unreduced matrices, as generalized
Hessenberg form, or in terms of their generalized Schur decomposition.
If they are given as general matrices or as a generalized Hessenberg form
their generalized Schur decomposition will be computed..fi
Remarks
This function is a wrapper for sla_ggcsylv.
See also
sla_ggcsylv
Parameters
FACTA
FACTA is String
Specifies how the matrix pencil (A,C) is given.
== 'N': The matrix pencil (A,C) is given as a general matrices and its Schur decomposition
A = QA*S*ZA**T, C = QA*R*ZA**T will be computed.
== 'F': The matrix pencil (A,C) is already in generalized Schur form and S, R, QA, and ZA
are given.
== 'H': The matrix pencil (A,C) is given in generalized Hessenberg form and its Schur decomposition
A = QA*S*ZA**T, C = QA*R*ZA**T will be computed.FACTB
FACTB is String
Specifies how the matrix pencil (B,D) is given.
== 'N': The matrix pencil (B,D) is given as a general matrices and its Schur decomposition
B = QB*U*ZB**T, D = QB*V*ZB**T will be computed.
== 'F': The matrix pencil (B,D) is already in generalized Schur form and U, V, QB, and ZB
are given.
== 'H': The matrix pencil (B,D) is given in generalized Hessenberg form and its Schur decomposition
B = QB*U*ZB**T, D = QB*V*ZB**T will be computed.TRANSA
TRANSA is String
Specifies the form of the system of equations with respect to A and C :
== 'N': op1(A) = A
== 'T': op1(A) = A**TTRANSB
TRANSB is String
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B,
== 'T': op2(B) = B**TSGN1
SGN1 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is DOUBLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is DOUBLE PRECISION array, dimension (LDA,M)
If FACT == 'N', the matrix A is a general matrix and it is overwritten with the
(quasi-) upper triangular factor S of the Schur decomposition of (A,C).
If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix S of
the Schur decomposition of (A,C).
If FACT == 'H', the matrix A is an upper Hessenberg matrix of the generalized
Hessenberg form (A,C) and it is overwritten with the (quasi-) upper triangular
factor S of the Schur decomposition of (A,C).LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is DOUBLE PRECISION array, dimension (LDB,N)
If FACT == 'N', the matrix B is a general matrix and it is overwritten with the
(quasi-) upper triangular factor U of the Schur decomposition of (B,D).
If FACT == 'F', the matrix B contains its (quasi-) upper triangular matrix U of
the Schur decomposition of (B,D).
If FACT == 'H', the matrix B is an upper Hessenberg matrix of the generalized
Hessenberg form (B,D) and it is overwritten with the (quasi-) upper triangular
factor U of the Schur decomposition of (B,D).LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is DOUBLE PRECISION array, dimension (LDC,M)
If FACT == 'N', the matrix C is a general matrix and it is overwritten with the
upper triangular factor R of the Schur decomposition of (A,C).
If FACT == 'F', the matrix C contains its upper triangular matrix R of
the Schur decomposition of (A,C).
If FACT == 'H', the matrix C is the upper triangular matrix of the generalized Hessenberg form
(A,C) and it is overwritten with the upper triangular factor R of the Schur decomposition of (A,C).LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is DOUBLE PRECISION array, dimension (LDD,N)
If FACT == 'N', the matrix D is a general matrix and it is overwritten with the
upper triangular factor V of the Schur decomposition of (B,D).
If FACT == 'F', the matrix D contains its upper triangular matrix V of
the Schur decomposition of (B,D).
If FACT == 'H', the matrix D is the upper triangular matrix of the generalized Hessenberg form
(B,D) and it is overwritten with the upper triangular factor V of the Schur decomposition of (B,D).LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).QA
QA is DOUBLE PRECISION array, dimension (LDQA,M)
If FACT == 'N', the matrix QA is an empty M-by-M matrix on input and contains the
left Schur vectors of (A,C) on output.
If FACT == 'F', the matrix QA contains the left Schur vectors of (A,C).
If FACT == 'H', the matrix QA is an empty M-by-M matrix on input and contains the
left Schur vectors of (A,C) on output.LDQA
LDQA is INTEGER
The leading dimension of the array QA. LDQA >= max(1,M).ZA
ZA is DOUBLE PRECISION array, dimension (LDZA,M)
If FACT == 'N', the matrix ZA is an empty M-by-M matrix on input and contains the
right Schur vectors of (A,C) on output.
If FACT == 'F', the matrix ZA contains the right Schur vectors of (A,C).
If FACT == 'H', the matrix ZA is an empty M-by-M matrix on input and contains the
right Schur vectors of (A,C) on output.LDZA
LDZA is INTEGER
The leading dimension of the array ZA. LDZA >= max(1,M).QB
QB is DOUBLE PRECISION array, dimension (LDQB,M)
If FACT == 'N', the matrix QB is an empty M-by-M matrix on input and contains the
left Schur vectors of (B,D) on output.
If FACT == 'F', the matrix QB contains the left Schur vectors of (B,D).
If FACT == 'H', the matrix QB is an empty M-by-M matrix on input and contains the
left Schur vectors of (B,D) on output.LDQB
LDQB is INTEGER
The leading dimension of the array QB. LDQB >= max(1,M).ZB
ZB is DOUBLE PRECISION array, dimension (LDZB,M)
If FACT == 'N', the matrix ZB is an empty M-by-M matrix on input and contains the
right Schur vectors of (B,D) on output.
If FACT == 'F', the matrix ZB contains the right Schur vectors of (B,D).
If FACT == 'H', the matrix ZB is an empty M-by-M matrix on input and contains the
right Schur vectors of (B,D) on output.LDZB
LDZB is INTEGER
The leading dimension of the array ZB. LDZB >= max(1,M).E
E is DOUBLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R.LDE
LDE is INTEGER
The leading dimension of the array X. LDE >= max(1,M).F
F is DOUBLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L.LDF
LDF is INTEGER
The leading dimension of the array X. LDF >= max(1,M).SCALE
SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory_frontend.LDWORK
LDWORK is INTEGER
Size of the workspace for the algorithm counted in floating point numbers of the actual precision.
The C interface does not support the workspace query by setting LDWORK == -1 on input. In this case,
the \ref mepack_memory_frontend function have to be used.INFO
INFO is INTEGER
== 0: successful exit
= 1: DHGGES failed
= 2: DLA_SORT_GEV failed
= 3: Inner solver failed
< 0: if INFO == -i, the i-Th argument had an illegal value- Attention
The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.
- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 644 of file cgsylv.c.
void mepack_single_ggcsylv_dual (const char * FACTA, const char * FACTB, const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * QA, int LDQA, float * ZA, int LDZA, float * QB, int LDQB, float * ZB, int LDZB, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, size_t LDWORK, int * INFO)
Frontend for the solution of the dual Coupled Generalized Sylvester Equations.
Purpose:
mepack_single_ggcsylv_dual solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * R = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A and C are M-by-M matrices and B and D are N-by-N matrices.
The right hand sides E, F and the solutions R, L are M-by-N matrices.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
| kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for sla_ggcsylv_dual.
See also
sla_ggcsylv_dual
Parameters
FACTA
FACTA is String
Specifies how the matrix pencil (A,C) is given.
== 'N': The matrix pencil (A,C) is given as a general matrices and its Schur decomposition
A = QA*S*ZA**T, C = QA*R*ZA**T will be computed.
== 'F': The matrix pencil (A,C) is already in generalized Schur form and S, R, QA, and ZA
are given.
== 'H': The matrix pencil (A,C) is given in generalized Hessenberg form and its Schur decomposition
A = QA*S*ZA**T, C = QA*R*ZA**T will be computed.FACTB
FACTB is String
Specifies how the matrix pencil (B,D) is given.
== 'N': The matrix pencil (B,D) is given as a general matrices and its Schur decomposition
B = QB*U*ZB**T, D = QB*V*ZB**T will be computed.
== 'F': The matrix pencil (B,D) is already in generalized Schur form and U, V, QB, and ZB
are given.
== 'H': The matrix pencil (B,D) is given in generalized Hessenberg form and its Schur decomposition
B = QB*U*ZB**T, D = QB*V*ZB**T will be computed.TRANSA
TRANSA is String
Specifies the form of the system of equations with respect to A and C :
== 'N': op1(A) = A
== 'T': op1(A) = A**TTRANSB
TRANSB is String
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B,
== 'T': op2(B) = B**TSGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M)
If FACT == 'N', the matrix A is a general matrix and it is overwritten with the
(quasi-) upper triangular factor S of the Schur decomposition of (A,C).
If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix S of
the Schur decomposition of (A,C).
If FACT == 'H', the matrix A is an upper Hessenberg matrix of the generalized
Hessenberg form (A,C) and it is overwritten with the (quasi-) upper triangular
factor S of the Schur decomposition of (A,C).LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N)
If FACT == 'N', the matrix B is a general matrix and it is overwritten with the
(quasi-) upper triangular factor U of the Schur decomposition of (B,D).
If FACT == 'F', the matrix B contains its (quasi-) upper triangular matrix U of
the Schur decomposition of (B,D).
If FACT == 'H', the matrix B is an upper Hessenberg matrix of the generalized
Hessenberg form (B,D) and it is overwritten with the (quasi-) upper triangular
factor U of the Schur decomposition of (B,D).LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M)
If FACT == 'N', the matrix C is a general matrix and it is overwritten with the
upper triangular factor R of the Schur decomposition of (A,C).
If FACT == 'F', the matrix C contains its upper triangular matrix R of
the Schur decomposition of (A,C).
If FACT == 'H', the matrix C is the upper triangular matrix of the generalized Hessenberg form
(A,C) and it is overwritten with the upper triangular factor R of the Schur decomposition of (A,C).LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N)
If FACT == 'N', the matrix D is a general matrix and it is overwritten with the
upper triangular factor V of the Schur decomposition of (B,D).
If FACT == 'F', the matrix D contains its upper triangular matrix V of
the Schur decomposition of (B,D).
If FACT == 'H', the matrix D is the upper triangular matrix of the generalized Hessenberg form
(B,D) and it is overwritten with the upper triangular factor V of the Schur decomposition of (B,D).LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).QA
QA is SINGLE PRECISION array, dimension (LDQA,M)
If FACT == 'N', the matrix QA is an empty M-by-M matrix on input and contains the
left Schur vectors of (A,C) on output.
If FACT == 'F', the matrix QA contains the left Schur vectors of (A,C).
If FACT == 'H', the matrix QA is an empty M-by-M matrix on input and contains the
left Schur vectors of (A,C) on output.LDQA
LDQA is INTEGER
The leading dimension of the array QA. LDQA >= max(1,M).ZA
ZA is SINGLE PRECISION array, dimension (LDZA,M)
If FACT == 'N', the matrix ZA is an empty M-by-M matrix on input and contains the
right Schur vectors of (A,C) on output.
If FACT == 'F', the matrix ZA contains the right Schur vectors of (A,C).
If FACT == 'H', the matrix ZA is an empty M-by-M matrix on input and contains the
right Schur vectors of (A,C) on output.LDZA
LDZA is INTEGER
The leading dimension of the array ZA. LDZA >= max(1,M).QB
QB is SINGLE PRECISION array, dimension (LDQB,M)
If FACT == 'N', the matrix QB is an empty M-by-M matrix on input and contains the
left Schur vectors of (B,D) on output.
If FACT == 'F', the matrix QB contains the left Schur vectors of (B,D).
If FACT == 'H', the matrix QB is an empty M-by-M matrix on input and contains the
left Schur vectors of (B,D) on output.LDQB
LDQB is INTEGER
The leading dimension of the array QB. LDQB >= max(1,M).ZB
ZB is SINGLE PRECISION array, dimension (LDZB,M)
If FACT == 'N', the matrix ZB is an empty M-by-M matrix on input and contains the
right Schur vectors of (B,D) on output.
If FACT == 'F', the matrix ZB contains the right Schur vectors of (B,D).
If FACT == 'H', the matrix ZB is an empty M-by-M matrix on input and contains the
right Schur vectors of (B,D) on output.LDZB
LDZB is INTEGER
The leading dimension of the array ZB. LDZB >= max(1,M).E
E is SINGLE PRECISION array, dimension (LDE,N)
On input, the matrix E contains the right hand side E.
On output, the matrix E contains the solution R.LDE
LDE is INTEGER
The leading dimension of the array X. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N)
On input, the matrix F contains the right hand side F.
On output, the matrix F contains the solution L.LDF
LDF is INTEGER
The leading dimension of the array X. LDF >= max(1,M).SCALE
SCALE is SINGLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.WORK
WORK is SINGLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory_frontend.LDWORK
LDWORK is INTEGER
Size of the workspace for the algorithm counted in floating point numbers of the actual precision.
The C interface does not support the workspace query by setting LDWORK == -1 on input. In this case,
the \ref mepack_memory_frontend function have to be used.INFO
INFO is INTEGER
== 0: successful exit
= 1: DHGGES failed
= 2: DLA_SORT_GEV failed
= 3: Inner solver failed
< 0: if INFO == -i, the i-Th argument had an illegal value- Attention
The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.
- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 677 of file cgsylv_dual.c.
void mepack_single_ggcsylv_dual_refine (const char * TRANSA, const char * TRANSB, const char * GUESS, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * R, int LDR, float * L, int LDL, float * E, int LDE, float * F, int LDF, float * AS, int LDAS, float * BS, int LDBS, float * CS, int LDCS, float * DS, int LDDS, float * Q, int LDQ, float * Z, int LDZ, float * U, int LDU, float * V, int LDV, int * MAXIT, float * TAU, float * CONVLOG, float * WORK, size_t LDWORK, int * INFO)
Iterative Refinement for the dual Coupled Generalized Sylvester Equations.
Purpose:
mepack_single_ggcsylv_dual_refine solves a generalized coupled Sylvester equation of the following form
op1(A)**T * R + op1(C)**T * R = SCALE * E (1)
SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F
where A and C are M-by-M matrices and B and D are N-by-N matrices.
The right hand sides E, F and the solutions R, L are M-by-N matrices.
The equation (1) is the dual to the generalized coupled Sylvester equation
op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2)
op1(C) * R + SGN2 * L * op2(D) = SCALE * F
The equation (1) is the dual one to equation (2) with respect to the underlying linear system.
Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields
| kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E |
Z X = | |*| | = | |
| kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F |
Regarding Z**T one obtains
| kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E |
Z**T X = | |*| | = | |
| SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F |
which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB
are expressed in terms of the Sylvester equation (2)..fi
Remarks
This function is a wrapper for sla_ggcsylv_dual_refine.
See also
sla_ggcsylv_dual_refine
Parameters
TRANSA
TRANSA is String
Specifies the form of the system of equations with respect to A and C :
== 'N': op1(A) = A
== 'T': op1(A) = A**TTRANSB
TRANSB is String
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B,
== 'T': op2(B) = B**TGUESS
GUESS is String
Specifies whether (R,L) contains an initial guess on input or not.
= 'I': (R,L) contains an initial guess for the solution
== 'N': No initial guess is provided. (R,L) are set to zero.SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the second equation.M
M is INTEGER
The order of the matrices A and C. M >= 0.N
N is INTEGER
The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M)
The array A contains the original matrix A defining the equation.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 array B contains the original matrix B defining the equation.LDB
LDB is INTEGER
The leading dimension of the array A. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M)
The array C contains the original matrix C defining the equation.LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDA,N)
The array D contains the original matrix D defining the equation.LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).R
R is SINGLE PRECISION array, dimension (LDR,N)
On input, the array R contains the initial guess for the first solution.
On output, the array R contains the solution R.LDR
LDR is INTEGER
The leading dimension of the array R. LDR >= max(1,M).L
L is SINGLE PRECISION array, dimension (LDL,N)
On input, the array L contains the initial guess for the second solution.
On output, the array L contains the second solution.LDL
LDL is INTEGER
The leading dimension of the array L. LDL >= max(1,M).E
E is SINGLE PRECISION array, dimension (LDE,N)
On input, the array E contains the right hand side E.LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N)
On input, the array F contains the right hand side F.LDF
LDF is INTEGER
The leading dimension of the array F. LDF >= max(1,M).AS
AS is SINGLE PRECISION array, dimension (LDAS,M)
The array AS contains the generalized Schur decomposition of the
A.LDAS
LDAS is INTEGER
The leading dimension of the array AS. LDAS >= max(1,M).BS
BS is SINGLE PRECISION array, dimension (LDBS,N)
The array AS contains the generalized Schur decomposition of the
B.LDBS
LDBS is INTEGER
The leading dimension of the array BS. LDBS >= max(1,N).CS
CS is SINGLE PRECISION array, dimension (LDCS,M)
The array CS contains the generalized Schur decomposition of the
C.LDCS
LDCS is INTEGER
The leading dimension of the array CS. LDCS >= max(1,M).DS
DS is SINGLE PRECISION array, dimension (LDDS,N)
The array DS contains the generalized Schur decomposition of the
D.LDDS
LDDS is INTEGER
The leading dimension of the array DS. LDAS >= max(1,N).Q
Q is SINGLE PRECISION array, dimension (LDQ,M)
The array Q contains the left generalized Schur vectors for (A,C) as returned by DGGES.LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,M).Z
Z is SINGLE PRECISION array, dimension (LDZ,M)
The array Z contains the right generalized Schur vectors for (A,C) as returned by DGGES.LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= max(1,M).U
U is SINGLE PRECISION array, dimension (LDU,N)
The array U contains the left generalized Schur vectors for (B,D) as returned by DGGES.LDU
LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,N).V
V is SINGLE PRECISION array, dimension (LDV,N)
The array V contains the right generalized Schur vectors for (B,D) as returned by DGGES.LDV
LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,N).MAXIT
MAXIT is INTEGER
On input, MAXIT contains the maximum number of iteration that are performed, MAXIT <= 100
On exit, MAXIT contains the number of iteration steps taken by the algorithm.TAU
TAU is SINGLE PRECISION
On input, TAU contains the additional security factor for the stopping criterion, typical values are 0.1
On exit, TAU contains the last relative residual when the stopping criterion got valid.CONVLOG
CONVLOG is SINGLE PRECISION array, dimension (MAXIT)
The CONVLOG array contains the convergence history of the iterative refinement. CONVLOG(I) contains the maximum
relative residual of both equations before it is solved for the I-Th time.WORK
WORK is SINGLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm.LDWORK
LDWORK is INTEGER
Size of the workspace for the algorithm. This can be determined by a call \ref mepack_memory_frontend.INFO
INFO is INTEGER
== 0: Success
> 0: Iteration failed in step INFO
< 0: if INFO == -i, the i-Th argument had an illegal value- Attention
The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.
- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 763 of file cgsylv_dual.c.
void mepack_single_ggcsylv_refine (const char * TRANSA, const char * TRANSB, const char * GUESS, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * R, int LDR, float * L, int LDL, float * E, int LDE, float * F, int LDF, float * AS, int LDAS, float * BS, int LDBS, float * CS, int LDCS, float * DS, int LDDS, float * Q, int LDQ, float * Z, int LDZ, float * U, int LDU, float * V, int LDV, int * MAXIT, float * TAU, float * CONVLOG, float * WORK, size_t LDWORK, int * INFO)
Iterative Refinement for the Coupled Generalized Sylvester Equations.
Purpose:
mepack_single_ggcsylv_refine solves a coupled generalized Sylvester equation of the following forms op1(A) * R + SGN1 * L * op2(B) = E (1) op1(C) * R + SGN2 * L * op2(D) = F with iterative refinement, Thereby (A,C) is a M-by-M matrix pencil and (B,D) is a N-by-N matrix pencil. The right hand side (E,F) and the solution (R,L) are M-by-N matrices. The matrix pencils (A,C) and (B,D) need to be given in the original form as well as in their generalized Schur decomposition since both are required in the iterative refinement procedure.
- Remarks
This function is a wrapper for sla_ggcsylv_refine
- See also
sla_ggcsylv_refine
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C : == 'N': op1(A) = A == 'T': op1(A) = A**TTRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, == 'T': op2(B) = B**TGUESS
GUESS is String Specifies whether (R,L) contains an initial guess on input or not. = 'I': (R,L) contains an initial guess for the solution == 'N': No initial guess is provided. (R,L) are set to zero.SGN1
SGN1 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.SGN2
SGN2 is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The array A contains the original matrix A defining the equation.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 array B contains the original matrix B defining the equation.LDB
LDB is INTEGER The leading dimension of the array A. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The array C contains the original matrix C defining the equation.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDA,N) The array D contains the original matrix D defining the equation.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).R
R is SINGLE PRECISION array, dimension (LDR,N) On input, the array R contains the initial guess for the first solution. On output, the array R contains the solution R.LDR
LDR is INTEGER The leading dimension of the array R. LDR >= max(1,M).L
L is SINGLE PRECISION array, dimension (LDL,N) On input, the array L contains the initial guess for the second solution. On output, the array L contains the second solution.LDL
LDL is INTEGER The leading dimension of the array L. LDL >= max(1,M).E
E is SINGLE PRECISION array, dimension (LDE,N) On input, the array E contains the right hand side E.LDE
LDE is INTEGER The leading dimension of the array E. LDE >= max(1,M).F
F is SINGLE PRECISION array, dimension (LDF,N) On input, the array F contains the right hand side F.LDF
LDF is INTEGER The leading dimension of the array F. LDF >= max(1,M).AS
AS is SINGLE PRECISION array, dimension (LDAS,M) The array AS contains the generalized Schur decomposition of the A.LDAS
LDAS is INTEGER The leading dimension of the array AS. LDAS >= max(1,M).BS
BS is SINGLE PRECISION array, dimension (LDBS,N) The array AS contains the generalized Schur decomposition of the B.LDBS
LDBS is INTEGER The leading dimension of the array BS. LDBS >= max(1,N).CS
CS is SINGLE PRECISION array, dimension (LDCS,M) The array CS contains the generalized Schur decomposition of the C.LDCS
LDCS is INTEGER The leading dimension of the array CS. LDCS >= max(1,M).DS
DS is SINGLE PRECISION array, dimension (LDDS,N) The array DS contains the generalized Schur decomposition of the D.LDDS
LDDS is INTEGER The leading dimension of the array DS. LDAS >= max(1,N).Q
Q is SINGLE PRECISION array, dimension (LDQ,M) The array Q contains the left generalized Schur vectors for (A,C) as returned by DGGES.LDQ
LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,M).Z
Z is SINGLE PRECISION array, dimension (LDZ,M) The array Z contains the right generalized Schur vectors for (A,C) as returned by DGGES.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= max(1,M).U
U is SINGLE PRECISION array, dimension (LDU,N) The array U contains the left generalized Schur vectors for (B,D) as returned by DGGES.LDU
LDU is INTEGER The leading dimension of the array U. LDU >= max(1,N).V
V is SINGLE PRECISION array, dimension (LDV,N) The array V contains the right generalized Schur vectors for (B,D) as returned by DGGES.LDV
LDV is INTEGER The leading dimension of the array V. LDV >= max(1,N).MAXIT
MAXIT is INTEGER On input, MAXIT contains the maximum number of iteration that are performed, MAXIT <= 100 On exit, MAXIT contains the number of iteration steps taken by the algorithm.TAU
TAU is SINGLE PRECISION On input, TAU contains the additional security factor for the stopping criterion, typical values are 0.1 On exit, TAU contains the last relative residual when the stopping criterion got valid.CONVLOG
CONVLOG is SINGLE PRECISION array, dimension (MAXIT) The CONVLOG array contains the convergence history of the iterative refinement. CONVLOG(I) contains the maximum relative residual of both equations before it is solved for the I-Th time.WORK
WORK is SINGLE PRECISION array, dimension (MAX(1,LDWORK)) Workspace for the algorithm.LDWORK
LDWORK is INTEGER Size of the workspace for the algorithm. This can be determined by a call \ref mepack_memory_frontend.INFO
INFO is INTEGER == 0: Success > 0: Iteration failed in step INFO < 0: if INFO == -i, the i-Th argument had an illegal value- Attention
The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.
- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 722 of file cgsylv.c.
void mepack_single_ggsylv (const char * FACTA, const char * FACTB, const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * QA, int LDQA, float * ZA, int LDZA, float * QB, int LDQB, float * ZB, int LDZB, float * X, int LDX, float * SCALE, float * WORK, size_t LDWORK, int * INFO)
Frontend for the solution of Generalized Sylvester Equations.
Purpose:
mepack_single_ggsylv solves a generalized Sylvester equation of the following forms
op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1)
or
op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2)
where (A,C) is a M-by-M matrix pencil and (B,D) is a N-by-N matrix pencil.
The right hand side Y and the solution X M-by-N matrices. The matrix pencils (A,C)
and (B,D) can be either given as general unreduced matrices, as generalized
Hessenberg form, or in terms of their generalized Schur decomposition.
If they are given as general matrices or as a generalized Hessenberg form
their generalized Schur decomposition will be computed..fi
Remarks
This function is a wrapper for sla_ggsylv.
See also
sla_ggsylv
Parameters
FACTA
FACTA is String
Specifies how the matrix pencil (A,C) is given.
== 'N': The matrix pencil (A,C) is given as a general matrices and its Schur decomposition
A = QA*S*ZA**T, C = QA*R*ZA**T will be computed.
== 'F': The matrix pencil (A,C) is already in generalized Schur form and S, R, QA, and ZA
are given.
== 'H': The matrix pencil (A,C) is given in generalized Hessenberg form and its Schur decomposition
A = QA*S*ZA**T, C = QA*R*ZA**T will be computed.FACTB
FACTB is String
Specifies how the matrix pencil (B,D) is given.
== 'N': The matrix pencil (B,D) is given as a general matrices and its Schur decomposition
B = QB*U*ZB**T, D = QB*V*ZB**T will be computed.
== 'F': The matrix pencil (B,D) is already in generalized Schur form and U, V, QB, and ZB
are given.
== 'H': The matrix pencil (B,D) is given in generalized Hessenberg form and its Schur decomposition
B = QB*U*ZB**T, D = QB*V*ZB**T will be computed.TRANSA
TRANSA is String
Specifies the form of the system of equations with respect to A and C :
== 'N': op1(A) = A
== 'T': op1(A) = A**TTRANSB
TRANSB is String
Specifies the form of the system of equations with respect to B and D:
== 'N': op2(B) = B,
== 'T': op2(B) = B**TSGN
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)
If FACT == 'N', the matrix A is a general matrix and it is overwritten with the
(quasi-) upper triangular factor S of the Schur decomposition of (A,C).
If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix S of
the Schur decomposition of (A,C).
If FACT == 'H', the matrix A is an upper Hessenberg matrix of the generalized
Hessenberg form (A,C) and it is overwritten with the (quasi-) upper triangular
factor S of the Schur decomposition of (A,C).LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).B
B is SINGLE PRECISION array, dimension (LDB,N)
If FACT == 'N', the matrix B is a general matrix and it is overwritten with the
(quasi-) upper triangular factor U of the Schur decomposition of (B,D).
If FACT == 'F', the matrix B contains its (quasi-) upper triangular matrix U of
the Schur decomposition of (B,D).
If FACT == 'H', the matrix B is an upper Hessenberg matrix of the generalized
Hessenberg form (B,D) and it is overwritten with the (quasi-) upper triangular
factor U of the Schur decomposition of (B,D).LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M)
If FACT == 'N', the matrix C is a general matrix and it is overwritten with the
upper triangular factor R of the Schur decomposition of (A,C).
If FACT == 'F', the matrix C contains its upper triangular matrix R of
the Schur decomposition of (A,C).
If FACT == 'H', the matrix C is the upper triangular matrix of the generalized Hessenberg form
(A,C) and it is overwritten with the upper triangular factor R of the Schur decomposition of (A,C).LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDD,N)
If FACT == 'N', the matrix D is a general matrix and it is overwritten with the
upper triangular factor V of the Schur decomposition of (B,D).
If FACT == 'F', the matrix D contains its upper triangular matrix V of
the Schur decomposition of (B,D).
If FACT == 'H', the matrix D is the upper triangular matrix of the generalized Hessenberg form
(B,D) and it is overwritten with the upper triangular factor V of the Schur decomposition of (B,D).LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1,N).QA
QA is SINGLE PRECISION array, dimension (LDQA,M)
If FACT == 'N', the matrix QA is an empty M-by-M matrix on input and contains the
left Schur vectors of (A,C) on output.
If FACT == 'F', the matrix QA contains the left Schur vectors of (A,C).
If FACT == 'H', the matrix QA is an empty M-by-M matrix on input and contains the
left Schur vectors of (A,C) on output.LDQA
LDQA is INTEGER
The leading dimension of the array QA. LDQA >= max(1,M).ZA
ZA is SINGLE PRECISION array, dimension (LDZA,M)
If FACT == 'N', the matrix ZA is an empty M-by-M matrix on input and contains the
right Schur vectors of (A,C) on output.
If FACT == 'F', the matrix ZA contains the right Schur vectors of (A,C).
If FACT == 'H', the matrix ZA is an empty M-by-M matrix on input and contains the
right Schur vectors of (A,C) on output.LDZA
LDZA is INTEGER
The leading dimension of the array ZA. LDZA >= max(1,M).QB
QB is SINGLE PRECISION array, dimension (LDQB,M)
If FACT == 'N', the matrix QB is an empty M-by-M matrix on input and contains the
left Schur vectors of (B,D) on output.
If FACT == 'F', the matrix QB contains the left Schur vectors of (B,D).
If FACT == 'H', the matrix QB is an empty M-by-M matrix on input and contains the
left Schur vectors of (B,D) on output.LDQB
LDQB is INTEGER
The leading dimension of the array QB. LDQB >= max(1,M).ZB
ZB is SINGLE PRECISION array, dimension (LDZB,M)
If FACT == 'N', the matrix ZB is an empty M-by-M matrix on input and contains the
right Schur vectors of (B,D) on output.
If FACT == 'F', the matrix ZB contains the right Schur vectors of (B,D).
If FACT == 'H', the matrix ZB is an empty M-by-M matrix on input and contains the
right Schur vectors of (B,D) on output.LDZB
LDZB is INTEGER
The leading dimension of the array ZB. LDZB >= max(1,M).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)
Right hand side Y and the solution X are symmetric M-by-M 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 (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory_frontend.LDWORK
LDWORK is INTEGER
Size of the workspace for the algorithm counted in floating point numbers of the actual precision.
The C interface does not support the workspace query by setting LDWORK == -1 on input. In this case,
the \ref mepack_memory_frontend function have to be used.INFO
INFO is INTEGER
== 0: successful exit
= 1: DHGGES failed
= 2: DLA_SORT_GEV failed
= 3: Inner solver failed
< 0: if INFO == -i, the i-Th argument had an illegal value- Attention
The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.
- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 614 of file ggsylv.c.
void mepack_single_ggsylv_refine (const char * TRANSA, const char * TRANSB, const char * GUESS, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * X, int LDX, float * Y, int LDY, float * AS, int LDAS, float * BS, int LDBS, float * CS, int LDCS, float * DS, int LDDS, float * Q, int LDQ, float * Z, int LDZ, float * U, int LDU, float * V, int LDV, int * MAXIT, float * TAU, float * CONVLOG, float * WORK, size_t LDWORK, int * INFO)
Iterative Refinement for the Generalized Sylvester Equations.
Purpose:
mepack_single_gesylv_refine solves a coupled generalized Sylvester equation of the following forms op1(A) * X * op2(B) + SGN * op1(C) * X * op2(D) = Y (1) with iterative refinement, Thereby (A,C) is a M-by-M matrix pencil and (B,D) is a N-by-N matrix pencil. The right hand side Y and the solution X are M-by-N matrices. The matrix pencils (A,C) and (B,D) need to be given in the original form as well as in their generalized Schur decomposition since both are required in the iterative refinement procedure.
- Remarks
This function is a wrapper for sla_ggsylv_refine
- See also
sla_ggsylv_refine
- Parameters
TRANSA
TRANSA is String Specifies the form of the system of equations with respect to A and C : == 'N': op1(A) = A == 'T': op1(A) = A**TTRANSB
TRANSB is String Specifies the form of the system of equations with respect to B and D: == 'N': op2(B) = B, == 'T': op2(B) = B**TGUESS
GUESS is String Specifies whether X contains an initial guess on input or not. = 'I': X contains an initial guess for the solution == 'N': No initial guess is provided. X is set to zero.SGN
SGN is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.M
M is INTEGER The order of the matrices A and C. M >= 0.N
N is INTEGER The order of the matrices B and D. N >= 0.A
A is SINGLE PRECISION array, dimension (LDA,M) The array A contains the original matrix A defining the equation.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 array B contains the original matrix B defining the equation.LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).C
C is SINGLE PRECISION array, dimension (LDC,M) The array C contains the original matrix C defining the equation.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).D
D is SINGLE PRECISION array, dimension (LDA,N) The array D contains the original matrix D defining the equation.LDD
LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).X
X is SINGLE PRECISION array, dimension (LDX,M) On input, the array X contains the initial guess. On output, the array X contains the solution X.LDX
LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).Y
Y is SINGLE PRECISION array, dimension (LDY,M) On input, the array Y contains the right hand side.LDY
LDY is INTEGER The leading dimension of the array Y. LDY >= max(1,M).AS
AS is SINGLE PRECISION array, dimension (LDAS,M) The array AS contains the generalized Schur decomposition of the A.LDAS
LDAS is INTEGER The leading dimension of the array AS. LDAS >= max(1,M).BS
BS is SINGLE PRECISION array, dimension (LDBS,N) The array AS contains the generalized Schur decomposition of the B.LDBS
LDBS is INTEGER The leading dimension of the array BS. LDBS >= max(1,N).CS
CS is SINGLE PRECISION array, dimension (LDCS,M) The array CS contains the generalized Schur decomposition of the C.LDCS
LDCS is INTEGER The leading dimension of the array CS. LDCS >= max(1,M).DS
DS is SINGLE PRECISION array, dimension (LDDS,N) The array DS contains the generalized Schur decomposition of the D.LDDS
LDDS is INTEGER The leading dimension of the array DS. LDAS >= max(1,N).Q
Q is SINGLE PRECISION array, dimension (LDQ,M) The array Q contains the left generalized Schur vectors for (A,C) as returned by DGGES.LDQ
LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,M).Z
Z is SINGLE PRECISION array, dimension (LDZ,M) The array Z contains the right generalized Schur vectors for (A,C) as returned by DGGES.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= max(1,M).U
U is SINGLE PRECISION array, dimension (LDU,N) The array U contains the left generalized Schur vectors for (B,D) as returned by DGGES.LDU
LDU is INTEGER The leading dimension of the array U. LDU >= max(1,N).V
V is SINGLE PRECISION array, dimension (LDV,N) The array V contains the right generalized Schur vectors for (B,D) as returned by DGGES.LDV
LDV is INTEGER The leading dimension of the array V. LDV >= max(1,N).MAXIT
MAXIT is INTEGER On input, MAXIT contains the maximum number of iteration that are performed, MAXIT <= 100 On exit, MAXIT contains the number of iteration steps taken by the algorithm.TAU
TAU is SINGLE PRECISION On input, TAU contains the additional security factor for the stopping criterion, typical values are 0.1 On exit, TAU contains the last relative residual when the stopping criterion got valid.CONVLOG
CONVLOG is SINGLE PRECISION array, dimension (MAXIT) The CONVLOG array contains the convergence history of the iterative refinement. CONVLOG(I) contains the maximum relative residual before it is solved for the I-Th time.WORK
WORK is SINGLE PRECISION array, dimension (MAX(1,LDWORK)) Workspace for the algorithm.LDWORK
LDWORK is INTEGER Size of the workspace for the algorithm. This can be determined by a call \ref mepack_memory_frontend.INFO
INFO is INTEGER == 0: Success > 0: Iteration failed in step INFO < 0: if INFO == -i, the i-Th argument had an illegal value- Attention
The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.
- Author
Martin Koehler, MPI Magdeburg
- Date
January 2024
Definition at line 660 of file ggsylv.c.
Author
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