# ctglyap - Man Page

## Name

ctglyap — C-Interface

— C-Interface for generalized Lyapunov and Stein equations with triangular coefficient matrices.

## Synopsis

### Functions

void **mepack_double_tglyap_dag** (const char *TRANS, int M, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Lyapunov equation with DAG scheduling.

void **mepack_single_tglyap_dag** (const char *TRANS, int M, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Lyapunov equation with DAG scheduling.

void **mepack_double_tgstein_dag** (const char *TRANS, int M, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Stein equation with DAG scheduling.

void **mepack_single_tgstein_dag** (const char *TRANS, int M, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Stein equation with DAG scheduling.

void **mepack_double_tglyap_level2** (const char *TRANS, int M, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

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

void **mepack_double_tglyap_level2_unopt** (const char *TRANS, int M, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Lyapunov equation (Unoptimized)

void **mepack_single_tglyap_level2** (const char *TRANS, int M, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

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

void **mepack_single_tglyap_level2_unopt** (const char *TRANS, int M, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Lyapunov equation (Unoptimized)

void **mepack_double_tgstein_level2** (const char *TRANS, int M, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

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

void **mepack_single_tgstein_level2** (const char *TRANS, int M, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

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

void **mepack_double_tglyap_level3** (const char *TRANS, int M, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

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

void **mepack_double_tglyap_level3_2stage** (const char *TRANS, int M, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

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

void **mepack_single_tglyap_level3** (const char *TRANS, int M, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

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

void **mepack_single_tglyap_level3_2stage** (const char *TRANS, int M, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

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

void **mepack_double_tgstein_level3** (const char *TRANS, int M, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

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

void **mepack_double_tgstein_level3_2stage** (const char *TRANS, int M, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

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

void **mepack_single_tgstein_level3** (const char *TRANS, int M, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

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

void **mepack_single_tgstein_level3_2stage** (const char *TRANS, int M, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

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

void **mepack_double_tglyap_recursive** (const char *TRANS, int M, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

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

void **mepack_single_tglyap_recursive** (const char *TRANS, int M, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

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

void **mepack_double_tgstein_recursive** (const char *TRANS, int M, double *A, int LDA, double *B, int LDB, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

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

void **mepack_single_tgstein_recursive** (const char *TRANS, int M, float *A, int LDA, float *B, int LDB, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

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

## Detailed Description

C-Interface for generalized Lyapunov and Stein equations with triangular coefficient matrices.

The Fortran routines to solve generalized Lyapunov and Stein equations with triangular coefficients are wrapped in C to provide an easier access to them. All wrapper routines are direct wrappers to the corresponding Fortran subroutines without sanity checks. These are performed by the Fortran routines. Since the routines are using **int** values to pass sizes the work_space query will fail for large scale problems. For this reason the function **mepack_memory** should be used to query the required work_space from a C code. This function is aware of 64 bit integers if MEPACK is compiled with it.

## Function Documentation

### void mepack_double_tglyap_dag (const char * TRANS, int M, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Lyapunov equation with DAG scheduling.

**Purpose:**

mepack_double_tglyap_dag solves a generalized Lyapunov equation of the following forms A * X * B^T + A * X * B^T = SCALE * Y (2) or A^T * X * B + A^T * X * B = SCALE * Y (1) where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular, and X and Y are symmetric M-by-M matrices. Typically the matrix pencil (A,B) is created by DGGES from LAPACK.

**Remarks**This function is a wrapper around

**dla_tglyap_dag**.**See also****dla_tglyap_dag****Parameters***TRANS*TRANS is a string Specifies the form of the system of equations with respect to A and B: == 'N': Equation (1) is solved. == 'T': Equation (2) is solved.

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

*A*A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.

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

*B*B is DOUBLE PRECISION array, dimension (LDB,M) The matrix B must be upper triangular.

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

*X*X is DOUBLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.

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

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

*WORK*WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.

*INFO*INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **146** of file **tglyap.c**.

### void mepack_double_tglyap_level2 (const char * TRANS, int M, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

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

**Purpose:**

mepack_double_tglyap_level2 solves a generalized Lyapunov equation of the following forms A * X * B^T + A * X * B^T = SCALE * Y (2) or A^T * X * B + A^T * X * B = SCALE * Y (1) where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular, and X and Y are symmetric M-by-M matrices. Typically the matrix pencil (A,B) is created by DGGES from LAPACK.

**Remarks**This function is a wrapper around

**dla_tglyap_l2**.**See also****dla_tglyap_l2****Parameters***TRANS*TRANS is a string Specifies the form of the system of equations with respect to A and B: == 'N': Equation (1) is solved. == 'T': Equation (2) is solved.

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

*A*A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.

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

*B*B is DOUBLE PRECISION array, dimension (LDB,M) The matrix B must be upper triangular.

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

*X*X is DOUBLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.

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

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

*WORK*WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.

*INFO*INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **146** of file **tglyap.c**.

### void mepack_double_tglyap_level2_unopt (const char * TRANS, int M, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Lyapunov equation (Unoptimized)

**Purpose:**

mepack_double_tglyap_level2_unopt solves a generalized Lyapunov equation of the following forms A * X * B^T + A * X * B^T = SCALE * Y (2) or A^T * X * B + A^T * X * B = SCALE * Y (1) where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular, and X and Y are symmetric M-by-M matrices. Typically the matrix pencil (A,B) is created by DGGES from LAPACK.

**Remarks**This function is a wrapper around

**dla_tglyap_l2_unopt**.**See also****dla_tglyap_l2_unopt****Parameters***TRANS*TRANS is a string Specifies the form of the system of equations with respect to A and B: == 'N': Equation (1) is solved. == 'T': Equation (2) is solved.

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

*A*A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.

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

*B*B is DOUBLE PRECISION array, dimension (LDB,M) The matrix B must be upper triangular.

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

*X*X is DOUBLE PRECISION array, dimension (LDX,N) On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.

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

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

*WORK*WORK is DOUBLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.

*INFO*INFO is INTEGER On input: == -1 : Perform a workspace query <> -1 : normal operation On exit, workspace query: < 0 : if INFO == -i, the i-Th argument had an illegal value >= 0: The value of INFO is the required number of elements in the workspace. On exit, normal operation: == 0: successful exit < 0: if INFO == -i, the i-Th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **281** of file **tglyap.c**.

### void mepack_double_tglyap_level3 (const char * TRANS, int M, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

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

**Purpose:**

mepack_double_tglyap_level3 solves a generalized Lyapunov equation of the following forms A * X * B^T + A * X * B^T = SCALE * Y (2) or A^T * X * B + A^T * X * B = SCALE * Y (1) where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular, and X and Y are symmetric M-by-M matrices. Typically the matrix pencil (A,B) is created by DGGES from LAPACK.

**Remarks**This function is a wrapper around

**dla_tglyap_l3**.**See also****dla_tglyap_l3****Parameters***TRANS**M*M is INTEGER The order of the matrices A and B. M >= 0.

*A*A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.

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

*B*B is DOUBLE PRECISION array, dimension (LDB,M) The matrix B must be upper triangular.

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

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

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

**Date**January 2024

Definition at line **146** of file **tglyap.c**.

### void mepack_double_tglyap_level3_2stage (const char * TRANS, int M, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

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

**Purpose:**

mepack_double_tglyap_level3_2stage solves a generalized Lyapunov equation of the following forms A * X * B^T + A * X * B^T = SCALE * Y (2) or A^T * X * B + A^T * X * B = SCALE * Y (1) where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular, and X and Y are symmetric M-by-M matrices. Typically the matrix pencil (A,B) is created by DGGES from LAPACK.

**Remarks**This function is a wrapper around

**dla_tglyap_l3_2s**.**See also****dla_tglyap_l3_2s****Parameters***TRANS**M*M is INTEGER The order of the matrices A and B. M >= 0.

*A*A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.

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

*B*B is DOUBLE PRECISION array, dimension (LDB,M) The matrix B must be upper triangular.

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

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

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

**Date**January 2024

Definition at line **281** of file **tglyap.c**.

### void mepack_double_tglyap_recursive (const char * TRANS, int M, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

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

**Purpose:**

mepack_double_tglyap_recursive solves a generalized Lyapunov equation of the following forms A * X * B^T + A * X * B^T = SCALE * Y (2) or A^T * X * B + A^T * X * B = SCALE * Y (1) where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular, and X and Y are symmetric M-by-M matrices. Typically the matrix pencil (A,B) is created by DGGES from LAPACK.

**Remarks**This function is a wrapper around

**dla_tglyap_recursive**.**See also****dla_tglyap_recursive****Parameters***TRANS**M*M is INTEGER The order of the matrices A and B. M >= 0.

*A*A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.

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

*B*B is DOUBLE PRECISION array, dimension (LDB,M) The matrix B must be upper triangular.

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

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

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

**Date**January 2024

Definition at line **146** of file **tglyap.c**.

### void mepack_double_tgstein_dag (const char * TRANS, int M, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Stein equation with DAG scheduling.

**Purpose:**

mepack_double_tgstein_dag solves a generalized Stein equation of the following forms A * X * A^T - B * X * B^T = SCALE * Y (1) or A^T * X * A - B^T * X * B = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular, and X and Y are symmetric M-by-M matrices. Typically the matrix pencil (A,B) is created by DGGES from LAPACK.

**Remarks**This function is a wrapper around

**dla_tgstein_dag**.**See also****dla_tgstein_dag****Parameters***TRANS**M*M is INTEGER The order of the matrices A and B. M >= 0.

*A*A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.

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

*B*B is DOUBLE PRECISION array, dimension (LDB,M) The matrix B must be upper triangular.

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

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

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

**Date**January 2024

Definition at line **146** of file **tgstein.c**.

### void mepack_double_tgstein_level2 (const char * TRANS, int M, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

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

**Purpose:**

mepack_double_tgstein_level2 solves a generalized Stein equation of the following forms A * X * A^T - B * X * B^T = SCALE * Y (1) or A^T * X * A - B^T * X * B = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular, and X and Y are symmetric M-by-M matrices. Typically the matrix pencil (A,B) is created by DGGES from LAPACK.

**Remarks**This function is a wrapper around

**dla_tgstein_l2**.**See also****dla_tgstein_l2****Parameters***TRANS**M*M is INTEGER The order of the matrices A and B. M >= 0.

*A*A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.

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

*B*B is DOUBLE PRECISION array, dimension (LDB,M) The matrix B must be upper triangular.

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

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

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

**Date**January 2024

Definition at line **146** of file **tgstein.c**.

### void mepack_double_tgstein_level3 (const char * TRANS, int M, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

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

**Purpose:**

mepack_double_tgstein_level3 solves a generalized Stein equation of the following forms A * X * A^T - B * X * B^T = SCALE * Y (1) or A^T * X * A - B^T * X * B = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular, and X and Y are symmetric M-by-M matrices. Typically the matrix pencil (A,B) is created by DGGES from LAPACK.

**Remarks**This function is a wrapper around

**dla_tgstein_l3**.**See also****dla_tgstein_l3****Parameters***TRANS**M*M is INTEGER The order of the matrices A and B. M >= 0.

*A*A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.

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

*B*B is DOUBLE PRECISION array, dimension (LDB,M) The matrix B must be upper triangular.

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

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

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

**Date**January 2024

Definition at line **146** of file **tgstein.c**.

### void mepack_double_tgstein_level3_2stage (const char * TRANS, int M, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

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

**Purpose:**

mepack_double_tgstein_level3_2stage solves a generalized Stein equation of the following forms A * X * A^T - B * X * B^T = SCALE * Y (1) or A^T * X * A - B^T * X * B = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular, and X and Y are symmetric M-by-M matrices. Typically the matrix pencil (A,B) is created by DGGES from LAPACK.

**Remarks**This function is a wrapper around

**dla_tgstein_l3_2s**.**See also****dla_tgstein_l3_2s****Parameters***TRANS**M*M is INTEGER The order of the matrices A and B. M >= 0.

*A*A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.

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

*B*B is DOUBLE PRECISION array, dimension (LDB,M) The matrix B must be upper triangular.

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

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

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

**Date**January 2024

Definition at line **281** of file **tgstein.c**.

### void mepack_double_tgstein_recursive (const char * TRANS, int M, double * A, int LDA, double * B, int LDB, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

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

**Purpose:**

mepack_double_tgstein_recursive solves a generalized Stein equation of the following forms A * X * A^T - B * X * B^T = SCALE * Y (1) or A^T * X * A - B^T * X * B = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular, and X and Y are symmetric M-by-M matrices. Typically the matrix pencil (A,B) is created by DGGES from LAPACK.

**Remarks**This function is a wrapper around

**dla_tgstein_recursive**.**See also****dla_tgstein_recursive****Parameters***TRANS**M*M is INTEGER The order of the matrices A and B. M >= 0.

*A*A is DOUBLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.

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

*B*B is DOUBLE PRECISION array, dimension (LDB,M) The matrix B must be upper triangular.

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

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

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

**Date**January 2024

Definition at line **146** of file **tgstein.c**.

### void mepack_single_tglyap_dag (const char * TRANS, int M, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Lyapunov equation with DAG scheduling.

**Purpose:**

mepack_single_tglyap_dag solves a generalized Lyapunov equation of the following forms A * X * B^T + A * X * B^T = SCALE * Y (2) or A^T * X * B + A^T * X * B = SCALE * Y (1) where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular, and X and Y are symmetric M-by-M matrices. Typically the matrix pencil (A,B) is created by SGGES from LAPACK.

**Remarks**This function is a wrapper around

**sla_tglyap_dag**.**See also****sla_tglyap_dag****Parameters***TRANS**M*M is INTEGER The order of the matrices A and B. M >= 0.

*A*A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.

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

*B*B is SINGLE PRECISION array, dimension (LDB,M) The matrix B must be upper triangular.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= 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) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.

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

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

*WORK*WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.

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

**Date**January 2024

Definition at line **281** of file **tglyap.c**.

### void mepack_single_tglyap_level2 (const char * TRANS, int M, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

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

**Purpose:**

mepack_single_tglyap_level2 solves a generalized Lyapunov equation of the following forms A * X * B^T + A * X * B^T = SCALE * Y (2) or A^T * X * B + A^T * X * B = SCALE * Y (1) where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular, and X and Y are symmetric M-by-M matrices. Typically the matrix pencil (A,B) is created by SGGES from LAPACK.

**Remarks**This function is a wrapper around

**sla_tglyap_l2**.**See also****sla_tglyap_l2****Parameters***TRANS**M*M is INTEGER The order of the matrices A and B. M >= 0.

*A*A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.

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

*B*B is SINGLE PRECISION array, dimension (LDB,M) The matrix B must be upper triangular.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= 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) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.

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

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

*WORK*WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.

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

**Date**January 2024

Definition at line **420** of file **tglyap.c**.

### void mepack_single_tglyap_level2_unopt (const char * TRANS, int M, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the generalized Lyapunov equation (Unoptimized)

**Purpose:**

mepack_single_tglyap_level2_unopt solves a generalized Lyapunov equation of the following forms A * X * B^T + A * X * B^T = SCALE * Y (2) or A^T * X * B + A^T * X * B = SCALE * Y (1) where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular, and X and Y are symmetric M-by-M matrices. Typically the matrix pencil (A,B) is created by SGGES from LAPACK.

**Remarks**This function is a wrapper around

**sla_tglyap_l2_unopt**.**See also****sla_tglyap_l2_unopt****Parameters***TRANS**M*M is INTEGER The order of the matrices A and B. M >= 0.

*A*A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.

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

*B*B is SINGLE PRECISION array, dimension (LDB,M) The matrix B must be upper triangular.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= 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) as selected by TRANSA, TRANSB, and SGN. Right hand side Y and the solution X are M-by-N matrices.

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

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

*WORK*WORK is SINGLE PRECISION array, dimension LDWORK Workspace for the algorithm. The workspace needs to queried before the running the computation. The query is performed by calling the subroutine with INFO == -1 on input. The required workspace is then returned in INFO.

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

**Date**January 2024

Definition at line **557** of file **tglyap.c**.

### void mepack_single_tglyap_level3 (const char * TRANS, int M, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

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

**Purpose:**

mepack_single_tglyap_level3 solves a generalized Lyapunov equation of the following forms A * X * B^T + A * X * B^T = SCALE * Y (2) or A^T * X * B + A^T * X * B = SCALE * Y (1) where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular, and X and Y are symmetric M-by-M matrices. Typically the matrix pencil (A,B) is created by SGGES from LAPACK.

**Remarks**This function is a wrapper around

**sla_tglyap_l3**.**See also****sla_tglyap_l3****Parameters***TRANS**M*M is INTEGER The order of the matrices A and B. M >= 0.

*A*A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.

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

*B*B is SINGLE PRECISION array, dimension (LDB,M) The matrix B must be upper triangular.

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

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

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

**Date**January 2024

Definition at line **416** of file **tglyap.c**.

### void mepack_single_tglyap_level3_2stage (const char * TRANS, int M, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

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

**Purpose:**

mepack_single_tglyap_level3_2stage solves a generalized Lyapunov equation of the following forms A * X * B^T + A * X * B^T = SCALE * Y (2) or A^T * X * B + A^T * X * B = SCALE * Y (1) where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular, and X and Y are symmetric M-by-M matrices. Typically the matrix pencil (A,B) is created by SGGES from LAPACK.

**Remarks**This function is a wrapper around

**sla_tglyap_l3_2s**.**See also****sla_tglyap_l3_2s****Parameters***TRANS**M*M is INTEGER The order of the matrices A and B. M >= 0.

*A*A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.

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

*B*B is SINGLE PRECISION array, dimension (LDB,M) The matrix B must be upper triangular.

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

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

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

**Date**January 2024

Definition at line **551** of file **tglyap.c**.

### void mepack_single_tglyap_recursive (const char * TRANS, int M, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

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

**Purpose:**

mepack_single_tglyap_recursive solves a generalized Lyapunov equation of the following forms A * X * B^T + A * X * B^T = SCALE * Y (2) or A^T * X * B + A^T * X * B = SCALE * Y (1) where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular, and X and Y are symmetric M-by-M matrices. Typically the matrix pencil (A,B) is created by SGGES from LAPACK.

**Remarks**This function is a wrapper around

**sla_tglyap_recursive**.**See also****sla_tglyap_recursive****Parameters***TRANS**M*M is INTEGER The order of the matrices A and B. M >= 0.

*A*A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.

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

*B*B is SINGLE PRECISION array, dimension (LDB,M) The matrix B must be upper triangular.

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

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

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

**Date**January 2024

Definition at line **281** of file **tglyap.c**.

### void mepack_single_tgstein_dag (const char * TRANS, int M, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm for the generalized Stein equation with DAG scheduling.

**Purpose:**

mepack_single_tgstein_dag solves a generalized Stein equation of the following forms A * X * A^T - B * X * B^T = SCALE * Y (1) or A^T * X * A - B^T * X * B = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular, and X and Y are symmetric M-by-M matrices. Typically the matrix pencil (A,B) is created by SGGES from LAPACK.

**Remarks**This function is a wrapper around

**sla_tgstein_dag**.**See also****sla_tgstein_dag****Parameters***TRANS**M*M is INTEGER The order of the matrices A and B. M >= 0.

*A*A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.

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

*B*B is SINGLE PRECISION array, dimension (LDB,M) The matrix B must be upper triangular.

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

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

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

**Date**January 2024

Definition at line **281** of file **tgstein.c**.

### void mepack_single_tgstein_level2 (const char * TRANS, int M, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

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

**Purpose:**

mepack_single_tgstein_level2 solves a generalized Stein equation of the following forms A * X * A^T - B * X * B^T = SCALE * Y (1) or A^T * X * A - B^T * X * B = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular, and X and Y are symmetric M-by-M matrices. Typically the matrix pencil (A,B) is created by SGGES from LAPACK.

**Remarks**This function is a wrapper around

**sla_tgstein_l2**.**See also****sla_tgstein_l2****Parameters***TRANS**M*M is INTEGER The order of the matrices A and B. M >= 0.

*A*A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.

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

*B*B is SINGLE PRECISION array, dimension (LDB,M) The matrix B must be upper triangular.

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

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

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

**Date**January 2024

Definition at line **282** of file **tgstein.c**.

### void mepack_single_tgstein_level3 (const char * TRANS, int M, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

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

**Purpose:**

mepack_single_tgstein_level3 solves a generalized Stein equation of the following forms A * X * A^T - B * X * B^T = SCALE * Y (1) or A^T * X * A - B^T * X * B = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular, and X and Y are symmetric M-by-M matrices. Typically the matrix pencil (A,B) is created by SGGES from LAPACK.

**Remarks**This function is a wrapper around

**sla_tgstein_l3**.**See also****sla_tgstein_l3****Parameters***TRANS**M*M is INTEGER The order of the matrices A and B. M >= 0.

*A*A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.

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

*B*B is SINGLE PRECISION array, dimension (LDB,M) The matrix B must be upper triangular.

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

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

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

**Date**January 2024

Definition at line **416** of file **tgstein.c**.

### void mepack_single_tgstein_level3_2stage (const char * TRANS, int M, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

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

**Purpose:**

mepack_single_tgstein_level3_2stage solves a generalized Stein equation of the following forms A * X * A^T - B * X * B^T = SCALE * Y (1) or A^T * X * A - B^T * X * B = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular, and X and Y are symmetric M-by-M matrices. Typically the matrix pencil (A,B) is created by SGGES from LAPACK.

**Remarks**This function is a wrapper around

**sla_tgstein_l3_2s**.**See also****sla_tgstein_l3_2s****Parameters***TRANS**M*M is INTEGER The order of the matrices A and B. M >= 0.

*A*A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.

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

*B*B is SINGLE PRECISION array, dimension (LDB,M) The matrix B must be upper triangular.

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

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

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

**Date**January 2024

Definition at line **551** of file **tgstein.c**.

### void mepack_single_tgstein_recursive (const char * TRANS, int M, float * A, int LDA, float * B, int LDB, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

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

**Purpose:**

mepack_single_tgstein_recursive solves a generalized Stein equation of the following forms A * X * A^T - B * X * B^T = SCALE * Y (1) or A^T * X * A - B^T * X * B = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular, and X and Y are symmetric M-by-M matrices. Typically the matrix pencil (A,B) is created by SGGES from LAPACK.

**Remarks**This function is a wrapper around

**sla_tgstein_recursive**.**See also****sla_tgstein_recursive****Parameters***TRANS**M*M is INTEGER The order of the matrices A and B. M >= 0.

*A*A is SINGLE PRECISION array, dimension (LDA,M) The matrix A must be (quasi-) upper triangular.

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

*B*B is SINGLE PRECISION array, dimension (LDB,M) The matrix B must be upper triangular.

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

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

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

**Date**January 2024

Definition at line **281** of file **tgstein.c**.

## Author

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