# dblgelyap - Man Page

## Name

dblgelyap — Double Precision

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

## Synopsis

### Functions

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

Frontend for the solution of Standard Lyapunov Equations.

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

Frontend for the solution of Standard Stein Equations.

subroutine **dla_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 **dla_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

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

## Function Documentation

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

Frontend for the solution of Standard Lyapunov Equations.

**Purpose:**

DLA_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. The right hand side Y and the solution X are M-by-M matrices. The matrix A can be either supplied as general matrix or 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

*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 DOUBLE PRECISION 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.

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

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

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

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

*WORK*WORK is DOUBLE PRECISION array, dimension (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: DGEES failed = 2: DLA_SORT_EV failed = 3: Internal solver failed < 0: if INFO = -i, the i-th argument had an illegal value

**See also**DLA_TRLYAP_L3

DLA_TRLYAP_L3_2S

DLA_TRLYAP_DAG

DLA_TRLYAP_L2

DLA_TRLYAP_RECURSIVE

**Author**Martin Koehler, MPI Magdeburg

**Date**Januar 2023

Definition at line **169** of file **dla_gelyap.f90**.

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

Iterative Refinement for the Standard Lyapunov Equation.

**Purpose:**

DLA_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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDQ,M) The array Q contains the Schur vectors for A as returned by DGEES.

*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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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**DLA_TRLYAP_L3

DLA_TRLYAP_L2

DLA_TRLYAP_L3_2S

DLA_TRLYAP_DAG

DLA_TRLYAP_RECURSIVE

**Author**Martin Koehler, MPI Magdeburg

**Date**Januar 2023

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

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

Frontend for the solution of Standard Stein Equations.

**Purpose:**

DLA_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. The right hand side Y and the solution X are M-by-M matrices. The matrix A can be either supplied as general matrix or 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

*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 DOUBLE PRECISION 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.

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

*Q*Q is DOUBLE PRECISION 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.

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

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

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

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

*WORK*WORK is DOUBLE PRECISION array, dimension (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: DGEES failed = 2: DLA_SORT_EV failed = 3: Inner solver failed < 0: if INFO = -i, the i-th argument had an illegal value

**See also**DLA_TRSTEIN_L3

DLA_TRSTEIN_L3_2S

DLA_TRSTEIN_DAG

DLA_TRSTEIN_L2

DLA_TRSTEIN_RECURSIVE

**Author**Martin Koehler, MPI Magdeburg

**Date**Januar 2023

Definition at line **169** of file **dla_gestein.f90**.

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

Iterative Refinement for the Standard Stein Equations.

**Purpose:**

DLA_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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDQ,M) The array Q contains the Schur vectors for A as returned by DGEES.

*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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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**DLA_TRSTEIN_L3

DLA_TRSTEIN_L2

DLA_TRSTEIN_L3_2S

DLA_TRSTEIN_DAG

DLA_TRSTEIN_RECURSIVE

**Author**Martin Koehler, MPI Magdeburg

**Date**Januar 2023

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

## Author

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