# cgesylv - Man Page

## Name

cgesylv — C-Interface

— C-Interface for standard Sylvester equations.

## Synopsis

### Functions

void **mepack_double_gesylv** (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 *QA, int LDQA, double *QB, int LDQB, double *X, int LDX, double *SCALE, double *WORK, size_t LDWORK, int *INFO)

Frontend for the solution of Standard Sylvester Equations.

void **mepack_single_gesylv** (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 *QA, int LDQA, float *QB, int LDQB, float *X, int LDX, float *SCALE, float *WORK, size_t LDWORK, int *INFO)

Frontend for the solution of Standard Sylvester Equations.

void **mepack_double_gesylv2** (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 *QA, int LDQA, double *QB, int LDQB, double *X, int LDX, double *SCALE, double *WORK, size_t LDWORK, int *INFO)

Frontend for the solution of Standard Sylvester Equations.

void **mepack_single_gesylv2** (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 *QA, int LDQA, float *QB, int LDQB, float *X, int LDX, float *SCALE, float *WORK, size_t LDWORK, int *INFO)

Frontend for the solution of Standard Sylvester Equations.

void **mepack_double_gesylv_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 *X, int LDX, double *Y, int LDY, double *AS, int LDAS, double *BS, int LDBS, double *Q, int LDQ, double *U, int LDU, int *MAXIT, double *TAU, double *CONVLOG, double *WORK, size_t LDWORK, int *INFO)

Iterative Refinement for the standard Sylvester Equations.

void **mepack_single_gesylv_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 *X, int LDX, float *Y, int LDY, float *AS, int LDAS, float *BS, int LDBS, float *Q, int LDQ, float *U, int LDU, int *MAXIT, float *TAU, float *CONVLOG, float *WORK, size_t LDWORK, int *INFO)

Iterative Refinement for the standard Sylvester Equations.

void **mepack_double_gesylv2_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 *X, int LDX, double *Y, int LDY, double *AS, int LDAS, double *BS, int LDBS, double *Q, int LDQ, double *U, int LDU, int *MAXIT, double *TAU, double *CONVLOG, double *WORK, size_t LDWORK, int *INFO)

Iterative Refinement for the standard Sylvester Equations.

void **mepack_single_gesylv2_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 *X, int LDX, float *Y, int LDY, float *AS, int LDAS, float *BS, int LDBS, float *Q, int LDQ, float *U, int LDU, int *MAXIT, float *TAU, float *CONVLOG, float *WORK, size_t LDWORK, int *INFO)

Iterative Refinement for the standard Sylvester Equations.

## Detailed Description

C-Interface for standard Sylvester equations.

The Fortran routines to solve the standard 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_gesylv (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 * QA, int LDQA, double * QB, int LDQB, double * X, int LDX, double * SCALE, double * WORK, size_t LDWORK, int * INFO)

Frontend for the solution of Standard Sylvester Equations.

**Purpose:**

mepack_double_gesylv solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M matrix and B is a N-by-N matrix. The right hand side Y and the solution X are M-by-N matrices. The matrices A and B can be either a general unreduced matrix or a (quasi-) upper triangular factor. In the later case QA and QB provide the Schur-vectors of the matrices A and B.

**Remarks**This function is a wrapper for

**dla_gesylv**.**See also****dla_gesylv****Parameters***FACTA*FACTA is String Specifies how the matrix A is given. == 'N': The matrix A is given as a general matrix and its Schur decomposition A = QA*S*QA**T will be computed. == 'F': The matrix A is given as its Schur decomposition in terms of S and QA form A = QA*S*QA**T

*FACTB*FACTB is String Specifies how the matrix B is given. == 'N': The matrix B is given as a general matrix and its Schur decomposition B = QB*R*QB**T will be computed. == 'F': The matrix A is given as its Schur decomposition in terms of R and QB form A = QB*R*QB**T

*TRANSA*TRANSA is String Specifies the form of the system of equations with respect to A: == 'N': op1(A) = A == 'T': op1(A) = A**T

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

*SGN*SGN is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)

*M*M is INTEGER The order of the matrices A and C. M >= 0.

*N*N is INTEGER The order of the matrices B and D. N >= 0.

*A*A is DOUBLE PRECISION array, dimension (LDA,M) If FACT == 'N', the matrix A is a general matrix and it is overwritten with its schur decomposition S. If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix S being the Schur decomposition of A.

*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 its schur decomposition R. If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix R being the Schur decomposition of B.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= 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 Schur vectors of A on output. If FACT == 'F', the matrix QA contains the Schur vectors of A.

*LDQA*LDQA is INTEGER The leading dimension of the array QA. LDQA >= 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 Schur vectors of B on output. If FACT == 'F', the matrix QB contains the Schur vectors of B.

*LDQB*LDQB is INTEGER The leading dimension of the array QB. LDQB >= 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 either 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: DGEES failed = 2: DLA_SORT_EV failed = 3: DLA_TRLYAP_DAG 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**Januar 2023

Definition at line **221** of file **gesylv.c**.

### void mepack_double_gesylv2 (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 * QA, int LDQA, double * QB, int LDQB, double * X, int LDX, double * SCALE, double * WORK, size_t LDWORK, int * INFO)

Frontend for the solution of Standard Sylvester Equations.

**Purpose:**

mepack_double_gesylv2 solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M matrix and B is a N-by-N matrix. The right hand side Y and the solution X are M-by-N matrices. The matrices A and B can be either a general unreduced matrix or a (quasi-) upper triangular factor. In the later case QA and QB provide the Schur-vectors of the matrices A and B.

**Remarks**This function is a wrapper for

**dla_gesylv2**.**See also****dla_gesylv2****Parameters***FACTA*FACTA is String Specifies how the matrix A is given. == 'N': The matrix A is given as a general matrix and its Schur decomposition A = QA*S*QA**T will be computed. == 'F': The matrix A is given as its Schur decomposition in terms of S and QA form A = QA*S*QA**T

*FACTB*FACTB is String Specifies how the matrix B is given. == 'N': The matrix B is given as a general matrix and its Schur decomposition B = QB*R*QB**T will be computed. == 'F': The matrix A is given as its Schur decomposition in terms of R and QB form A = QB*R*QB**T

*TRANSA*TRANSA is String Specifies the form of the system of equations with respect to A: == 'N': op1(A) = A == 'T': op1(A) = A**T

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

*SGN*SGN is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)

*M*M is INTEGER The order of the matrices A and C. M >= 0.

*N*N is INTEGER The order of the matrices B and D. N >= 0.

*A*A is DOUBLE PRECISION array, dimension (LDA,M) If FACT == 'N', the matrix A is a general matrix and it is overwritten with its schur decomposition S. If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix S being the Schur decomposition of A.

*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 its schur decomposition R. If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix R beeping the Schur decomposition of B.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= 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 Schur vectors of A on output. If FACT == 'F', the matrix QA contains the Schur vectors of A.

*LDQA*LDQA is INTEGER The leading dimension of the array QA. LDQA >= 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 Schur vectors of B on output. If FACT == 'F', the matrix QB contains the Schur vectors of B.

*LDQB*LDQB is INTEGER The leading dimension of the array QB. LDQB >= 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 either 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: DGEES failed = 2: DLA_SORT_EV failed = 3: DLA_TRLYAP_DAG 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**Januar 2023

Definition at line **222** of file **gesylv2.c**.

### void mepack_double_gesylv2_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 * X, int LDX, double * Y, int LDY, double * AS, int LDAS, double * BS, int LDBS, double * Q, int LDQ, double * U, int LDU, int * MAXIT, double * TAU, double * CONVLOG, double * WORK, size_t LDWORK, int * INFO)

Iterative Refinement for the standard Sylvester Equations.

**Purpose:**

mepack_double_gesylv2_refine solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = Y (1) or op1(A) * X * op2(B) - X = Y (2) where A is a M-by-M matrix and B is a N-by-N matrix using iterative refinement. The right hand side Y and the solution X are M-by-N matrices. The matrix A and B need to be given in the original form as well as in their Schur decomposition since both are required in the iterative refinement procedure.

**Remarks**This function is a wrapper for

**dla_gesylv2_refine****See also****dla_gesylv2_refine****Parameters***TRANSA*TRANSA is String Specifies the form of the system of equations with respect to A: == 'N': op1(A) = A == 'T': op1(A) = A**T

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

*GUESS*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 both terms.

*M*M is INTEGER The order of the matrix A. M >= 0.

*N*N is INTEGER The order of the matrix B. 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).

*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 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 Schur decomposition of B.

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

*Q*Q is DOUBLE PRECISION array, dimension (LDQ,M) The array Q contains the Schur vectors of A as returned by DGEES.

*LDQ*LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,M).

*U*U is DOUBLE PRECISION array, dimension (LDU,N) The array U contains the Schur vectors of B as returned by DGEES.

*LDU*LDU is INTEGER The leading dimension of the array U. LDU >= 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**Januar 2023

Definition at line **245** of file **gesylv2.c**.

### void mepack_double_gesylv_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 * X, int LDX, double * Y, int LDY, double * AS, int LDAS, double * BS, int LDBS, double * Q, int LDQ, double * U, int LDU, int * MAXIT, double * TAU, double * CONVLOG, double * WORK, size_t LDWORK, int * INFO)

Iterative Refinement for the standard Sylvester Equations.

**Purpose:**

mepack_double_gesylv_refine solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = Y (1) or op1(A) * X - X * op2(B) = Y (2) where A is a M-by-M matrix and B is a N-by-N matrix using iterative refinement. The right hand side Y and the solution X are M-by-N matrices. The matrix A and B need to be given in the original form as well as in their Schur decomposition since both are required in the iterative refinement procedure.

**Remarks**This function is a wrapper for

**dla_gesylv_refine****See also****dla_gesylv_refine****Parameters***TRANSA**TRANSB**GUESS*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 both terms.

*M*M is INTEGER The order of the matrix A. M >= 0.

*N*N is INTEGER The order of the matrix B. 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).

*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 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 Schur decomposition of B.

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

*Q*Q is DOUBLE PRECISION array, dimension (LDQ,M) The array Q contains the Schur vectors of A as returned by DGEES.

*LDQ*LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,M).

*U*U is DOUBLE PRECISION array, dimension (LDU,N) The array U contains the Schur vectors of B as returned by DGEES.

*LDU*LDU is INTEGER The leading dimension of the array U. LDU >= 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****mepack_memory_frontend**function for this purpose.**Author**Martin Koehler, MPI Magdeburg

**Date**Januar 2023

Definition at line **246** of file **gesylv.c**.

### void mepack_single_gesylv (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 * QA, int LDQA, float * QB, int LDQB, float * X, int LDX, float * SCALE, float * WORK, size_t LDWORK, int * INFO)

Frontend for the solution of Standard Sylvester Equations.

**Purpose:**

mepack_single_gesylv solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = SCALE * Y (1) or op1(A) * X - X * op2(B) = SCALE * Y (2) where A is a M-by-M matrix and B is a N-by-N matrix. The right hand side Y and the solution X are M-by-N matrices. The matrices A and B can be either a general unreduced matrix or a (quasi-) upper triangular factor. In the later case QA and QB provide the Schur-vectors of the matrices A and B.

**Remarks**This function is a wrapper for

**sla_gesylv**.**See also****dla_gesylv****Parameters***FACTA*FACTA is String Specifies how the matrix A is given. == 'N': The matrix A is given as a general matrix and its Schur decomposition A = QA*S*QA**T will be computed. == 'F': The matrix A is given as its Schur decomposition in terms of S and QA form A = QA*S*QA**T

*FACTB*FACTB is String Specifies how the matrix B is given. == 'N': The matrix B is given as a general matrix and its Schur decomposition B = QB*R*QB**T will be computed. == 'F': The matrix A is given as its Schur decomposition in terms of R and QB form A = QB*R*QB**T

*TRANSA**TRANSB**SGN*SGN is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between the two parts of the Sylvester equation. = 1 : Solve Equation (1) == -1: Solve Equation (2)

*M*M is INTEGER The order of the matrices A and C. M >= 0.

*N*N is INTEGER The order of the matrices B and D. N >= 0.

*A*A is DOUBLE PRECISION array, dimension (LDA,M) If FACT == 'N', the matrix A is a general matrix and it is overwritten with its schur decomposition S. If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix S being the Schur decomposition of A.

*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 its schur decomposition R. If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix R being the Schur decomposition of B.

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

*QA*QA is DOUBLE PRECISION array, dimension (LDA,M) If FACT == 'N', the matrix QA is an empty M-by-M matrix on input and contains the Schur vectors of A on output. If FACT == 'F', the matrix QA contains the Schur vectors of A.

*LDQA*LDQA is INTEGER The leading dimension of the array QA. LDQA >= max(1,M).

*QB*QB is DOUBLE PRECISION array, dimension (LDA,M) If FACT == 'N', the matrix QB is an empty M-by-M matrix on input and contains the Schur vectors of B on output. If FACT == 'F', the matrix QB contains the Schur vectors of B.

*LDQB*LDQB is INTEGER The leading dimension of the array QB. LDQB >= 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 symmetric M-by-M matrices.

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

*SCALE*SCALE is DOUBLE PRECISION SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.

*WORK*WORK is DOUBLE PRECISION array, dimension (MAX(1,LDWORK)) Workspace for the algorithm. The optimal workspace is given either 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: DGEES failed = 2: DLA_SORT_EV failed = 3: DLA_TRLYAP_DAG failed < 0: if INFO == -i, the i-Th argument had an illegal value

**Attention****mepack_memory_frontend**function for this purpose.**Author**Martin Koehler, MPI Magdeburg

**Date**Januar 2023

Definition at line **443** of file **gesylv.c**.

### void mepack_single_gesylv2 (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 * QA, int LDQA, float * QB, int LDQB, float * X, int LDX, float * SCALE, float * WORK, size_t LDWORK, int * INFO)

Frontend for the solution of Standard Sylvester Equations.

**Purpose:**

mepack_single_gesylv2 solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = SCALE * Y (1) or op1(A) * X * op2(B) - X = SCALE * Y (2) where A is a M-by-M matrix and B is a N-by-N matrix. The right hand side Y and the solution X are M-by-N matrices. The matrices A and B can be either a general unreduced matrix or a (quasi-) upper triangular factor. In the later case QA and QB provide the Schur-vectors of the matrices A and B.

**Remarks**This function is a wrapper for

**sla_gesylv2**.**See also****dla_gesylv2****Parameters***FACTA**FACTB**TRANSA**TRANSB**SGN**M*M is INTEGER The order of the matrices A and C. M >= 0.

*N*N is INTEGER The order of the matrices B and D. N >= 0.

*A**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 its schur decomposition R. If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix R being the Schur decomposition of B.

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

*QA*QA is DOUBLE PRECISION array, dimension (LDA,M) If FACT == 'N', the matrix QA is an empty M-by-M matrix on input and contains the Schur vectors of A on output. If FACT == 'F', the matrix QA contains the Schur vectors of A.

*LDQA*LDQA is INTEGER The leading dimension of the array QA. LDQA >= max(1,M).

*QB*QB is DOUBLE PRECISION array, dimension (LDA,M) If FACT == 'N', the matrix QB is an empty M-by-M matrix on input and contains the Schur vectors of B on output. If FACT == 'F', the matrix QB contains the Schur vectors of B.

*LDQB*LDQB is INTEGER The leading dimension of the array QB. LDQB >= 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 symmetric M-by-M matrices.

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

*SCALE**WORK**LDWORK**INFO***Attention****mepack_memory_frontend**function for this purpose.**Author**Martin Koehler, MPI Magdeburg

**Date**Januar 2023

Definition at line **444** of file **gesylv2.c**.

### void mepack_single_gesylv2_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 * X, int LDX, float * Y, int LDY, float * AS, int LDAS, float * BS, int LDBS, float * Q, int LDQ, float * U, int LDU, int * MAXIT, float * TAU, float * CONVLOG, float * WORK, size_t LDWORK, int * INFO)

Iterative Refinement for the standard Sylvester Equations.

**Purpose:**

mepack_single_gesylv2_refine solves a Sylvester equation of the following forms op1(A) * X * op2(B) + X = Y (1) or op1(A) * X * op2(B) - X = Y (2) where A is a M-by-M matrix and B is a N-by-N matrix using iterative refinement. The right hand side Y and the solution X are M-by-N matrices. The matrix A and B need to be given in the original form as well as in their Schur decomposition since both are required in the iterative refinement procedure.

**Remarks**This function is a wrapper for

**sla_gesylv2_refine****See also****sla_gesylv2_refine****Parameters***TRANSA**TRANSB**GUESS*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 both terms.

*M*M is INTEGER The order of the matrix A. M >= 0.

*N*N is INTEGER The order of the matrix B. 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).

*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 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 Schur decomposition of B.

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

*Q*Q is SINGLE PRECISION array, dimension (LDQ,M) The array Q contains the Schur vectors of A as returned by DGEES.

*LDQ*LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,M).

*U*U is SINGLE PRECISION array, dimension (LDU,N) The array U contains the Schur vectors of B as returned by DGEES.

*LDU*LDU is INTEGER The leading dimension of the array U. LDU >= 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****mepack_memory_frontend**function for this purpose.**Author**Martin Koehler, MPI Magdeburg

**Date**Januar 2023

Definition at line **496** of file **gesylv2.c**.

### void mepack_single_gesylv_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 * X, int LDX, float * Y, int LDY, float * AS, int LDAS, float * BS, int LDBS, float * Q, int LDQ, float * U, int LDU, int * MAXIT, float * TAU, float * CONVLOG, float * WORK, size_t LDWORK, int * INFO)

Iterative Refinement for the standard Sylvester Equations.

**Purpose:**

mepack_single_gesylv_refine solves a Sylvester equation of the following forms op1(A) * X + X * op2(B) = Y (1) or op1(A) * X - X * op2(B) = Y (2) where A is a M-by-M matrix and B is a N-by-N matrix using iterative refinement. The right hand side Y and the solution X are M-by-N matrices. The matrix A and B need to be given in the original form as well as in their Schur decomposition since both are required in the iterative refinement procedure.

**Remarks**This function is a wrapper for

**sla_gesylv_refine****See also****sla_gesylv_refine****Parameters***TRANSA**TRANSB**GUESS**SGN*SGN is SINGLE PRECISION, allowed values: +/-1 Specifies the sign between both terms.

*M*M is INTEGER The order of the matrix A. M >= 0.

*N*N is INTEGER The order of the matrix B. 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).

*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 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 Schur decomposition of B.

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

*Q*Q is SINGLE PRECISION array, dimension (LDQ,M) The array Q contains the Schur vectors of A as returned by DGEES.

*LDQ*LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,M).

*U*U is SINGLE PRECISION array, dimension (LDU,N) The array U contains the Schur vectors of B as returned by DGEES.

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

*MAXIT**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**INFO***Attention****mepack_memory_frontend**function for this purpose.**Author**Martin Koehler, MPI Magdeburg

**Date**Januar 2023

Definition at line **498** of file **gesylv.c**.

## Author

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