# sgltrsylv - Man Page

## Name

sgltrsylv — Single Precision

— Single Precision routines for triangular standard Sylvester equations.

## Synopsis

### Functions

subroutine **sla_trsylv2_dag** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-3 Bartels-Stewart Algorithm for the discrete time Sylvester equation with DAG based parallelization.

subroutine **sla_trsylv_dag** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation with DAG parallelization.

subroutine **sla_trsylv2_kernel_44nn** (sgn, m, n, a, lda, b, ldb, x, ldx, scale, info)

Solver for a 4x4 discrete time Sylvester equation (TRANSA = N, TRANSB = N)

subroutine **sla_trsylv2_kernel_44nt** (sgn, m, n, a, lda, b, ldb, x, ldx, scale, info)

Solver for a 4x4 discrete time Sylvester equation (TRANSA = N, TRANSB = T)

subroutine **sla_trsylv2_kernel_44tn** (sgn, m, n, a, lda, b, ldb, x, ldx, scale, info)

Solver for a 4x4 discrete time Sylvester equation (TRANSA = T, TRANSB = N)

subroutine **sla_trsylv2_kernel_44tt** (sgn, m, n, a, lda, b, ldb, x, ldx, scale, info)

Solver for a 4x4 discrete time Sylvester equation (TRANSA = T, TRANSB = T)

subroutine **sla_trsylv_kernel_44nn** (sgn, m, n, a, lda, b, ldb, x, ldx, scale, info)

Solver for a 4x4 Sylvester equation (TRANSA = N, TRANSB = N)

subroutine **sla_trsylv_kernel_44nt** (sgn, m, n, a, lda, b, ldb, x, ldx, scale, info)

Solver for a 4x4 Sylvester equation (TRANSA = N, TRANSB = T)

subroutine **sla_trsylv_kernel_44tn** (sgn, m, n, a, lda, b, ldb, x, ldx, scale, info)

Solver for a 4x4 Sylvester equation (TRANSA = T, TRANSB = N)

subroutine **sla_trsylv_kernel_44tt** (sgn, m, n, a, lda, b, ldb, x, ldx, scale, info)

Solver for a 4x4 Sylvester equation (TRANSA = T, TRANSB = T)

subroutine **sla_trsylv2_l2** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the discrete time Sylvester equation (Optimized)

subroutine **sla_trsylv2_l2_local_copy** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the discrete time Sylvester equation (Optimized)

subroutine **sla_trsylv2_l2_local_copy_128** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the discrete time Sylvester equation (Optimized)

subroutine **sla_trsylv2_l2_local_copy_32** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the discrete time Sylvester equation (Optimized)

subroutine **sla_trsylv2_l2_local_copy_64** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the discrete time Sylvester equation (Optimized)

subroutine **sla_trsylv2_l2_local_copy_96** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the discrete time Sylvester equation (Optimized)

subroutine **sla_trsylv2_l2_reorder** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the discrete time Sylvester equation (Optimized)

subroutine **sla_trsylv2_l2_unopt** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the discrete time Sylvester equation (unoptimized)

subroutine **sla_trsylv_l2** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the Sylvester equation (optimized)

subroutine **sla_trsylv_l2_local_copy** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the Sylvester equation (optimized)

subroutine **sla_trsylv_l2_local_copy_128** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Optimized)

subroutine **sla_trsylv_l2_local_copy_32** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Optimized)

subroutine **sla_trsylv_l2_local_copy_64** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Optimized)

subroutine **sla_trsylv_l2_local_copy_96** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Optimized)

subroutine **sla_trsylv_l2_reorder** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the Sylvester equation (optimized)

subroutine **sla_trsylv_l2_unopt** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the Sylvester equation (unoptimized)

subroutine **sla_trsylv2_l3_2s** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Two level blocked Bartels-Stewart Algorithm for the discrete time Sylvester equation (Optimized)

subroutine **sla_trsylv2_l3** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-3 Bartels-Stewart Algorithm for the discrete time Sylvester equation (Optimized)

subroutine **sla_trsylv2_l3_unopt** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-3 Bartels-Stewart Algorithm for the discrete time Sylvester equation (Not Optimized)

subroutine **sla_trsylv_l3_2s** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

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

subroutine **sla_trsylv_l3** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

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

subroutine **sla_trsylv_l3_unopt** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation (unoptimized)

recursive subroutine **sla_trsylv2_recursive** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-3 Recursive Blocked Algorithm for the discrete time Sylvester equation.

recursive subroutine **sla_trsylv_recursive** (transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-3 Recursive Blocked Solver for the Sylvester equation.

## Detailed Description

Single Precision routines for triangular standard Sylvester equations.

This section contains the solvers for the standard Sylvester equation with (quasi) triangular coefficient matrices. The coefficient matrices are normally generated with the help of the Schur decomposition from LAPACK. All codes use single precision arithmetic.

## Function Documentation

### subroutine sla_trsylv2_dag (character, dimension(1) transa, character, dimension(1) transb, real sgn, integer m, integer n, real, dimension(lda, m) a, integer lda, real, dimension(ldb, n) b, integer ldb, real, dimension(ldx, n) x, integer ldx, real scale, real, dimension(*) work, integer info)

Level-3 Bartels-Stewart Algorithm for the discrete time Sylvester equation with DAG based parallelization.

**Purpose:**

!> !> SLA_TRSYLV2_DAG solves a discrete time Sylvester equation of the following forms !> !> op1(A) * X * op2(B) + X = SCALE * Y (1) !> !> or !> !> op1(A) * X * op2(B) - X = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix and B is a N-by-N upper !> quasi triangular matrices. The right hand side Y and the solution X !> M-by-N matrices. Typically the matrices A and B are created by SGEES from LAPACK. !>

**Remarks**The algorithm is implemented using BLAS level 3 operations and OpenMP 4.0 DAG scheduling.

**Attention**Due to the parallel nature of the algorithm the scaling is not applied to the right hand. If the problem is ill-posed and scaling appears you have to solve the equation again with a solver with complete scaling support like the SLA_TRSYLV2_L3 routine.

**Parameters***TRANSA*!> TRANSA is CHARACTER(1) !> Specifies the form of the system of equations with respect to A and C: !> == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) !> == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C) !>

*TRANSB*!> TRANSB is CHARACTER(1) !> Specifies the form of the system of equations with respect to B and D: !> == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) !> == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D) !>

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

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

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

*A*!> A is REAL 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 REAL array, dimension (LDB,N) !> The matrix B must be (quasi-) upper triangular. If the matrix D is already !> quasi-upper triangular the matrix B must be upper triangular. !>

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

*X*!> X is REAL 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 REAL !> SCALE is a scaling factor to prevent the overflow in the result. !> If INFO == 0 then SCALE is 1.0 otherwise if one of the inner systems !> could not be solved correctly, 0 < SCALE <= 1 holds true. !> If SCALE .NE. 1 the problem is no solved correctly in this case !> one have to use an other solver. !>

*WORK*!> WORK is REAL 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**January 2024

Definition at line **181** of file **sla_trsylv2_dag.f90**.

### subroutine sla_trsylv2_kernel_44nn (real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, integer info)

Solver for a 4x4 discrete time Sylvester equation (TRANSA = N, TRANSB = N)

**Purpose:**

!> !> SLA_TRSYLV2_KERNEL_44NN solves a discrete time Sylvester equation of the following form !> !> A * X * B + SGN * X = SCALE * Y (1) !> !> where A is a M-by-M quasi upper triangular matrix and B is a N-by-N quasi upper !> triangular matrix. The right hand side Y and the solution X !> M-by-N matrices. Typically the matrices A and B are created by SGEES from LAPACK. !> The algorithm is implemented using 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***SGN*!> SGN is REAL, allowed values: +/-1 !> Specifies the sign between the two parts of the Sylvester equation. !>

*M*!> M is INTEGER !> The order of the matrix A. 4 >= M >= 0. !>

*N*!> N is INTEGER !> The order of the matrix B. 4 >= N >= 0. !>

*A*!> A is REAL 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 REAL array, dimension (LDB,N) !> The matrix B must be (quasi-) upper triangular. !>

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

*X*!> X is REAL 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) !> as selected by 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 REAL !> SCALE is a scaling factor to prevent the overflow in the result. !> If INFO == 0 then SCALE is 1.0 otherwise if one of the inner systems !> could not be solved correctly, 0 < SCAL <= 1 holds true. !>

*INFO*!> INFO is INTEGER !> On input: !> = 1 : Skip the input data checks !> <> 1: Check input data like normal LAPACK like routines. !> 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**January 2024

Definition at line **135** of file **sla_trsylv2_kernel_44_nn.f90**.

### subroutine sla_trsylv2_kernel_44nt (real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, integer info)

Solver for a 4x4 discrete time Sylvester equation (TRANSA = N, TRANSB = T)

**Purpose:**

!> !> SLA_TRSYLV2_KERNEL_44NT solves a discrete time Sylvester equation of the following form !> !> A * X * B**T + SGN * X = SCALE * Y (1) !> !> where A is a M-by-M quasi upper triangular matrix and B is a N-by-N quasi upper !> triangular matrix. The right hand side Y and the solution X !> M-by-N matrices. Typically the matrices A and B are created by SGEES from LAPACK. !> The algorithm is implemented using 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***SGN*!> SGN is REAL, allowed values: +/-1 !> Specifies the sign between the two parts of the Sylvester equation. !>

*M*!> M is INTEGER !> The order of the matrix A. 4 >= M >= 0. !>

*N*!> N is INTEGER !> The order of the matrix B. 4 >= N >= 0. !>

*A*!> A is REAL 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 REAL array, dimension (LDB,N) !> The matrix B must be (quasi-) upper triangular. !>

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

*X*!> X is REAL 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) !> as selected by 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 REAL !> SCALE is a scaling factor to prevent the overflow in the result. !> If INFO == 0 then SCALE is 1.0 otherwise if one of the inner systems !> could not be solved correctly, 0 < SCAL <= 1 holds true. !>

*INFO*!> INFO is INTEGER !> On input: !> = 1 : Skip the input data checks !> <> 1: Check input data like normal LAPACK like routines. !> 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**January 2024

Definition at line **134** of file **sla_trsylv2_kernel_44_nt.f90**.

### subroutine sla_trsylv2_kernel_44tn (real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, integer info)

Solver for a 4x4 discrete time Sylvester equation (TRANSA = T, TRANSB = N)

**Purpose:**

!> !> SLA_TRSYLV2_KERNEL_44TN solves a discrete time Sylvester equation of the following form !> !> A**T * X * B + SGN * X = SCALE * Y (1) !> !> where A is a M-by-M quasi upper triangular matrix and B is a N-by-N quasi upper !> triangular matrix. The right hand side Y and the solution X !> M-by-N matrices. Typically the matrices A and B are created by SGEES from LAPACK. !> The algorithm is implemented using 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***SGN*!> SGN is REAL, allowed values: +/-1 !> Specifies the sign between the two parts of the Sylvester equation. !>

*M*!> M is INTEGER !> The order of the matrix A. 4 >= M >= 0. !>

*N*!> N is INTEGER !> The order of the matrix B. 4 >= N >= 0. !>

*A*!> A is REAL 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 REAL array, dimension (LDB,N) !> The matrix B must be (quasi-) upper triangular. !>

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

*X*!> X is REAL 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) !> as selected by 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 REAL !> SCALE is a scaling factor to prevent the overflow in the result. !> If INFO == 0 then SCALE is 1.0 otherwise if one of the inner systems !> could not be solved correctly, 0 < SCAL <= 1 holds true. !>

*INFO*!> INFO is INTEGER !> On input: !> = 1 : Skip the input data checks !> <> 1: Check input data like normal LAPACK like routines. !> 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**January 2024

Definition at line **135** of file **sla_trsylv2_kernel_44_tn.f90**.

### subroutine sla_trsylv2_kernel_44tt (real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, integer info)

Solver for a 4x4 discrete time Sylvester equation (TRANSA = T, TRANSB = T)

**Purpose:**

!> !> SLA_TRSYLV2_KERNEL_44TT solves a discrete time Sylvester equation of the following form !> !> A**T * X * B**T + SGN * X = SCALE * Y (1) !> !> where A is a M-by-M quasi upper triangular matrix and B is a N-by-N quasi upper !> triangular matrix. The right hand side Y and the solution X !> M-by-N matrices. Typically the matrices A and B are created by SGEES from LAPACK. !> The algorithm is implemented using 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***SGN**M*!> M is INTEGER !> The order of the matrix A. 4 >= M >= 0. !>

*N*!> N is INTEGER !> The order of the matrix B. 4 >= N >= 0. !>

*A*!> A is REAL 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 REAL array, dimension (LDB,N) !> The matrix B must be (quasi-) upper triangular. !>

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

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

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

**Date**January 2024

Definition at line **135** of file **sla_trsylv2_kernel_44_tt.f90**.

### subroutine sla_trsylv2_l2 (character, dimension(1) transa, character, dimension(1) transb, real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, real, dimension(*) work, integer info)

Level-2 Bartels-Stewart Algorithm for the discrete time Sylvester equation (Optimized)

**Purpose:**

!> !> SLA_TRSYLV2_L2 solves a generalized Sylvester equation of the following forms !> !> op1(A) * X * op2(B) + X = SCALE * Y (1) !> !> or !> !> op1(A) * X * op2(B) - X = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix and B is a N-by-N upper !> quasi triangular matrices. The right hand side Y and the solution X !> M-by-N matrices. Typically the matrices A and B are created by SGEES from LAPACK. !>

**Parameters***TRANSA*!> TRANSA is CHARACTER(1) !> Specifies the form of the system of equations with respect to A and C: !> == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) !> == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C) !>

*TRANSB*!> TRANSB is CHARACTER(1) !> Specifies the form of the system of equations with respect to B and D: !> == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) !> == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D) !>

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

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

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

*A*!> A is REAL 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 REAL array, dimension (LDB,N) !> The matrix B must be (quasi-) upper triangular. If the matrix D is already !> quasi-upper triangular the matrix B must be upper triangular. !>

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

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

*WORK*!> WORK is REAL 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. !>

**Optimizations:**- Replaced level-3 by level-2 and level-1 calls,
- Replaced SAXPY operation by Fortran intrinsic
- Replaced all BLAS calls except of STRMV and SGER by Fortran intrinsic,

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **182** of file **sla_trsylv2_l2.f90**.

### subroutine sla_trsylv2_l2_local_copy (character, dimension(1) transa, character, dimension(1) transb, real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, real, dimension(*) work, integer info)

Level-2 Bartels-Stewart Algorithm for the discrete time Sylvester equation (Optimized)

**Purpose:**

!> !> SLA_TRSYLV2_L2_LOCAL_COPY solves a generalized Sylvester equation of the following forms !> !> op1(A) * X * op2(B) + X = SCALE * Y (1) !> !> or !> !> op1(A) * X * op2(B) - X = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix and B is a N-by-N upper !> quasi triangular matrices. The right hand side Y and the solution X !> M-by-N matrices. Typically the matrices A and B are created by SGEES from LAPACK. !>

**Parameters***TRANSA*!> TRANSA is CHARACTER(1) !> Specifies the form of the system of equations with respect to A and C: !> == 'N': op1(A) = A, op1(C) = C (No transpose for A and C) !> == 'T': op1(A) = A**T, op1(C) = C **T (Transpose A and C) !>

*TRANSB*!> TRANSB is CHARACTER(1) !> Specifies the form of the system of equations with respect to B and D: !> == 'N': op2(B) = B, op2(D) = D (No transpose for B and D) !> == 'T': op2(B) = B**T, op2(D) = D **T (Transpose B and D) !>

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

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

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

*A*!> A is REAL 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 REAL array, dimension (LDB,N) !> The matrix B must be (quasi-) upper triangular. If the matrix D is already !> quasi-upper triangular the matrix B must be upper triangular. !>

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

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

*WORK*!> WORK is REAL 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. !>

**Optimizations:**- Replaced level-3 by level-2 and level-1 calls,
- Reorder the solution order to column first style,
- Replaced SAXPY operation by Fortran intrinsic
- Replaced all BLAS calls except of STRMV and SGER by Fortran intrinsic,
- Use local copies of A, B, C, D, and X (M, N <=128) .

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **184** of file **sla_trsylv2_l2_opt_local_copy.f90**.

### subroutine sla_trsylv2_l2_local_copy_128 ( transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the discrete time Sylvester equation (Optimized)

**Purpose:**

!> !> SLA_TRSYLV2_L2_LOCAL_COPY_128 solves a generalized Sylvester equation of the following forms !> !> op1(A) * X * op2(B) + X = SCALE * Y (1) !> !> or !> !> op1(A) * X * op2(B) - X = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix and B is a N-by-N upper !> quasi triangular matrices. The right hand side Y and the solution X !> M-by-N matrices. Typically the matrices A and B are created by SGEES from LAPACK. !>

**Attention**The size of the Problem is limited by M,N <= 128

**Parameters***TRANSA**TRANSB**SGN**M*!> M is INTEGER !> The order of the matrices A and C. M >= 0. !>

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

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

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

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

*WORK**INFO***Optimizations:**- Replaced level-3 by level-2 and level-1 calls,
- Reorder the solution order to column first style,
- Replaced SAXPY operation by Fortran intrinsic
- Replaced all BLAS calls except of STRMV by Fortran intrinsic,
- Use local copies of A, B, C, D, and X.
- Align local copies and fix problem size to <= 128

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **184** of file **sla_trsylv2_l2_opt_local_copy_128.f90**.

### subroutine sla_trsylv2_l2_local_copy_32 ( transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the discrete time Sylvester equation (Optimized)

**Purpose:**

!> !> SLA_TRSYLV2_L2_LOCAL_COPY_32 solves a generalized Sylvester equation of the following forms !> !> op1(A) * X * op2(B) + X = SCALE * Y (1) !> !> or !> !> op1(A) * X * op2(B) - X = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix and B is a N-by-N upper !> quasi triangular matrices. The right hand side Y and the solution X !> M-by-N matrices. Typically the matrices A and B are created by SGEES from LAPACK. !>

**Attention**The size of the Problem is limited by M,N <= 32

**Parameters***TRANSA**TRANSB**SGN**M*!> M is INTEGER !> The order of the matrices A and C. M >= 0. !>

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

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

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

*SCALE**WORK**INFO***Optimizations:**- Replaced level-3 by level-2 and level-1 calls,
- Reorder the solution order to column first style,
- Replaced SAXPY operation by Fortran intrinsic
- Replaced all BLAS calls except of STRMV by Fortran intrinsic,
- Use local copies of A, B, C, D, and X.
- Align local copies and fix problem size to <= 32

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **184** of file **sla_trsylv2_l2_opt_local_copy_32.f90**.

### subroutine sla_trsylv2_l2_local_copy_64 ( transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the discrete time Sylvester equation (Optimized)

**Purpose:**

!> !> SLA_TRSYLV2_L2_LOCAL_COPY_64 solves a generalized Sylvester equation of the following forms !> !> op1(A) * X * op2(B) + X = SCALE * Y (1) !> !> or !> !> op1(A) * X * op2(B) - X = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix and B is a N-by-N upper !> quasi triangular matrices. The right hand side Y and the solution X !> M-by-N matrices. Typically the matrices A and B are created by SGEES from LAPACK. !>

**Attention**The size of the Problem is limited by M,N <= 64

**Parameters***TRANSA**TRANSB**SGN**M*!> M is INTEGER !> The order of the matrices A and C. M >= 0. !>

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

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

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

*SCALE**WORK**INFO***Optimizations:**- Replaced level-3 by level-2 and level-1 calls,
- Reorder the solution order to column first style,
- Replaced SAXPY operation by Fortran intrinsic
- Replaced all BLAS calls except of STRMV by Fortran intrinsic,
- Use local copies of A, B, C, D, and X.
- Align local copies and fix problem size to <= 64

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **184** of file **sla_trsylv2_l2_opt_local_copy_64.f90**.

### subroutine sla_trsylv2_l2_local_copy_96 ( transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the discrete time Sylvester equation (Optimized)

**Purpose:**

!> !> SLA_TRSYLV2_L2_LOCAL_COPY_96 solves a generalized Sylvester equation of the following forms !> !> op1(A) * X * op2(B) + X = SCALE * Y (1) !> !> or !> !> op1(A) * X * op2(B) - X = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix and B is a N-by-N upper !> quasi triangular matrices. The right hand side Y and the solution X !> M-by-N matrices. Typically the matrices A and B are created by SGEES from LAPACK. !>

**Attention**The size of the Problem is limited by M,N <= 96

**Parameters***TRANSA**TRANSB**SGN**M*!> M is INTEGER !> The order of the matrices A and C. M >= 0. !>

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

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

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

*SCALE**WORK**INFO***Optimizations:**- Replaced level-3 by level-2 and level-1 calls,
- Reorder the solution order to column first style,
- Replaced SAXPY operation by Fortran intrinsic
- Replaced all BLAS calls except of STRMV by Fortran intrinsic,
- Use local copies of A, B, C, D, and X.
- Align local copies and fix problem size to <= 96

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **184** of file **sla_trsylv2_l2_opt_local_copy_96.f90**.

### subroutine sla_trsylv2_l2_reorder (character, dimension(1) transa, character, dimension(1) transb, real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, real, dimension(*) work, integer info)

Level-2 Bartels-Stewart Algorithm for the discrete time Sylvester equation (Optimized)

**Purpose:**

!> !> SLA_TRSYLV2_L2_UNOPT solves a generalized Sylvester equation of the following forms !> !> op1(A) * X * op2(B) + X = SCALE * Y (1) !> !> or !> !> op1(A) * X * op2(B) - X = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix and B is a N-by-N upper !> quasi triangular matrices. The right hand side Y and the solution X !> M-by-N matrices. Typically the matrices A and B are created by SGEES from LAPACK. !>

**Parameters***TRANSA**TRANSB**SGN**M*!> M is INTEGER !> The order of the matrices A and C. M >= 0. !>

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

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

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

*SCALE**WORK**INFO***Optimizations:**- Replaced level-3 by level-2 and level-1 calls,
- Reorder the solution order to column first style,
- Replaced SAXPY operation by Fortran intrinsic
- Replaced all BLAS calls except of STRMV and SGER by Fortran intrinsic,

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **183** of file **sla_trsylv2_l2_opt_reorder.f90**.

### subroutine sla_trsylv2_l2_unopt (character, dimension(1) transa, character, dimension(1) transb, real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, real, dimension(*) work, integer info)

Level-2 Bartels-Stewart Algorithm for the discrete time Sylvester equation (unoptimized)

**Purpose:**

!> !> SLA_TRSYLV2_L2_UNOPT solves a generalized Sylvester equation of the following forms !> !> op1(A) * X * op2(B) + X = SCALE * Y (1) !> !> or !> !> op1(A) * X * op2(B) - X = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix and B is a N-by-N upper !> quasi triangular matrices. The right hand side Y and the solution X !> M-by-N matrices. Typically the matrices A and B are created by SGEES from LAPACK. !>

**Attention**The algorithm is implemented using BLAS level 2 operations without further optimizations. For a faster implementation see SLA_TRSYLV2_L2.

**Parameters***TRANSA**TRANSB**SGN**M*!> M is INTEGER !> The order of the matrices A and C. M >= 0. !>

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

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

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

*SCALE**WORK**INFO***Optimizations:**- Nothing.

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **181** of file **sla_trsylv2_l2_unopt.f90**.

### subroutine sla_trsylv2_l3 (character, dimension(1) transa, character, dimension(1) transb, real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, real, dimension(*) work, integer info)

Level-3 Bartels-Stewart Algorithm for the discrete time Sylvester equation (Optimized)

**Purpose:**

!> !> SLA_TRSYLV2_L3 solves a generalized Sylvester equation of the following forms !> !> op1(A) * X * op2(B) + X = SCALE * Y (1) !> !> or !> !> op1(A) * X * op2(B) - X = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix and B is a N-by-N upper !> quasi triangular matrices. The right hand side Y and the solution X !> M-by-N matrices. Typically the matrices A and B are created by SGEES from LAPACK. !>

**Parameters***TRANSA**TRANSB**SGN**M*!> M is INTEGER !> The order of the matrices A and C. M >= 0. !>

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

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

*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**January 2024

Definition at line **173** of file **sla_trsylv2_l3_opt.f90**.

### subroutine sla_trsylv2_l3_2s (character, dimension(1) transa, character, dimension(1) transb, real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, real, dimension(*) work, integer info)

Two level blocked Bartels-Stewart Algorithm for the discrete time Sylvester equation (Optimized)

**Purpose:**

!> !> SLA_TRSYLV2_L3_2S solves a generalized Sylvester equation of the following forms !> !> op1(A) * X * op2(B) + X = SCALE * Y (1) !> !> or !> !> op1(A) * X * op2(B) - X = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix and B is a N-by-N upper !> quasi triangular matrices. The right hand side Y and the solution X !> M-by-N matrices. Typically the matrices A and B are created by SGEES from LAPACK. !>

**Remarks**The algorithm used level-3 BLAS operations and a DAG scheduled inner solver.

**Attention**Due to the parallel nature of the inner solvers the scaling is turned off and SCALE is set to ONE.

**Parameters***TRANSA**TRANSB**SGN**M*!> M is INTEGER !> The order of the matrices A and C. M >= 0. !>

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

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

*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**January 2024

Definition at line **174** of file **sla_trsylv2_l3_2stage.f90**.

### subroutine sla_trsylv2_l3_unopt (character, dimension(1) transa, character, dimension(1) transb, real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, real, dimension(*) work, integer info)

Level-3 Bartels-Stewart Algorithm for the discrete time Sylvester equation (Not Optimized)

**Purpose:**

!> !> SLA_TRSYLV2_L3_UNOPT solves a generalized Sylvester equation of the following forms !> !> op1(A) * X * op2(B) + X = SCALE * Y (1) !> !> or !> !> op1(A) * X * op2(B) - X = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix and B is a N-by-N upper !> quasi triangular matrices. The right hand side Y and the solution X !> M-by-N matrices. Typically the matrices A and B are created by SGEES from LAPACK. !>

**Attention**This function iterates column first over the result and from this fact it will be a bit slower than SLA_TRSYLV2_L3.

**Parameters***TRANSA**TRANSB**SGN**M*!> M is INTEGER !> The order of the matrices A and C. M >= 0. !>

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

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

*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**January 2024

Definition at line **175** of file **sla_trsylv2_l3_unopt.f90**.

### recursive subroutine sla_trsylv2_recursive (character, dimension(1) transa, character, dimension(1) transb, real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, real, dimension(*) work, integer info)

Level-3 Recursive Blocked Algorithm for the discrete time Sylvester equation.

**Purpose:**

!> !> SLA_TRSYLV2_RECURSIVE solves a generalized Sylvester equation of the following forms !> !> op1(A) * X * op2(B) + X = SCALE * Y (1) !> !> or !> !> op1(A) * X * op2(B) - X = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix and B is a N-by-N upper !> quasi triangular matrices. The right hand side Y and the solution X !> M-by-N matrices. Typically the matrices A and B are created by SGEES from LAPACK. !>

**Remarks**The algorithm uses recursive blocking instead of the Bartels-Stewart approach.

**Parameters***TRANSA**TRANSB**SGN**M*!> M is INTEGER !> The order of the matrices A and C. M >= 0. !>

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

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

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

*SCALE**WORK*!> WORK is REAL 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**January 2024

Definition at line **163** of file **sla_trsylv2_recursive.f90**.

### subroutine sla_trsylv_dag (character, dimension(1) transa, character, dimension(1) transb, real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, n) x, integer ldx, real scale, real, dimension(*) work, integer info)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation with DAG parallelization.

**Purpose:**

!> SLA_TRSYLV_DAG solves a Sylvester equation of the following forms !> !> op1(A) * X + X * op2(B) = SCALE * Y (1) !> !> or !> !> op1(A) * X - X * op2(B) = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular !> matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices !> A and B are generated via SGEES form LAPACK. !>

**Remarks**The algorithm is implemented using BLAS level 3 operations and OpenMP 4.0 DAG Scheduling.

**Attention**Due to the parallel nature of the algorithm the scaling is not applied to the right hand. If the problem is ill-posed and scaling appears you have to solve the equation again with a solver with complete scaling support like SLA_TRSYLV_L3.

**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 !>

*TRANSB*!> TRANSB is CHARACTER(1) !> Specifies the form of the system of equations with respect to B: !> == 'N': op2(B) = B, !> == 'T': op2(B) = B**T !>

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

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

*A*!> A is REAL 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 REAL array, dimension (LDB,N) !> The matrix B must be (quasi-) upper triangular. !>

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

*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**January 2024

Definition at line **177** of file **sla_trsylv_dag.f90**.

### subroutine sla_trsylv_kernel_44nn (real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, integer info)

Solver for a 4x4 Sylvester equation (TRANSA = N, TRANSB = N)

**Purpose:**

!> !> SLA_TRSYLV_KERNEL_44NN solves a Sylvester equation of the following form !> !> A * X + SGN * X * B = SCALE * Y (1) !> !> where A is a M-by-M quasi upper triangular matrix and B is N-by-N quasi !> upper triangular matrices. The right hand side Y and the solution X !> M-by-N matrices. Typically the matrices A and B are create by SGEES from LAPACK. !> The algorithm is implemented without using 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***SGN**M*!> M is INTEGER !> The order of the matrices A and C. 4 >= M >= 0. !>

*N*!> N is INTEGER !> The order of the matrices B and D. 4 >= N >= 0. !>

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

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

*SCALE**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**January 2024

Definition at line **134** of file **sla_trsylv_kernel_44_nn.f90**.

### subroutine sla_trsylv_kernel_44nt (real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, integer info)

Solver for a 4x4 Sylvester equation (TRANSA = N, TRANSB = T)

**Purpose:**

!> !> SLA_TRSYLV_KERNEL_44NT solves a Sylvester equation of the following form !> !> A * X + SGN * X * B**T = SCALE * Y (1) !> !> where A is a M-by-M quasi upper triangular matrix and B is N-by-N quasi !> upper triangular matrices. The right hand side Y and the solution X !> M-by-N matrices. Typically the matrices A and B are create by SGEES from LAPACK. !> The algorithm is implemented using 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***SGN**M*!> M is INTEGER !> The order of the matrices A and C. 4 >= M >= 0. !>

*N*!> N is INTEGER !> The order of the matrices B and D. 4 >= N >= 0. !>

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

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

*SCALE**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**January 2024

Definition at line **133** of file **sla_trsylv_kernel_44_nt.f90**.

### subroutine sla_trsylv_kernel_44tn (real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, integer info)

Solver for a 4x4 Sylvester equation (TRANSA = T, TRANSB = N)

**Purpose:**

!> !> SLA_TRSYLV_KERNEL_44TN solves a Sylvester equation of the following form !> !> A**T * X + SGN * X * B = SCALE * Y (1) !> !> where A is a M-by-M quasi upper triangular matrix and B is N-by-N quasi !> upper triangular matrices. The right hand side Y and the solution X !> M-by-N matrices. Typically the matrices A and B are create by SGEES from LAPACK. !> The algorithm is implemented without using 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***SGN**M*!> M is INTEGER !> The order of the matrices A and C. 4 >= M >= 0. !>

*N*!> N is INTEGER !> The order of the matrices B and D. 4 >= N >= 0. !>

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

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

*SCALE**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**January 2024

Definition at line **133** of file **sla_trsylv_kernel_44_tn.f90**.

### subroutine sla_trsylv_kernel_44tt (real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, integer info)

Solver for a 4x4 Sylvester equation (TRANSA = T, TRANSB = T)

**Purpose:**

!> !> SLA_TRSYLV_KERNEL_44TT solves a Sylvester equation of the following form !> !> A**T * X + SGN * X * B**T = SCALE * Y (1) !> !> where A is a M-by-M quasi upper triangular matrix and B is N-by-N quasi !> upper triangular matrices. The right hand side Y and the solution X !> M-by-N matrices. Typically the matrices A and B are create by SGEES from LAPACK. !> The algorithm is implemented without using 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***SGN**M*!> M is INTEGER !> The order of the matrices A and C. 4 >= M >= 0. !>

*N*!> N is INTEGER !> The order of the matrices B and D. 4 >= N >= 0. !>

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

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

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

**Date**January 2024

Definition at line **133** of file **sla_trsylv_kernel_44_tt.f90**.

### subroutine sla_trsylv_l2 (character, dimension(1) transa, character, dimension(1) transb, real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, real, dimension(*) work, integer info)

Level-2 Bartels-Stewart Algorithm for the Sylvester equation (optimized)

**Purpose:**

!> !> SLA_TRSYLV_L2 solves a Sylvester equation of the following forms !> !> op1(A) * X + X * op2(B) = SCALE * Y (1) !> !> or !> !> op1(A) * X - X * op2(B) = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular !> matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices !> A and B are generated via SGEES form LAPACK. !>

**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 !>

*TRANSB*!> TRANSB is CHARACTER(1) !> Specifies the form of the system of equations with respect to B: !> == 'N': op2(B) = B, !> == 'T': op2(B) = B**T !>

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

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

*A*!> A is REAL 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 REAL array, dimension (LDB,N) !> The matrix B must be (quasi-) upper triangular. !>

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

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

*SCALE**WORK**INFO***Optimizations:**- Replaced level-3 by level-2 and level-1 calls,
- Replaced SAXPY operation by Fortran intrinsic
- Replaced all BLAS calls except of STRMV and SGER by Fortran intrinsic,

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **180** of file **sla_trsylv_l2.f90**.

### subroutine sla_trsylv_l2_local_copy (character, dimension(1) transa, character, dimension(1) transb, real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, real, dimension(*) work, integer info)

Level-2 Bartels-Stewart Algorithm for the Sylvester equation (optimized)

**Purpose:**

!> !> SLA_TRSYLV_L2_LOCAL_COPY solves a Sylvester equation of the following forms !> !> op1(A) * X + X * op2(B) = SCALE * Y (1) !> !> or !> !> op1(A) * X - X * op2(B) = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular !> matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices !> A and B are generated via SGEES form LAPACK. !>

**Parameters***TRANSA**TRANSB**SGN**M*!> M is INTEGER !> The order of the matrices A and C. M >= 0. !>

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

*A*!> A is REAL 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 REAL array, dimension (LDB,N) !> The matrix B must be (quasi-) upper triangular. !>

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

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

*SCALE**WORK**INFO***Optimizations:**- Replaced level-3 by level-2 and level-1 calls,
- Reorder the solution order to column first style,
- Replaced SAXPY operation by Fortran intrinsic
- Replaced all BLAS calls except of STRMV and SGER by Fortran intrinsic,
- Use local copies of A, B, C, D, and X (M, N <=128) .

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **181** of file **sla_trsylv_l2_opt_local_copy.f90**.

### subroutine sla_trsylv_l2_local_copy_128 ( transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Optimized)

**Purpose:**

!> !> SLA_TRSYLV_L2_LOCAL_COPY_128 solves a Sylvester equation of the following forms !> !> op1(A) * X + X * op2(B) = SCALE * Y (1) !> !> or !> !> op1(A) * X - X * op2(B) = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular !> matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices !> A and B are generated via SGEES form LAPACK. !>

**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 !>

*TRANSB*!> TRANSB is CHARACTER(1) !> Specifies the form of the system of equations with respect to B: !> == 'N': op2(B) = B, !> == 'T': op2(B) = B**T !>

*SGN**M*!> M is INTEGER !> The order of the matrices A and C. 128 >= M >= 0. !>

*N*!> N is INTEGER !> The order of the matrices B and D. 128 >= N >= 0. !>

*A*!> A is REAL 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 REAL array, dimension (LDB,N) !> The matrix B must be (quasi-) upper triangular. !>

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

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

*SCALE**WORK**INFO***Optimizations:**- Replaced level-3 by level-2 and level-1 calls,
- Reorder the solution order to column first style,
- Replaced SAXPY operation by Fortran intrinsic
- Replaced all BLAS calls except of STRMV by Fortran intrinsic,
- Use local copies of A, B, C, D, and X.
- Align local copies and fix problem size to <= 64

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **183** of file **sla_trsylv_l2_opt_local_copy_128.f90**.

### subroutine sla_trsylv_l2_local_copy_32 ( transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Optimized)

**Purpose:**

!> !> SLA_TRSYLV_L2_LOCAL_COPY_32 solves a Sylvester equation of the following forms !> !> op1(A) * X + X * op2(B) = SCALE * Y (1) !> !> or !> !> op1(A) * X - X * op2(B) = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular !> matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices !> A and B are generated via SGEES form LAPACK. !>

**Parameters***TRANSA**TRANSB**SGN**M*!> M is INTEGER !> The order of the matrices A and C. 32 >= M >= 0. !>

*N*!> N is INTEGER !> The order of the matrices B and D. 32 >= N >= 0. !>

*A*!> A is REAL 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 REAL array, dimension (LDB,N) !> The matrix B must be (quasi-) upper triangular. !>

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

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

*SCALE**WORK**INFO***Optimizations:**- Replaced level-3 by level-2 and level-1 calls,
- Reorder the solution order to column first style,
- Replaced SAXPY operation by Fortran intrinsic
- Replaced all BLAS calls except of STRMV by Fortran intrinsic,
- Use local copies of A, B, C, D, and X.
- Align local copies and fix problem size to <= 64

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **183** of file **sla_trsylv_l2_opt_local_copy_32.f90**.

### subroutine sla_trsylv_l2_local_copy_64 ( transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Optimized)

**Purpose:**

!> !> SLA_TRSYLV_L2_LOCAL_COPY_64 solves a Sylvester equation of the following forms !> !> op1(A) * X + X * op2(B) = SCALE * Y (1) !> !> or !> !> op1(A) * X - X * op2(B) = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular !> matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices !> A and B are generated via SGEES form LAPACK. !>

**Parameters***TRANSA**TRANSB**SGN**M*!> M is INTEGER !> The order of the matrices A and C. 64 >= M >= 0. !>

*N*!> N is INTEGER !> The order of the matrices B and D. 64 >= N >= 0. !>

*A*!> A is REAL 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 REAL array, dimension (LDB,N) !> The matrix B must be (quasi-) upper triangular. !>

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

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

*SCALE**WORK**INFO***Optimizations:**- Replaced level-3 by level-2 and level-1 calls,
- Reorder the solution order to column first style,
- Replaced SAXPY operation by Fortran intrinsic
- Replaced all BLAS calls except of STRMV by Fortran intrinsic,
- Use local copies of A, B, C, D, and X.
- Align local copies and fix problem size to <= 64

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **182** of file **sla_trsylv_l2_opt_local_copy_64.f90**.

### subroutine sla_trsylv_l2_local_copy_96 ( transa, transb, sgn, m, n, a, lda, b, ldb, x, ldx, scale, work, info)

Level-2 Bartels-Stewart Algorithm for the generalized Sylvester equation (Optimized)

**Purpose:**

!> !> SLA_TRSYLV_L2_LOCAL_COPY_96 solves a Sylvester equation of the following forms !> !> op1(A) * X + X * op2(B) = SCALE * Y (1) !> !> or !> !> op1(A) * X - X * op2(B) = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular !> matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices !> A and B are generated via SGEES form LAPACK. !>

**Parameters***TRANSA**TRANSB**SGN**M*!> M is INTEGER !> The order of the matrices A and C. 96 >= M >= 0. !>

*N*!> N is INTEGER !> The order of the matrices B and D. 96 >= N >= 0. !>

*A*!> A is REAL 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 REAL array, dimension (LDB,N) !> The matrix B must be (quasi-) upper triangular. !>

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

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

*SCALE**WORK**INFO***Optimizations:**- Replaced level-3 by level-2 and level-1 calls,
- Reorder the solution order to column first style,
- Replaced SAXPY operation by Fortran intrinsic
- Replaced all BLAS calls except of STRMV by Fortran intrinsic,
- Use local copies of A, B, C, D, and X.
- Align local copies and fix problem size to <= 64

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **181** of file **sla_trsylv_l2_opt_local_copy_96.f90**.

### subroutine sla_trsylv_l2_reorder (character, dimension(1) transa, character, dimension(1) transb, real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, real, dimension(*) work, integer info)

Level-2 Bartels-Stewart Algorithm for the Sylvester equation (optimized)

**Purpose:**

!> !> SLA_TRSYLV_L2_REORDER solves a Sylvester equation of the following forms !> !> op1(A) * X + X * op2(B) = SCALE * Y (1) !> !> or !> !> op1(A) * X - X * op2(B) = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular !> matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices !> A and B are generated via SGEES form LAPACK. !>

**Attention**The algorithm is implemented using BLAS level 2 operations without further optimizations. For a faster implementation see SLA_TRSYLV_L2 .

**Parameters***TRANSA**TRANSB**SGN**M*!> M is INTEGER !> The order of the matrices A and C. M >= 0. !>

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

*A*!> A is REAL 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 REAL array, dimension (LDB,N) !> The matrix B must be (quasi-) upper triangular. !>

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

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

*SCALE**WORK**INFO***Optimizations:**- Replaced level-3 by level-2 and level-1 calls,
- Reorder the solution order to column first style,
- Replaced SAXPY operation by Fortran intrinsic
- Replaced all BLAS calls except of STRMV and SGER by Fortran intrinsic,

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **182** of file **sla_trsylv_l2_opt_reorder.f90**.

### subroutine sla_trsylv_l2_unopt (character, dimension(1) transa, character, dimension(1) transb, real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, real, dimension(*) work, integer info)

Level-2 Bartels-Stewart Algorithm for the Sylvester equation (unoptimized)

**Purpose:**

!> !> SLA_TRSYLV_L2_UNOPT solves a Sylvester equation of the following forms !> !> op1(A) * X + X * op2(B) = SCALE * Y (1) !> !> or !> !> op1(A) * X - X * op2(B) = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular !> matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices !> A and B are generated via SGEES form LAPACK. !>

**Attention**The algorithm is implemented using BLAS level 2 operations without further optimizations. For a faster implementation see SLA_TRSYLV_L2 .

**Parameters***TRANSA**TRANSB**SGN**M*!> M is INTEGER !> The order of the matrices A and C. M >= 0. !>

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

*A*!> A is REAL 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 REAL array, dimension (LDB,N) !> The matrix B must be (quasi-) upper triangular. !>

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

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

*SCALE**WORK**INFO***Optimizations:**- Nothing.

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **180** of file **sla_trsylv_l2_unopt.f90**.

### subroutine sla_trsylv_l3 (character, dimension(1) transa, character, dimension(1) transb, real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, real, dimension(*) work, integer info)

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

**Purpose:**

!> SLA_TRSYLV_L3 solves a Sylvester equation of the following forms !> !> op1(A) * X + X * op2(B) = SCALE * Y (1) !> !> or !> !> op1(A) * X - X * op2(B) = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular !> matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices !> A and B are generated via SGEES form LAPACK. !>

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

**Parameters***TRANSA**TRANSB**SGN**M*!> M is INTEGER !> The order of the matrices A and C. M >= 0. !>

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

*A*!> A is REAL 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 REAL array, dimension (LDB,N) !> The matrix B must be (quasi-) upper triangular. !>

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

*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**January 2024

Definition at line **172** of file **sla_trsylv_l3_opt.f90**.

### subroutine sla_trsylv_l3_2s (character, dimension(1) transa, character, dimension(1) transb, real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, real, dimension(*) work, integer info)

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

**Purpose:**

!> SLA_TRSYLV_L3_2S solves a Sylvester equation of the following forms !> !> op1(A) * X + X * op2(B) = SCALE * Y (1) !> !> or !> !> op1(A) * X - X * op2(B) = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular !> matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices !> A and B are generated via SGEES form LAPACK. !>

**Remarks**The algorithm is implemented using BLAS level 3 operations and a two level inner solver consisting of a massive parallel DAG scheduled solver from SLA_TRSYLV_DAG and optimized kernel solvers.

**Attention**Due to the parallel nature of the inner solvers the scaling is turned off and SCALE is set to ONE.

**Parameters***TRANSA**TRANSB**SGN**M*!> M is INTEGER !> The order of the matrices A and C. M >= 0. !>

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

*A*!> A is REAL 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 REAL array, dimension (LDB,N) !> The matrix B must be (quasi-) upper triangular. !>

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

*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**January 2024

Definition at line **174** of file **sla_trsylv_l3_2stage.f90**.

### subroutine sla_trsylv_l3_unopt (character, dimension(1) transa, character, dimension(1) transb, real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, real, dimension(*) work, integer info)

Level-3 Bartels-Stewart Algorithm for the generalized Sylvester equation (unoptimized)

**Purpose:**

!> SLA_TRSYLV_L3_UNOPT solves a Sylvester equation of the following forms !> !> op1(A) * X + X * op2(B) = SCALE * Y (1) !> !> or !> !> op1(A) * X - X * op2(B) = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular !> matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices !> A and B are generated via SGEES form LAPACK. !>

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

**Parameters***TRANSA**TRANSB**SGN**M*!> M is INTEGER !> The order of the matrices A and C. M >= 0. !>

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

*A*!> A is REAL 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 REAL array, dimension (LDB,N) !> The matrix B must be (quasi-) upper triangular. !>

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

*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**January 2024

Definition at line **172** of file **sla_trsylv_l3_unopt.f90**.

### recursive subroutine sla_trsylv_recursive (character, dimension(1) transa, character, dimension(1) transb, real sgn, integer m, integer n, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldx, *) x, integer ldx, real scale, real, dimension(*) work, integer info)

Level-3 Recursive Blocked Solver for the Sylvester equation.

**Purpose:**

!> SLA_TRSYLV_RECURSIVE solves a Sylvester equation of the following forms !> !> op1(A) * X + X * op2(B) = SCALE * Y (1) !> !> or !> !> op1(A) * X - X * op2(B) = SCALE * Y (2) !> !> where A is a M-by-M quasi upper triangular matrix, B is a N-by-N quasi upper triangular !> matrix. The right hand side Y and the solution X are M-by-N matrices. Typically the matrices !> A and B are generated via SGEES form LAPACK. !>

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

**Parameters***TRANSA**TRANSB**SGN**M*!> M is INTEGER !> The order of the matrices A and C. M >= 0. !>

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

*A*!> A is REAL 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 REAL array, dimension (LDB,N) !> The matrix B must be (quasi-) upper triangular. !>

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

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

*SCALE**WORK*!> WORK is REAL 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**January 2024

Definition at line **160** of file **sla_trsylv_recursive.f90**.

## Author

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