# dblgglyap - Man Page

## Name

dblgglyap — Double Precision

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

## Synopsis

### Functions

subroutine **dla_gglyap** (fact, trans, m, a, lda, b, ldb, q, ldq, z, ldz, x, ldx, scale, work, ldwork, info)

Frontend for the solution of Generalized Lyapunov Equations.

subroutine **dla_ggstein** (fact, trans, m, a, lda, b, ldb, q, ldq, z, ldz, x, ldx, scale, work, ldwork, info)

Frontend for the solution of Generalized Stein Equations.

subroutine **dla_gglyap_refine** (trans, guess, m, a, lda, b, ldb, x, ldx, y, ldy, as, ldas, bs, ldbs, q, ldq, z, ldz, maxit, tau, convlog, work, ldwork, info)

Iterative Refinement for the Generalized Lyapunov Equations.

## Detailed Description

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

## Function Documentation

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

Frontend for the solution of Generalized Lyapunov Equations.

**Purpose:**

DLA_GGLYAP solves a generalized Lyapunov equation of the following forms A * X * B^T + B * X * A^T = SCALE * Y (1) or A^T * X * B + B^T * X * A = SCALE * Y (2) where (A,B) is a M-by-M matrix pencil. The right hand side Y and the solution X are M-by-M matrices. The matrix pencil (A,B) is either in general form, in generalized Hessenberg form, or in generalized Schur form where Q and Z also need to be provided.

**Parameters***FACT*FACT is CHARACTER Specifies how the matrix A is given. == 'N': The matrix pencil (A,B) is given as a general matrix pencil and its Schur decomposition A = Q*S*Z**T, B = Q*R*Z**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 pencil (A,B) is given in generalized Hessenberg form and its Schur decomposition A = Q*S*Z**T, B = Q*R*Z**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, B, Y and X. 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 the quasi upper triangular matrix S of the generalized schur decomposition. If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix S of the generalized Schur decomposition of (A,B). If FACT == 'H', the matrix A is an upper Hessenberg matrix of the generalized Hessenberg form (A,B) and it is overwritten with the quasi upper triangular matrix S of the generalized Schur decomposition.

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

*B*B is DOUBLE PRECISION array, dimension (LDB,M) If FACT == 'N', the matrix B a general matrix and it is overwritten with the upper triangular matrix of the generalized Schur decomposition. If FACT == 'F', the matrix B contains its upper triangular matrix R of the generalized schur Schur decomposition of (A,B). If FACT == 'H', the matrix B is the upper triangular matrix of the generalized Hessenberg form (A,B) and it is overwritten with the upper triangular matrix of the generalized Schur decomposition.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= 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 left Schur vectors of (A,B) on output. If FACT == 'F', the matrix Q contains the left Schur vectors of (A,B). If FACT == 'H', the matrix Q is an empty M-by-M matrix on input and contains the left Schur vectors of (A,B) on output.

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

*Z*Z is DOUBLE PRECISION array, dimension (LDZ,M) If FACT == 'N', the matrix Z is an empty M-by-M matrix on input and contains the right Schur vectors of (A,B) on output. If FACT == 'F', the matrix Z contains the right Schur vectors of (A,B). If FACT == 'H', the matrix Z is an empty M-by-M matrix on input and contains the right Schur vectors of (A,B) on output.

*LDZ*LDZ is INTEGER The leading dimension of the array Z. LDZ >= 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. LDX >= 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 without performing any computations.

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

**See also**DLA_TGLYAP_L3

DLA_TGLYAP_L3_2S

DLA_TGLYAP_DAG

DLA_TGLYAP_L2

DLA_TGLYAP_RECURSIVE

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **207** of file **dla_gglyap.f90**.

### subroutine dla_gglyap_refine (character, dimension(1) trans, character, dimension(1) guess, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldb, *) b, integer ldb, double precision, dimension( ldx, *) x, integer ldx, double precision, dimension(ldy, *) y, integer ldy, double precision, dimension(ldas, *) as, integer ldas, double precision, dimension(ldbs,*) bs, integer ldbs, double precision, dimension(ldq, *) q, integer ldq, double precision, dimension(ldz, *) z, integer ldz, integer maxit, double precision tau, double precision, dimension(*) convlog, double precision, dimension(*) work, integer ldwork, integer info)

Iterative Refinement for the Generalized Lyapunov Equations.

**Purpose:**

DLA_GGLYAP_REFINE solves a generalized Lyapunov equation of the following forms A * X * B^T + B * X * A^T = SCALE * Y (1) or A^T * X * B + B^T * X * A = SCALE * Y (2) where (A,B) is a M-by-M matrix pencil using iterative refinement. The right hand side Y and the solution X are M-by-M matrices. The matrix pencil (A,B) needs to provide as the original data as well as in generalized 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).

*B*B is DOUBLE PRECISION array, dimension (LDB,M) The array B contains the original matrix B defining the eqaution.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= 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 generalized Schur decomposition of the A.

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

*BS*BS is DOUBLE PRECISION array, dimension (LDBS,M) The array AS contains the generalized Schur decomposition of the B.

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

*Q*Q is DOUBLE PRECISION array, dimension (LDQ,M) The array Q contains the left generalized Schur vectors for (A,B) as returned by DGGES.

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

*Z*Z is DOUBLE PRECISION array, dimension (LDZ,M) The array Z contains the right generalized Schur vectors for (A,B) as returned by DGGES.

*LDZ*LDZ is INTEGER The leading dimension of the array Z. LDZ >= 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_TGLYAP_L3

DLA_TGLYAP_L2

DLA_TGLYAP_L3_2S

DLA_TGLYAP_DAG

DLA_TGLYAP_RECURSIVE

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **244** of file **dla_gglyap_refine.f90**.

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

Frontend for the solution of Generalized Stein Equations.

**Purpose:**

DLA_GGSTEIN solves a generalized Stein equation of the following forms A * X * A^T - B * X * B^T = SCALE * Y (1) or A^T * X * A - B^T * X * B = SCALE * Y (2) where (A,B) is a M-by-M matrix pencil. The right hand side Y and the solution X M-by-M matrices. The matrix pencil (A,B) is either in general form, in generalized Hessenberg form, or in generalized Schur form where Q and Z also need to be provided.

**Parameters***FACT*FACT is CHARACTER Specifies how the matrix A is given. == 'N': The matrix pencil (A,B) is given as a general matrix pencil and its Schur decomposition A = Q*S*Z**T, B = Q*R*Z**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 pencil (A,B) is given in generalized Hessenberg form and its Schur decomposition A = Q*S*Z**T, B = Q*R*Z**T will be computed.

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

*M*M is INTEGER The order of the matrices A, B, Y and X. 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 the quasi upper triangular matrix S of the generalized schur decomposition. If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix S of the generalized Schur decomposition of (A,B). If FACT == 'H', the matrix A is an upper Hessenberg matrix of the generalized Hessenberg form (A,B) and it is overwritten with the quasi upper triangular matrix S of the generalized Schur decomposition.

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

*B*B is DOUBLE PRECISION array, dimension (LDB,M) If FACT == 'N', the matrix B a general matrix and it is overwritten with the upper triangular matrix of the generalized Schur decomposition. If FACT == 'F', the matrix B contains its upper triangular matrix R of the generalized schur Schur decomposition of (A,B). If FACT == 'H', the matrix B is the upper triangular matrix of the generalized Hessenberg form (A,B) and it is overwritten with the upper triangular matrix of the generalized Schur decomposition.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= 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 left Schur vectors of (A,B) on output. If FACT == 'F', the matrix Q contains the left Schur vectors of (A,B). If FACT == 'H', the matrix Q is an empty M-by-M matrix on input and contains the left Schur vectors of (A,B) on output.

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

*Z*Z is DOUBLE PRECISION array, dimension (LDZ,M) If FACT == 'N', the matrix Z is an empty M-by-M matrix on input and contains the right Schur vectors of (A,B) on output. If FACT == 'F', the matrix Z contains the right Schur vectors of (A,B). If FACT == 'H', the matrix Z is an empty M-by-M matrix on input and contains the right Schur vectors of (A,B) on output.

*LDZ*LDZ is INTEGER The leading dimension of the array Z. LDZ >= 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. LDX >= 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 without performing any computations.

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

**See also**DLA_TGSTEIN_L3

DLA_TGSTEIN_L3_2S

DLA_TGSTEIN_DAG

DLA_TGSTEIN_L2

DLA_TGSTEIN_RECURSIVE

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **208** of file **dla_ggstein.f90**.

## Author

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