# ctrlyap - Man Page

## Name

ctrlyap — C-Interface

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

## Synopsis

### Functions

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

DAG Scheduled Bartels-Stewart Algorithm for the standard Lyapunov Equation.

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

DAG Scheduled Bartels-Stewart Algorithm for the standard Lyapunov Equation.

void **mepack_double_trstein_dag** (const char *TRANSA, int M, double *A, int LDA, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm for the Stein equation with OpenMP 4.

void **mepack_single_trstein_dag** (const char *TRANSA, int M, float *A, int LDA, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm for the Stein equation with OpenMP 4.

void **mepack_double_trlyap_level2** (const char *TRANS, int M, double *A, int LDA, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the Lyapunov Equation.

void **mepack_double_trlyap_level2_opt** (const char *TRANS, int M, double *A, int LDA, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the Lyapunov Equation (Optimized for small problems)

void **mepack_single_trlyap_level2** (const char *TRANS, int M, float *A, int LDA, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the Lyapunov Equation.

void **mepack_single_trlyap_level2_opt** (const char *TRANS, int M, float *A, int LDA, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-2 Bartels-Stewart Algorithm for the Lyapunov Equation (Optimized for small problems)

void **mepack_double_trstein_level2** (const char *TRANSA, int M, double *A, int LDA, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

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

void **mepack_single_trstein_level2** (const char *TRANSA, int M, float *A, int LDA, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

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

void **mepack_double_trlyap_level3** (const char *TRANS, int M, double *A, int LDA, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm for the standard Lyapunov Equation.

void **mepack_double_trlyap_level3_2stage** (const char *TRANS, int M, double *A, int LDA, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm with sub-blocking for the standard Lyapunov Equation.

void **mepack_single_trlyap_level3** (const char *TRANS, int M, float *A, int LDA, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm for the standard Lyapunov Equation.

void **mepack_single_trlyap_level3_2stage** (const char *TRANS, int M, float *A, int LDA, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

Level-3 Bartels-Stewart Algorithm with sub-blocking for the standard Lyapunov Equation.

void **mepack_double_trstein_level3** (const char *TRANSA, int M, double *A, int LDA, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

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

void **mepack_double_trstein_level3_2stage** (const char *TRANSA, int M, double *A, int LDA, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

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

void **mepack_single_trstein_level3** (const char *TRANSA, int M, float *A, int LDA, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

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

void **mepack_single_trstein_level3_2stage** (const char *TRANSA, int M, float *A, int LDA, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

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

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

Level-3 Recursive Blocked Algorithm for the standard Lyapunov Equation.

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

Level-3 Recursive Blocked Algorithm for the standard Lyapunov Equation.

void **mepack_double_trstein_recursive** (const char *TRANSA, int M, double *A, int LDA, double *X, int LDX, double *SCALE, double *WORK, int *INFO)

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

void **mepack_single_trstein_recursive** (const char *TRANSA, int M, float *A, int LDA, float *X, int LDX, float *SCALE, float *WORK, int *INFO)

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

## Detailed Description

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

The Fortran routines to solve standard 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_trlyap_dag (const char * TRANS, int M, double * A, int LDA, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

DAG Scheduled Bartels-Stewart Algorithm for the standard Lyapunov Equation.

**Purpose:**

mepack_double_trlyap_dag solves a Lyapunov equation of the following forms A * X + X * A**T = SCALE * Y (1) or A ** T * X + X * A = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrix A is generated by DGEES from LAPACK.

**Remarks**The algorithm is implemented using DAG Scheduling

This function is a wrapper around

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

*M*M is INTEGER The order of the matrices A and C. 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).

*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 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**June 2023

Definition at line **132** of file **trlyap.c**.

### void mepack_double_trlyap_level2 (const char * TRANS, int M, double * A, int LDA, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the Lyapunov Equation.

**Purpose:**

mepack_double_trlyap_level2 solves a Lyapunov equation of the following forms A * X + X * A**T = SCALE * Y (1) or A **T * X - X * A = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrix A is generated by DGEES from LAPACK.

**Remarks**The algorithm is implemented using BLAS level 2 operations.

This is a wrapper around

**dla_trlyap_l2****See also****dla_trlyap_l2****Parameters***TRANS*TRANS is string Specifies the form of the system of equations with respect to A: == 'N': op1(A) = A == 'T': op1(A) = A**T

*M*M is INTEGER The order of the matrices A and C. 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).

*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 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**June 2023

Definition at line **132** of file **trlyap.c**.

### void mepack_double_trlyap_level2_opt (const char * TRANS, int M, double * A, int LDA, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the Lyapunov Equation (Optimized for small problems)

**Purpose:**

mepack_double_trlyap_level2 solves a Lyapunov equation of the following forms A * X + X * A**T = SCALE * Y (1) or A **T * X - X * A = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrix A is generated by DGEES from LAPACK.

**Remarks**The algorithm is implemented using BLAS level 2 operations.

This is a wrapper around

**dla_trlyap_l2_opt****See also****dla_trlyap_l2_opt****Parameters***TRANS*TRANS is string Specifies the form of the system of equations with respect to A: == 'N': op1(A) = A == 'T': op1(A) = A**T

*M*M is INTEGER The order of the matrices A and C. 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).

*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 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**June 2023

Definition at line **249** of file **trlyap.c**.

### void mepack_double_trlyap_level3 (const char * TRANS, int M, double * A, int LDA, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm for the standard Lyapunov Equation.

**Purpose:**

mepack_double_trlyap_level3 solves a Lyapunov equation of the following forms A * X + X * A**T = SCALE * Y (1) or A ** T * X + X * A = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrix A is generated by DGEES from LAPACK.

**Remarks**The algorithm is implemented a level-3 block approach.

This function is a wrapper around

**dla_trlyap_l3****See also****dla_trlyap_l3****Parameters***TRANS*TRANS is a string Specifies the form of the system of equations with respect to A: == 'N': op1(A) = A == 'T': op1(A) = A**T

*M*M is INTEGER The order of the matrices A and C. 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).

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

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

**Date**June 2023

Definition at line **132** of file **trlyap.c**.

### MEPACK_EXPORT void mepack_double_trlyap_level3_2stage (const char * TRANS, int M, double * A, int LDA, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm with sub-blocking for the standard Lyapunov Equation.

**Purpose:**

mepack_double_trlyap_level3_2stage solves a Lyapunov equation of the following forms A * X + X * A**T = SCALE * Y (1) or A ** T * X + X * A = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrix A is generated by DGEES from LAPACK.

**Remarks**The algorithm is implemented a level-3 block approach.

This function is a wrapper around

**dla_trlyap_l3_2s****See also****dla_trlyap_l3_2s****Parameters***TRANS*TRANS is a string Specifies the form of the system of equations with respect to A: == 'N': op1(A) = A == 'T': op1(A) = A**T

*M*M is INTEGER The order of the matrices A and C. 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).

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

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

**Date**June 2023

Definition at line **248** of file **trlyap.c**.

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

Level-3 Recursive Blocked Algorithm for the standard Lyapunov Equation.

**Purpose:**

mepack_double_trlyap_recursive solves a Lyapunov equation of the following forms A * X + X * A**T = SCALE * Y (1) or A ** T * X + X * A = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrix A is generated by DGEES from LAPACK.

**Remarks**The algorithm is implemented using BLAS level 3 operations and recursive blocking.

This function is a wrapper around

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

*M*M is INTEGER The order of the matrices A and C. 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).

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

*SCALE**WORK*WORK is DOUBLE PRECISION array, dimension 1 Workspace for the algorithm

*INFO*INFO is INTEGER == 0: successful exit < 0: if INFO == -i, the i-the 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**June 2023

Definition at line **118** of file **trlyap.c**.

### void mepack_double_trstein_dag (const char * TRANSA, int M, double * A, int LDA, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm for the Stein equation with OpenMP 4.

**Purpose:**

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

**Remarks**The algorithm is implemented using DAG Scheduling

The routine is a wrapper around

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

*M*M is INTEGER The order of the matrices A and C. 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).

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

**Date**June 2023

Definition at line **136** of file **trstein.c**.

### void mepack_double_trstein_level2 (const char * TRANSA, int M, double * A, int LDA, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

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

**Purpose:**

mepack_double_trstein_level2 solves a Stein equation of the following forms A * X * A^T - X = SCALE * Y (2) or A^T * X * A - X = SCALE * Y (1) where A is a M-by-M quasi upper triangular matrix. and X and Y are symmetric M-by-M matrices. Typically the matrix A is created by DGEES from LAPACK.

**Remarks**The routine is a wrapper around

**dla_trstein_l2**.**See also****dla_trstein_l2****Parameters***TRANSA*TRANSA is string Specifies the form of the system of equations with respect to A and C: == 'N': Equation (1) is solved. == 'T': Equation (2) is solved.

*M*M is INTEGER The order of the matrices A and C. 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).

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

**Date**June 2023

Definition at line **135** of file **trstein.c**.

### void mepack_double_trstein_level3 (const char * TRANSA, int M, double * A, int LDA, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

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

**Purpose:**

mepack_double_trstein_level3 solves a Stein equation of the following forms A * X * A^T - X = SCALE * Y (2) or A^T * X * A - X = SCALE * Y (1) where A is a M-by-M quasi upper triangular matrix. and X and Y are symmetric M-by-M matrices. Typically the matrix A is created by DGEES from LAPACK.

**Remarks**The routine is a wrapper around

**dla_trstein_l3**.**See also****dla_trstein_l3****Parameters***TRANSA*TRANSA is string Specifies the form of the system of equations with respect to A and C: == 'N': Equation (1) is solved. == 'T': Equation (2) is solved.

*M*M is INTEGER The order of the matrices A and C. 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).

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

**Date**June 2023

Definition at line **135** of file **trstein.c**.

### void mepack_double_trstein_level3_2stage (const char * TRANSA, int M, double * A, int LDA, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

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

**Purpose:**

mepack_double_trstein_level3_2stage solves a Stein equation of the following forms A * X * A^T - X = SCALE * Y (2) or A^T * X * A - X = SCALE * Y (1) where A is a M-by-M quasi upper triangular matrix. and X and Y are symmetric M-by-M matrices. Typically the matrix A is created by DGEES from LAPACK.

**Remarks**The routine is a wrapper around

**dla_trstein_l3_2s**.**See also****dla_trstein_l3_2s****Parameters***TRANSA**M*M is INTEGER The order of the matrices A and C. 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).

*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**June 2023

Definition at line **255** of file **trstein.c**.

### void mepack_double_trstein_recursive (const char * TRANSA, int M, double * A, int LDA, double * X, int LDX, double * SCALE, double * WORK, int * INFO)

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

**Purpose:**

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

**Remarks**The routine is a wrapper around

**dla_trstein_recursive**.**See also****dla_trstein_recursive****Parameters***TRANSA**M*M is INTEGER The order of the matrices A and C. 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).

*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**June 2023

Definition at line **135** of file **trstein.c**.

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

DAG Scheduled Bartels-Stewart Algorithm for the standard Lyapunov Equation.

**Purpose:**

mepack_single_trlyap_dag solves a Lyapunov equation of the following forms A * X + X * A**T = SCALE * Y (1) or A ** T * X + X * A = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrix A is generated by DGEES from LAPACK.

**Remarks**The algorithm is implemented using DAG Scheduling

This function is a wrapper around

**sla_trlyap_dag****See also****sla_trlyap_dag****Parameters***TRANS**M*M is INTEGER The order of the matrices A and C. 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).

*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 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**June 2023

Definition at line **250** of file **trlyap.c**.

### void mepack_single_trlyap_level2 (const char * TRANS, int M, float * A, int LDA, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the Lyapunov Equation.

**Purpose:**

mepack_single_trlyap_level2 solves a Lyapunov equation of the following forms A * X + X * A**T = SCALE * Y (1) or A **T * X - X * A = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrix A is generated by SGEES from LAPACK.

**Remarks**The algorithm is implemented using BLAS level 2 operations.

This is a wrapper around

**sla_trlyap_l2****See also****sla_trlyap_l2****Parameters***TRANS**M*M is INTEGER The order of the matrices A and C. 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).

*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 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**June 2023

Definition at line **365** of file **trlyap.c**.

### void mepack_single_trlyap_level2_opt (const char * TRANS, int M, float * A, int LDA, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-2 Bartels-Stewart Algorithm for the Lyapunov Equation (Optimized for small problems)

**Purpose:**

mepack_single_trlyap_level2_opt solves a Lyapunov equation of the following forms A * X + X * A**T = SCALE * Y (1) or A **T * X - X * A = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrix A is generated by SGEES from LAPACK.

**Remarks**The algorithm is implemented using BLAS level 2 operations.

This is a wrapper around

**sla_trlyap_l2_opt****See also****sla_trlyap_l2_opt****Parameters***TRANS**M*M is INTEGER The order of the matrices A and C. 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).

*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 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**June 2023

Definition at line **482** of file **trlyap.c**.

### void mepack_single_trlyap_level3 (const char * TRANS, int M, float * A, int LDA, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm for the standard Lyapunov Equation.

**Purpose:**

mepack_single_trlyap_level3 solves a Lyapunov equation of the following forms A * X + X * A**T = SCALE * Y (1) or A ** T * X + X * A = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrix A is generated by DGEES from LAPACK.

**Remarks**The algorithm is implemented a level-3 block approach.

This function is a wrapper around

**sla_trlyap_l3****See also****sla_trlyap_l3****Parameters***TRANS**M*M is INTEGER The order of the matrices A and C. 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).

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

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

**Date**June 2023

Definition at line **365** of file **trlyap.c**.

### MEPACK_EXPORT void mepack_single_trlyap_level3_2stage (const char * TRANS, int M, float * A, int LDA, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm with sub-blocking for the standard Lyapunov Equation.

**Purpose:**

mepack_single_trlyap_level3_2stage solves a Lyapunov equation of the following forms A * X + X * A**T = SCALE * Y (1) or A ** T * X + X * A = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrix A is generated by DGEES from LAPACK.

**Remarks**The algorithm is implemented a level-3 block approach.

This function is a wrapper around

**sla_trlyap_l3_2s****See also****sla_trlyap_l3_2s****Parameters***TRANS**M*M is INTEGER The order of the matrices A and C. 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).

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

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

**Date**June 2023

Definition at line **481** of file **trlyap.c**.

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

Level-3 Recursive Blocked Algorithm for the standard Lyapunov Equation.

**Purpose:**

mepack_single_trlyap_recursive solves a Lyapunov equation of the following forms A * X + X * A**T = SCALE * Y (1) or A ** T * X + X * A = SCALE * Y (2) where A is a M-by-M quasi upper triangular matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrix A is generated by SGEES from LAPACK.

**Remarks**The algorithm is implemented using BLAS level 3 operations and recursive blocking.

This function is a wrapper around

**sla_trlyap_recursive**.**See also****sla_trlyap_recursive****Parameters***TRANS**M*M is INTEGER The order of the matrices A and C. 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).

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

*SCALE**WORK*WORK is SINGLE PRECISION array, dimension 1 Workspace for the algorithm

*INFO*INFO is INTEGER == 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**June 2023

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

### void mepack_single_trstein_dag (const char * TRANSA, int M, float * A, int LDA, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

Level-3 Bartels-Stewart Algorithm for the Stein equation with OpenMP 4.

**Purpose:**

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

**Remarks**The algorithm is implemented using DAG Scheduling

The routine is a wrapper around

**sla_trstein_dag**.**See also****sla_trstein_dag****Parameters***TRANSA**M*M is INTEGER The order of the matrices A and C. 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).

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

**Date**June 2023

Definition at line **257** of file **trstein.c**.

### void mepack_single_trstein_level2 (const char * TRANSA, int M, float * A, int LDA, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

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

**Purpose:**

mepack_single_trstein_level2 solves a Stein equation of the following forms A * X * A^T - X = SCALE * Y (2) or A^T * X * A - X = SCALE * Y (1) where A is a M-by-M quasi upper triangular matrix. and X and Y are symmetric M-by-M matrices. Typically the matrix A is created by SGEES from LAPACK.

**Remarks**The routine is a wrapper around

**sla_trstein_l2**.**See also****sla_trstein_l2****Parameters***TRANSA**M*M is INTEGER The order of the matrices A and C. 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).

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

**Date**June 2023

Definition at line **255** of file **trstein.c**.

### void mepack_single_trstein_level3 (const char * TRANSA, int M, float * A, int LDA, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

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

**Purpose:**

mepack_single_trstein_level3 solves a Stein equation of the following forms A * X * A^T - X = SCALE * Y (2) or A^T * X * A - X = SCALE * Y (1) where A is a M-by-M quasi upper triangular matrix. and X and Y are symmetric M-by-M matrices. Typically the matrix A is created by SGEES from LAPACK.

**Remarks**The routine is a wrapper around

**sla_trstein_l3**.**See also****sla_trstein_l3****Parameters***TRANSA**M*M is INTEGER The order of the matrices A and C. 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).

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

**Date**June 2023

Definition at line **375** of file **trstein.c**.

### void mepack_single_trstein_level3_2stage (const char * TRANSA, int M, float * A, int LDA, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

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

**Purpose:**

mepack_single_trstein_level3_2stage solves a Stein equation of the following forms A * X * A^T - X = SCALE * Y (2) or A^T * X * A - X = SCALE * Y (1) where A is a M-by-M quasi upper triangular matrix. and X and Y are symmetric M-by-M matrices. Typically the matrix A is created by SGEES from LAPACK.

**Remarks**The routine is a wrapper around

**sla_trstein_l3_2s**.**See also****sla_trstein_l3_2s****Parameters***TRANSA**M*M is INTEGER The order of the matrices A and C. 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).

*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**June 2023

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

### void mepack_single_trstein_recursive (const char * TRANSA, int M, float * A, int LDA, float * X, int LDX, float * SCALE, float * WORK, int * INFO)

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

**Purpose:**

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

**Remarks**The routine is a wrapper around

**sla_trstein_recursive**.**See also****sla_trstein_recursive****Parameters***TRANSA**M*M is INTEGER The order of the matrices A and C. 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).

*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**June 2023

Definition at line **255** of file **trstein.c**.

## Author

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