# sglgelyap - Man Page

## Name

sglgelyap — Single Precision

— Single precision solvers for standard Lyapunov and Stein equations with general coefficient matrices.

## Synopsis

### Functions

subroutine **sla_gelyap** (fact, trans, m, a, lda, q, ldq, x, ldx, scale, work, ldwork, info)

Frontend for the solution of Standard Lyapunov Equations.

subroutine **sla_gestein** (fact, trans, m, a, lda, q, ldq, x, ldx, scale, work, ldwork, info)

Frontend for the solution of Standard Stein Equations.

subroutine **sla_gelyap_refine** (trans, guess, m, a, lda, x, ldx, y, ldy, as, ldas, q, ldq, maxit, tau, convlog, work, ldwork, info)

Iterative Refinement for the Standard Lyapunov Equation.

subroutine **sla_gestein_refine** (trans, guess, m, a, lda, x, ldx, y, ldy, as, ldas, q, ldq, maxit, tau, convlog, work, ldwork, info)

Iterative Refinement for the Standard Stein Equations.

## Detailed Description

Single precision solvers for standard Lyapunov and Stein equations with general coefficient matrices.

## Function Documentation

### subroutine sla_gelyap (character, dimension(1) fact, character, dimension(1) trans, integer m, real, dimension(lda,*) a, integer lda, real, dimension(ldq, *) q, integer ldq, real, dimension(ldx, *) x, integer ldx, real scale, real, dimension(*) work, integer ldwork, integer info)

Frontend for the solution of Standard Lyapunov Equations.

**Purpose:**

SLA_GELYAP 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 general matrix or a matrix in upper Hessenberg form. The right hand side Y and the solution X are M-by-M matrices. The general matrix A can be supplied factorized in terms of its Schur decomposition.

**Parameters***FACT*FACT is CHARACTER Specifies how the matrix A is given. == 'N': The matrix A is given as a general matrix and its Schur decomposition A = Q*S*Q**T will be computed. == 'F': The matrix A is given as its Schur decomposition in terms of S and Q form A = Q*S*Q**T == 'H': The matrix A is given in upper Hessenberg form and its Schur decomposition A = Q*S*Q**T will be computed

*TRANS*TRANS is CHARACTER Specifies the form of the system of equations with respect to A: == 'N': Equation (1) is solved. == 'T': Equation (2) is solved.

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

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

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

*Q*Q is REAL array, dimension (LDQ,M) If FACT == 'N', the matrix Q is an empty M-by-M matrix on input and contains the Schur vectors of A on output. If FACT == 'H', the matrix Q is an empty M-by-M matrix on input and contains the Schur vectors of A on output. If FACT == 'F', the matrix Q contains the Schur vectors of A.

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

*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) 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 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 (MAX(1,LDWORK)) Workspace for the algorithm. The optmimal workspace is given either by \ref mepack_memory_frontend or a previous call to the this routine with LDWORK === -1.

*LDWORK*LDWORK is INTEGER Size of the workspace for the algorithm. This can be determined by a call \ref mepack_memory_frontend . Alternatively, if LDWORK == -1 on input the subroutine will return the required size of the workspace in LDWORK again without performing any computations.

*INFO*INFO is INTEGER == 0: successful exit = 1: SHGEES failed = 2: SLA_SORT_EV failed = 3: Internal solver failed < 0: if INFO = -i, the i-th argument had an illegal value

**See also**SLA_TRLYAP_L3

SLA_TRLYAP_L3_2S

SLA_TRLYAP_DAG

SLA_TRLYAP_L2

SLA_TRLYAP_RECURSIVE

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

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

### subroutine sla_gelyap_refine (character, dimension(1) trans, character, dimension(1) guess, integer m, real, dimension(lda, *) a, integer lda, real, dimension( ldx, *) x, integer ldx, real, dimension(ldy, *) y, integer ldy, real, dimension(ldas, *) as, integer ldas, real, dimension(ldq, *) q, integer ldq, integer maxit, real tau, real, dimension(*) convlog, real, dimension(*) work, integer ldwork, integer info)

Iterative Refinement for the Standard Lyapunov Equation.

**Purpose:**

SLA_GELYAP_REFINE solves a standard 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 matrix using iterative refinement. The right hand side Y and the solution X are M-by-M matrices. The matrix A needs to be provided as the original data as well as in Schur decomposition since both are required in the iterative refinement process.

**Parameters***TRANS*TRANS is CHARACTER Specifies the form of the system of equations with respect to A : == 'N': Equation (1) is solved == 'T': Equation (2) is solved

*GUESS*GUESS is CHARACTER Specifies whether X contains an initial guess or nor not. = 'I': X contains an initial guess = 'N': No initial guess, X is set to zero at the begin of the iteration.

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

*A*A is REAL array, dimension (LDA,M) The array A contains the original matrix A defining the eqaution.

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

*X*X is REAL array, dimension (LDX,M) On input, the array X contains the initial guess, if GUESS = 'I'. On output, the array X contains the solution X.

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

*Y*Y is REAL array, dimension (LDY,M) On input, the array Y contains the right hand side Y. The array stays unchanged during the iteration.

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

*AS*AS is REAL array, dimension (LDAS,M) The array AS contains the Schur decomposition of A.

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

*Q*Q is REAL array, dimension (LDQ,M) The array Q contains the Schur vectors for A as returned by SGEES.

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

*MAXIT*MAXIT is INTEGER On input, MAXIT contains the maximum number of iteration that are performed, 2 <= MAXIT <= 100 On exit, MAXIT contains the number of iteration steps taken by the algorithm.

*TAU*TAU is REAL On input, TAU contains the additional security factor for the stopping criterion, typical values are 0.1 On exit, TAU contains the last relative residual when the stopping criterion got valid.

*CONVLOG*CONVLOG is REAL array, dimension (MAXIT) The CONVLOG array contains the convergence history of the iterative refinement. CONVLOG(I) contains the maximum relative residual before it is solved for the I-th time.

*WORK*WORK is REAL array, dimension (MAX(1,LDWORK)) Workspace for the algorithm. The optmimal workspace is returned in LDWORK, if LDWORK == -1 on input. In this case no computations are performed.

*LDWORK*LDWORK is INTEGER If LDWORK == -1 the subroutine will return the required size of the workspace in LDWORK on exit. No computations are performed and none of the arrays are referenced.

*INFO*INFO is INTEGER == 0: Success > 0: Iteration failed in step INFO < 0: if INFO = -i, the i-th argument had an illegal value = -50: Some of the internal settings like NB,... are incorrect.

**See also**SLA_TRLYAP_L3

SLA_TRLYAP_L2

SLA_TRLYAP_L3_2S

SLA_TRLYAP_DAG

SLA_TRLYAP_RECURSIVE

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **201** of file **sla_gelyap_refine.f90**.

### subroutine sla_gestein (character, dimension(1) fact, character, dimension(1) trans, integer m, real, dimension(lda,*) a, integer lda, real, dimension(ldq, *) q, integer ldq, real, dimension(ldx, *) x, integer ldx, real scale, real, dimension(*) work, integer ldwork, integer info)

Frontend for the solution of Standard Stein Equations.

**Purpose:**

SLA_GESTEIN 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 general matrix or a matrix in upper Hessenberg form. The right hand side Y and the solution X are M-by-M matrices. The general matrix A can be supplied factorized in terms of its Schur decomposition.

**Parameters***FACT*FACT is CHARACTER Specifies how the matrix A is given. == 'N': The matrix A is given as a general matrix and its Schur decomposition A = Q*S*Q**T will be computed. == 'F': The matrix A is given as its Schur decomposition in terms of S and Q form A = Q*S*Q**T == 'H': The matrix A is given in upper Hessenberg form and its Schur decomposition A = Q*S*Q**T will be computed

*TRANS*TRANS is CHARACTER Specifies the form of the system of equations with respect to A: == '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 REAL array, dimension (LDA,M) If FACT == 'N', the matrix A is a general matrix and it is overwritten with its schur decomposition S. If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix S being the Schur decomposition of A. If FACT == 'H', the matrix A is an upper Hessenberg matrix and it is overwritten with its schur decomposition S.

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

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

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

*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) 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 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 (MAX(1,LDWORK)) Workspace for the algorithm. The optmimal workspace is given either by \ref mepack_memory_frontend or a previous call to the this routine with LDWORK === -1.

*LDWORK*LDWORK is INTEGER Size of the workspace for the algorithm. This can be determined by a call \ref mepack_memory_frontend . Alternatively, if LDWORK == -1 on input the subroutine will return the required size of the workspace in LDWORK again without performing any computations.

*INFO*INFO is INTEGER == 0: successful exit = 1: SHGEES failed = 2: SLA_SORT_EV failed = 3: Inner solver failed < 0: if INFO = -i, the i-th argument had an illegal value

**See also**SLA_TRSTEIN_L3

SLA_TRSTEIN_L3_2S

SLA_TRSTEIN_DAG

SLA_TRSTEIN_L2

SLA_TRSTEIN_RECURSIVE

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

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

### subroutine sla_gestein_refine (character, dimension(1) trans, character, dimension(1) guess, integer m, real, dimension(lda, *) a, integer lda, real, dimension( ldx, *) x, integer ldx, real, dimension(ldy, *) y, integer ldy, real, dimension(ldas, *) as, integer ldas, real, dimension(ldq, *) q, integer ldq, integer maxit, real tau, real, dimension(*) convlog, real, dimension(*) work, integer ldwork, integer info)

Iterative Refinement for the Standard Stein Equations.

**Purpose:**

SLA_GESTEIN_REFINE solves a standard Stein 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 matrix using iterative refinement. The right hand side Y and the solution X are M-by-M matrices. The matrix A needs to be provided as the original data as well as in Schur decomposition since both are required in the iterative refinement process.

**Parameters***TRANS*TRANS is CHARACTER Specifies the form of the system of equations with respect to A : == 'N': Equation (1) is solved == 'T': Equation (2) is solved

*GUESS*GUESS is CHARACTER Specifies whether X contains an initial guess or nor not. = 'I': X contains an initial guess = 'N': No initial guess, X is set to zero at the begin of the iteration.

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

*A*A is REAL array, dimension (LDA,M) The array A contains the original matrix A defining the eqaution.

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

*X*X is REAL array, dimension (LDX,M) On input, the array X contains the initial guess, if GUESS = 'I'. On output, the array X contains the solution X.

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

*Y*Y is REAL array, dimension (LDY,M) On input, the array Y contains the right hand side Y. The array stays unchanged during the iteration.

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

*AS*AS is REAL array, dimension (LDAS,M) The array AS contains the Schur decomposition of A.

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

*Q*Q is REAL array, dimension (LDQ,M) The array Q contains the Schur vectors for A as returned by SGEES.

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

*MAXIT*MAXIT is INTEGER On input, MAXIT contains the maximum number of iteration that are performed, 2 <= MAXIT <= 100 On exit, MAXIT contains the number of iteration steps taken by the algorithm.

*TAU*TAU is REAL On input, TAU contains the additional security factor for the stopping criterion, typical values are 0.1 On exit, TAU contains the last relative residual when the stopping criterion got valid.

*CONVLOG*CONVLOG is REAL array, dimension (MAXIT) The CONVLOG array contains the convergence history of the iterative refinement. CONVLOG(I) contains the maximum relative residual before it is solved for the I-th time.

*WORK*WORK is REAL array, dimension (MAX(1,LDWORK)) Workspace for the algorithm. The optmimal workspace is returned in LDWORK, if LDWORK == -1 on input. In this case no computations are performed.

*LDWORK*LDWORK is INTEGER If LDWORK == -1 the subroutine will return the required size of the workspace in LDWORK on exit. No computations are performed and none of the arrays are referenced.

*INFO*INFO is INTEGER == 0: Success > 0: Iteration failed in step INFO < 0: if INFO = -i, the i-th argument had an illegal value = -50: Some of the internal settings like NB,... are incorrect.

**See also**SLA_TRSTEIN_L3

SLA_TRSTEIN_L2

SLA_TRSTEIN_L3_2S

SLA_TRSTEIN_DAG

SLA_TRSTEIN_RECURSIVE

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **202** of file **sla_gestein_refine.f90**.

## Author

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