# dbltrlyap - Man Page

## Name

dbltrlyap — Double Precision routines.

— Double precision routines for standard Lyapunov and Stein equations with triangular coefficient matrices.

## Synopsis

### Functions

subroutine **dla_trlyap_dag** (trans, m, a, lda, x, ldx, scale, work, info)

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

subroutine **dla_trstein_dag** (trans, m, a, lda, x, ldx, scale, work, info)

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

subroutine **dla_trlyap_kernel_44n** (m, a, lda, x, ldx, scale, info)

Solver for a 4x4 standard Lyapunov equation (TRANS = N)

subroutine **dla_trlyap_kernel_44t** (m, a, lda, x, ldx, scale, info)

Solver for a 4x4 standard Lyapunov equation (TRANS = T)

subroutine **dla_trlyap_l2** (transa, m, a, lda, x, ldx, scale, work, info)

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

subroutine **dla_trlyap_l2_opt** (transa, m, a, lda, x, ldx, scale, work, info)

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

subroutine **dla_trstein_l2** (trans, m, a, lda, x, ldx, scale, work, info)

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

subroutine **dla_trlyap_l3** (trans, m, a, lda, x, ldx, scale, work, info)

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

subroutine **dla_trlyap_l3_2s** (trans, m, a, lda, x, ldx, scale, work, info)

Level-3 Bartels-Stewart Algorithm for the standard Lyapunov Equation with 2 stage blocking.

subroutine **dla_trstein_l3** (trans, m, a, lda, x, ldx, scale, work, info)

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

subroutine **dla_trstein_l3_2s** (trans, m, a, lda, x, ldx, scale, work, info)

Level-3 Bartels-Stewart Algorithm for the Stein equation with 2 stage blocking.

recursive subroutine **dla_trlyap_recursive** (trans, m, a, lda, x, ldx, scale, work, info)

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

recursive subroutine **dla_trstein_recursive** (trans, m, a, lda, x, ldx, scale, work, info)

Recursive Blocked Algorithm for the Stein equation.

## Detailed Description

Double precision routines for standard Lyapunov and Stein equations with triangular coefficient matrices.

## Function Documentation

### subroutine dla_trlyap_dag (character, dimension(1) trans, integer m, double precision, dimension(lda, m) a, integer lda, double precision, dimension(ldx, m) x, integer ldx, double precision scale, double precision, dimension(*) work, integer info)

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

**Purpose:**

DLA_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

**Parameters***TRANS*TRANS is CHARACTER(1) 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 LWORK 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 **137** of file **dla_trlyap_dag.f90**.

### subroutine dla_trlyap_kernel_44n (integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldx, *) x, integer ldx, double precision scale, integer info)

Solver for a 4x4 standard Lyapunov equation (TRANS = N)

**Purpose:**

DLA_TRLYAP_KERNEL_44N solves a Lyapunov equation of the following form A * X + X * A**T = SCALE * Y (1) where A is a M-by-M quasi upper triangular matrix. The right hand side Y and the solution X M-by-M matrices. Typically the matrix A is created by DGEES from LAPACK. The algorithm is implemented without BLAS level 2 operations. Thereby the order of M and N is at most 4. Furthermore, for fast execution the function does not check the input arguments.

**Parameters***M*M is INTEGER The order of the matrices A and C. 4 >= 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. 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 < SCAL <= 1 holds true.

*INFO*INFO is INTEGER On output: == 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 **109** of file **dla_trlyap_kernel_44_n.f90**.

### subroutine dla_trlyap_kernel_44t (integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldx, *) x, integer ldx, double precision scale, integer info)

Solver for a 4x4 standard Lyapunov equation (TRANS = T)

**Purpose:**

DLA_TRLYAP_KERNEL_44T solves a Lyapunov equation of the following form A **T * X + X * A = SCALE * Y (1) where A is a M-by-M quasi upper triangular matrix. The right hand side Y and the solution X M-by-M matrices. Typically the matrix A is created by DGEES from LAPACK. The algorithm is implemented without BLAS level 2 operations. Thereby the order of M and N is at most 4. Furthermore, for fast execution the function does not check the input arguments.

**Parameters***M*M is INTEGER The order of the matrices A and C. 4 >= 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. 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 < SCAL <= 1 holds true.

*INFO*INFO is INTEGER On output: == 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 **109** of file **dla_trlyap_kernel_44_t.f90**.

### subroutine dla_trlyap_l2 (character, dimension(1) transa, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer info)

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

**Purpose:**

DLA_TRLYAP_L2 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.

The transposed case (2) is optimized w.r.t. to the usage of the DSYR2 operation.

**Parameters***TRANSA*TRANSA is CHARACTER(1) 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 LWORK 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 **138** of file **dla_trlyap_l2.f90**.

### subroutine dla_trlyap_l2_opt (character, dimension(1) transa, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer info)

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

**Purpose:**

DLA_TRLYAP_L2 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.

The transposed case (2) is optimized w.r.t. to the usage of the DSYR2 operation.

**Parameters***TRANSA*TRANSA is CHARACTER(1) 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 LWORK 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 **138** of file **dla_trlyap_l2_opt.f90**.

### subroutine dla_trlyap_l3 (character, dimension(1) trans, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer info)

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

**Purpose:**

DLA_TRLYAP_L3 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.

**Parameters***TRANS*TRANS is CHARACTER 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 **137** of file **dla_trlyap_l3.f90**.

### subroutine dla_trlyap_l3_2s (character, dimension(1) trans, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer info)

Level-3 Bartels-Stewart Algorithm for the standard Lyapunov Equation with 2 stage blocking.

**Purpose:**

DLA_TRLYAP_L3_2S 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 DAG schedule block solves.

**Parameters***TRANS*TRANS is CHARACTER 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 **137** of file **dla_trlyap_l3_2stage.f90**.

### recursive subroutine dla_trlyap_recursive (character, dimension(1) trans, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer info)

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

**Purpose:**

DLA_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.

**Parameters***TRANS*TRANS is CHARACTER(1) 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-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 **125** of file **dla_trlyap_recursive.f90**.

### subroutine dla_trstein_dag (character, dimension(1) trans, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer info)

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

**Purpose:**

DLA_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.

**Attention**The algorithm is implemented using BLAS level 3 operations.

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

**Parameters***TRANS*TRANS is CHARACTER(1) 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 **142** of file **dla_trstein_dag.f90**.

### subroutine dla_trstein_l2 (character, dimension(1) trans, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer info)

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

**Purpose:**

DLA_TRSTEIN_L2 solves a generalized Lyapunov 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.

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

**Parameters***TRANS*TRANS is CHARACTER(1) 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**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 **143** of file **dla_trstein_l2.f90**.

### subroutine dla_trstein_l3 (character, dimension(1) trans, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer info)

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

**Purpose:**

DLA_TRSTEIN_L3 solves a standard 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.

**Attention**The algorithm is implemented using BLAS level 3 operations.

**Parameters***TRANS*TRANS is CHARACTER 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**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 **141** of file **dla_trstein_l3.f90**.

### subroutine dla_trstein_l3_2s (character, dimension(1) trans, integer m, double precision, dimension(lda, m) a, integer lda, double precision, dimension(ldx, m) x, integer ldx, double precision scale, double precision, dimension(*) work, integer info)

Level-3 Bartels-Stewart Algorithm for the Stein equation with 2 stage blocking.

**Purpose:**

DLA_TRSTEIN_L3_2S solves a generalized Lyapunov equation of the following forms A * X * A^T - X = SCALE * Y (1) or A^T * X * A - X = SCALE * Y (2) 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.

**Attention**The algorithm is implemented using BLAS level 3 operations and a DAG scheduled inner solver.

**Parameters***TRANS*TRANS is CHARACTER(1) 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**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 **141** of file **dla_trstein_l3_2stage.f90**.

### recursive subroutine dla_trstein_recursive (character, dimension(1) trans, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer info)

Recursive Blocked Algorithm for the Stein equation.

**Purpose:**

DLA_TRSTEIN_RECURSIVE solves a generalized Lyapunov 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.

**Attention**The algorithm is implemented using recursive blocking.

**Parameters***TRANS**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*WORK is DOUBLE PRECISION array, dimension M*N 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 **129** of file **dla_trstein_recursive.f90**.

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