# dblggsylv - Man Page

## Name

dblggsylv — Double Precision

— Double Precision routines for standard Sylvester equations.

## Synopsis

### Functions

subroutine **dla_ggcsylv** (facta, factb, transa, transb, sgn1, sgn2, m, n, a, lda, b, ldb, c, ldc, d, ldd, qa, ldqa, za, ldza, qb, ldqb, zb, ldzb, e, lde, f, ldf, scale, work, ldwork, info)

Frontend for the solution of Coupled Generalized Sylvester Equations.

subroutine **dla_ggcsylv_dual** (facta, factb, transa, transb, sgn1, sgn2, m, n, a, lda, b, ldb, c, ldc, d, ldd, qa, ldqa, za, ldza, qb, ldqb, zb, ldzb, e, lde, f, ldf, scale, work, ldwork, info)

Frontend for the solution of the dual Coupled Generalized Sylvester Equations.

subroutine **dla_ggsylv** (facta, factb, transa, transb, sgn, m, n, a, lda, b, ldb, c, ldc, d, ldd, qa, ldqa, za, ldza, qb, ldqb, zb, ldzb, x, ldx, scale, work, ldwork, info)

Frontend for the solution of Generalized Sylvester Equations.

subroutine **dla_ggcsylv_dual_refine** (transa, transb, guess, sgn1, sgn2, m, n, a, lda, b, ldb, c, ldc, d, ldd, r, ldr, l, ldl, e, lde, f, ldf, as, ldas, bs, ldbs, cs, ldcs, ds, ldds, q, ldq, z, ldz, u, ldu, v, ldv, maxit, tau, convlog, work, ldwork, info)

Iterative Refinement for the dual Coupled Generalized Sylvester Equations.

subroutine **dla_ggcsylv_refine** (transa, transb, guess, sgn1, sgn2, m, n, a, lda, b, ldb, c, ldc, d, ldd, r, ldr, l, ldl, e, lde, f, ldf, as, ldas, bs, ldbs, cs, ldcs, ds, ldds, q, ldq, z, ldz, u, ldu, v, ldv, maxit, tau, convlog, work, ldwork, info)

Iterative Refinement for the Coupled Generalized Sylvester Equations.

subroutine **dla_ggsylv_refine** (transa, transb, guess, sgn, m, n, a, lda, b, ldb, c, ldc, d, ldd, x, ldx, y, ldy, as, ldas, bs, ldbs, cs, ldcs, ds, ldds, q, ldq, z, ldz, u, ldu, v, ldv, maxit, tau, convlog, work, ldwork, info)

Iterative Refinement for the Generalized Sylvester Equations.

## Detailed Description

Double Precision routines for standard Sylvester equations.

This subsection contains the solvers for generalized Sylvester equations with general coefficient matrices in double precision arithmetic. The generalized Schur decompositions are computed in double precision with the help of LAPACK.

## Function Documentation

### subroutine dla_ggcsylv (character, dimension(1) facta, character, dimension(1) factb, character, dimension(1) transa, character, dimension(1) transb, double precision sgn1, double precision sgn2, integer m, integer n, double precision, dimension(lda,*) a, integer lda, double precision, dimension(ldb, *) b, integer ldb, double precision, dimension(ldc, *) c, integer ldc, double precision, dimension(ldd,*) d, integer ldd, double precision, dimension(ldqa, *) qa, integer ldqa, double precision, dimension(ldza, *) za, integer ldza, double precision, dimension(ldqb, *) qb, integer ldqb, double precision, dimension(ldzb, *) zb, integer ldzb, double precision, dimension(lde, *) e, integer lde, double precision, dimension(ldf,*) f, integer ldf, double precision scale, double precision, dimension(*) work, integer ldwork, integer info)

Frontend for the solution of Coupled Generalized Sylvester Equations.

**Purpose:**

DLA_GGCSYLV solves a coupled generalized Sylvester equation of the following forms op1(A) * R + SGN1 * L * op2(B) = SCALE * E (1) op1(C) * R + SGN2 * L * op2(D) = SCALE * F where (A,C) is a M-by-M matrix pencil and (B,D) is a N-by-N matrix pencil. The right hand side (E,F) and the solution (R,L) are M-by-N matrix pencils. The pencils (A,C) and (B,D) can be either given as general unreduced matrices, as generalized Hessenberg form, or in terms of their generalized Schur decomposition. If they are given as general matrices or as a generalized Hessenberg form their generalized Schur decomposition will be computed.

**Parameters***FACTA*FACTA is CHARACTER Specifies how the matrix pencil (A,C) is given. == 'N': The matrix pencil (A,C) is given as a general matrices and its Schur decomposition A = QA*S*ZA**T, C = QA*R*ZA**T will be computed. == 'F': The matrix pencil (A,C) is already in generalized Schur form and S, R, QA, and ZA are given. == 'H': The matrix pencil (A,C) is given in generalized Hessenberg form and its Schur decomposition A = QA*S*ZA**T, C = QA*R*ZA**T will be computed.

*FACTB*FACTB is CHARACTER Specifies how the matrix pencil (B,D) is given. == 'N': The matrix pencil (B,D) is given as a general matrices and its Schur decomposition B = QB*U*ZB**T, D = QB*V*ZB**T will be computed. == 'F': The matrix pencil (B,D) is already in generalized Schur form and U, V, QB, and ZB are given. == 'H': The matrix pencil (B,D) is given in generalized Hessenberg form and its Schur decomposition B = QB*U*ZB**T, D = QB*V*ZB**T will be computed.

*TRANSA*TRANSA is CHARACTER Specifies the form of the system of equations with respect to A and C : == 'N': op1(A) = A == 'T': op1(A) = A**T

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

*SGN1*SGN1 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.

*SGN2*SGN2 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.

*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 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 factor S of the Schur decomposition of (A,C). If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix S of the Schur decomposition of (A,C). If FACT == 'H', the matrix A is an upper Hessenberg matrix of the generalized Hessenberg form (A,C) and it is overwritten with the (quasi-) upper triangular factor S of the Schur decomposition of (A,C).

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

*B*B is DOUBLE PRECISION array, dimension (LDB,N) If FACT == 'N', the matrix B is a general matrix and it is overwritten with the (quasi-) upper triangular factor U of the Schur decomposition of (B,D). If FACT == 'F', the matrix B contains its (quasi-) upper triangular matrix U of the Schur decomposition of (B,D). If FACT == 'H', the matrix B is an upper Hessenberg matrix of the generalized Hessenberg form (B,D) and it is overwritten with the (quasi-) upper triangular factor U of the Schur decomposition of (B,D).

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

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

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

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

*LDD*LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).

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

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

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

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

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

*LDQB*LDQB is INTEGER The leading dimension of the array QB. LDQB >= max(1,N).

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

*LDZB*LDZB is INTEGER The leading dimension of the array ZB. LDZB >= max(1,N).

*E*E is DOUBLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R.

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

*F*F is DOUBLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L.

*LDF*LDF is INTEGER The leading dimension of the array F. LDF >= 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 returned in LDWORK, if LDWORK == -1 on input. In this case no computations are performed.

*LDWORK*LDWORK is INTEGER If LDWORK == -1 on input 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: successful exit = 1: DHGGES 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_TGCSYLV_DAG

DLA_TGCSYLV_LEVEL3

DLA_TGCSYLV_L3_2S

DLA_TGCSYLV_L2_UNOPT

DLA_TGCSYLV_L2

DLA_TGCSYLV_L2_REORDER

DLA_TGCSYLV_L2_LOCAL_COPY_32

DLA_TGCSYLV_L2_LOCAL_COPY_64

DLA_TGCSYLV_L2_LOCAL_COPY_96

DLA_TGCSYLV_L2_LOCAL_COPY_128

DLA_TGCSYLV_L2_LOCAL_COPY

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **333** of file **dla_ggcsylv.f90**.

### subroutine dla_ggcsylv_dual (character, dimension(1) facta, character, dimension(1) factb, character, dimension(1) transa, character, dimension(1) transb, double precision sgn1, double precision sgn2, integer m, integer n, double precision, dimension(lda,*) a, integer lda, double precision, dimension(ldb, *) b, integer ldb, double precision, dimension(ldc, *) c, integer ldc, double precision, dimension(ldd,*) d, integer ldd, double precision, dimension(ldqa, *) qa, integer ldqa, double precision, dimension(ldza, *) za, integer ldza, double precision, dimension(ldqb, *) qb, integer ldqb, double precision, dimension(ldzb, *) zb, integer ldzb, double precision, dimension(lde, *) e, integer lde, double precision, dimension(ldf,*) f, integer ldf, double precision scale, double precision, dimension(*) work, integer ldwork, integer info)

Frontend for the solution of the dual Coupled Generalized Sylvester Equations.

**Purpose:**

DLA_GGCSYLV_DUAL_L3 solves a generalized coupled Sylvester equation of the following form op1(A)**T * R + op1(C)**T * L = SCALE * E (1) SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F where A and C are M-by-M matrices and B and D are N-by-N matrices. The right hand sides E, F and the solutions R, L are M-by-N matrices. The equation (1) is the dual to the generalized coupled Sylvester equation op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2) op1(C) * R + SGN2 * L * op2(D) = SCALE * F The equation (1) is the dual one to equation (2) with respect to the underlying linear system. Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields | kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E | Z X = | |*| | = | | | kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F | Regarding Z**T one obtains | kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E | Z**T X = | |*| | = | | | SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F | which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB are expressed in terms of the Sylvester equation (2).

**Parameters***FACTA*FACTA is CHARACTER Specifies how the matrix pencil (A,C) is given. == 'N': The matrix pencil (A,C) is given as a general matrices and its Schur decomposition A = QA*S*ZA**T, C = QA*R*ZA**T will be computed. == 'F': The matrix pencil (A,C) is already in generalized Schur form and S, R, QA, and ZA are given. == 'H': The matrix pencil (A,C) is given in generalized Hessenberg form and its Schur decomposition A = QA*S*ZA**T, C = QA*R*ZA**T will be computed.

*FACTB*FACTB is CHARACTER Specifies how the matrix pencil (B,D) is given. == 'N': The matrix pencil (B,D) is given as a general matrices and its Schur decomposition B = QB*U*ZB**T, D = QB*V*ZB**T will be computed. == 'F': The matrix pencil (B,D) is already in generalized Schur form and U, V, QB, and ZB are given. == 'H': The matrix pencil (B,D) is given in generalized Hessenberg form and its Schur decomposition B = QB*U*ZB**T, D = QB*V*ZB**T will be computed.

*TRANSA*TRANSA is CHARACTER Specifies the form of the system of equations with respect to A and C : == 'N': op1(A) = A == 'T': op1(A) = A**T

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

*SGN1*SGN1 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.

*SGN2*SGN2 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.

*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 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 factor S of the Schur decomposition of (A,C). If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix S of the Schur decomposition of (A,C). If FACT == 'H', the matrix A is an upper Hessenberg matrix of the generalized Hessenberg form (A,C) and it is overwritten with the (quasi-) upper triangular factor S of the Schur decomposition of (A,C).

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

*B*B is DOUBLE PRECISION array, dimension (LDB,N) If FACT == 'N', the matrix B is a general matrix and it is overwritten with the (quasi-) upper triangular factor U of the Schur decomposition of (B,D). If FACT == 'F', the matrix B contains its (quasi-) upper triangular matrix U of the Schur decomposition of (B,D). If FACT == 'H', the matrix B is an upper Hessenberg matrix of the generalized Hessenberg form (B,D) and it is overwritten with the (quasi-) upper triangular factor U of the Schur decomposition of (B,D).

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

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

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

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

*LDD*LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).

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

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

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

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

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

*LDQB*LDQB is INTEGER The leading dimension of the array QB. LDQB >= max(1,N).

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

*LDZB*LDZB is INTEGER The leading dimension of the array ZB. LDZB >= max(1,N).

*E*E is DOUBLE PRECISION array, dimension (LDE,N) On input, the matrix E contains the right hand side E. On output, the matrix E contains the solution R.

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

*F*F is DOUBLE PRECISION array, dimension (LDF,N) On input, the matrix F contains the right hand side F. On output, the matrix F contains the solution L.

*LDF*LDF is INTEGER The leading dimension of the array F. LDF >= 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)) If FACT == 'H', the matrix ZB is an empty M-by-M matrix on input and contains the right Schur vectors of (B,D) on output. 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 on input, 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: successful exit = 1: DHGGES 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_TGCSYLV_DUAL_DAG

DLA_TGCSYLV_DUAL_LEVEL3

DLA_TGCSYLV_DUAL_L3_2S

DLA_TGCSYLV_DUAL_L2

DLA_TGCSYLV_DUAL_L2_LOCAL_COPY_32

DLA_TGCSYLV_DUAL_L2_LOCAL_COPY_64

DLA_TGCSYLV_DUAL_L2_LOCAL_COPY_96

DLA_TGCSYLV_DUAL_L2_LOCAL_COPY_128

DLA_TGCSYLV_DUAL_L2_LOCAL_COPY

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **350** of file **dla_ggcsylv_dual.f90**.

### subroutine dla_ggcsylv_dual_refine (character, dimension(1) transa, character, dimension(1) transb, character, dimension(1) guess, double precision sgn1, double precision sgn2, integer m, integer n, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldb, *) b, integer ldb, double precision, dimension(ldc, *) c, integer ldc, double precision, dimension(ldd, *) d, integer ldd, double precision, dimension(ldr, *) r, integer ldr, double precision, dimension(ldl, *) l, integer ldl, double precision, dimension ( lde , * ) e, integer lde, double precision, dimension( ldf, *) f, integer ldf, double precision, dimension(ldas, *) as, integer ldas, double precision, dimension(ldbs,*) bs, integer ldbs, double precision, dimension(ldcs, *) cs, integer ldcs, double precision, dimension(ldds, *) ds, integer ldds, double precision, dimension(ldq, *) q, integer ldq, double precision, dimension(ldz, *) z, integer ldz, double precision, dimension(ldu, *) u, integer ldu, double precision, dimension(ldv, *) v, integer ldv, integer maxit, double precision tau, double precision, dimension(*) convlog, double precision, dimension(*) work, integer ldwork, integer info)

Iterative Refinement for the dual Coupled Generalized Sylvester Equations.

**Purpose:**

DLA_GGCSYLV_DUAL_REFINE solves a coupled generalized Sylvester equation of the following forms op1(A)**T * R + op1(C)**T * L = SCALE * E (1) SGN1 * R * op2(B)**T + SGN2 * L * op2(D)** T = SCALE * F where A and C are M-by-M matrices and B and D are N-by-N matrices. The right hand sides E, F and the solutions R, L are M-by-N matrices. The equation (1) is the dual to the generalized coupled Sylvester equation op1(A) * R + SGN1 * L * op2(B) = SCALE * E (2) op1(C) * R + SGN2 * L * op2(D) = SCALE * F The equation (1) is the dual one to equation (2) with respect to the underlying linear system. Let Z be the matrix formed by rewriting (2) into its Kronecker form. This yields | kron(I, op1(A)) SGN1*kron(op2(B)**T, I) | | Vec R | | Vec E | Z X = | |*| | = | | | kron(I, op1(C)) SGN2*kron(op2(D)**T, I) | | Vec L | | Vec F | Regarding Z**T one obtains | kron(I, op1(A)**T ) kron(I, op1(C)**T) | | Vec R | | Vec E | Z**T X = | |*| | = | | | SGN1*kron(op2(B), I) SGN2*kron(op2(D), I) | | Vec L | | Vec F | which belongs to the Sylvester equation (1). For this reason the parameters TRANSA and TRANSB are expressed in terms of the Sylvester equation (2). The pencils (A,C) and (B,D) need to be given in the original form as well as in their generalized Schur decomposition since both are required in the iterative refinement procedure.

**Parameters***TRANSA*TRANSA is CHARACTER Specifies the form of the system of equations with respect to A and C : == 'N': op1(A) = A == 'T': op1(A) = A**T

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

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

*SGN1*SGN1 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.

*SGN2*SGN2 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.

*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 DOUBLE PRECISION array, dimension (LDA,M) The array A contains the original matrix A defining the equation.

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

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

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

*C*C is DOUBLE PRECISION array, dimension (LDC,M) The array C contains the original matrix C defining the equation.

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

*D*D is DOUBLE PRECISION array, dimension (LDD,N) The array D contains the original matrix D defining the equation.

*LDD*LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).

*R*R is DOUBLE PRECISION array, dimension (LDR,N) On input, the array R contains the initial guess R0 for the first solution matrix. On output, the array R contains the refine solution matrix R.

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

*L*L is DOUBLE PRECISION array, dimension (LDL,N) On input, the array L contains the initial guess for the second solution matrix. On output, the array L contains the solution L.

*LDL*LDL is INTEGER The leading dimension of the array L. LDF >= max(1,M).

*E*E is DOUBLE PRECISION array, dimension (LDE,N) On input, the array E contains the right hand side E.

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

*F*F is DOUBLE PRECISION array, dimension (LDF,N) On input, the array F contains the right hand side F.

*LDF*LDF is INTEGER The leading dimension of the array F. LDF >= 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,N) The array BS contains the generalized Schur decomposition of the B.

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

*CS*CS is DOUBLE PRECISION array, dimension (LDCS,M) The array CS contains the generalized Schur decomposition of the C.

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

*DS*DS is DOUBLE PRECISION array, dimension (LDDS,N) The array DS contains the generalized Schur decomposition of the D.

*LDDS*LDDS is INTEGER The leading dimension of the array DS. LDDS >= max(1,N).

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

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

*U*U is DOUBLE PRECISION array, dimension (LDU,N) The array U contains the left generalized Schur vectors for (B,D) as returned by DGGES.

*LDU*LDU is INTEGER The leading dimension of the array U. LDU >= max(1,N).

*V*V is DOUBLE PRECISION array, dimension (LDV,N) The array V contains the right generalized Schur vectors for (B,D) as returned by DGGES.

*LDV*LDV is INTEGER The leading dimension of the array V. LDV >= max(1,N).

*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 of both equations 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_TGCSYLV_DUAL_DAG

DLA_TGCSYLV_DUAL_LEVEL3

DLA_TGCSYLV_DUAL_L3_2S

DLA_TGCSYLV_DUAL_L2

DLA_TGCSYLV_DUAL_L2_LOCAL_COPY_32

DLA_TGCSYLV_DUAL_L2_LOCAL_COPY_64

DLA_TGCSYLV_DUAL_L2_LOCAL_COPY_96

DLA_TGCSYLV_DUAL_L2_LOCAL_COPY_128

DLA_TGCSYLV_DUAL_L2_LOCAL_COPY

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **399** of file **dla_ggcsylv_dual_refine.f90**.

### subroutine dla_ggcsylv_refine (character, dimension(1) transa, character, dimension(1) transb, character, dimension(1) guess, double precision sgn1, double precision sgn2, integer m, integer n, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldb, *) b, integer ldb, double precision, dimension(ldc, *) c, integer ldc, double precision, dimension(ldd, *) d, integer ldd, double precision, dimension(ldr, *) r, integer ldr, double precision, dimension(ldl, *) l, integer ldl, double precision, dimension ( lde , * ) e, integer lde, double precision, dimension( ldf, *) f, integer ldf, double precision, dimension(ldas, *) as, integer ldas, double precision, dimension(ldbs,*) bs, integer ldbs, double precision, dimension(ldcs, *) cs, integer ldcs, double precision, dimension(ldds, *) ds, integer ldds, double precision, dimension(ldq, *) q, integer ldq, double precision, dimension(ldz, *) z, integer ldz, double precision, dimension(ldu, *) u, integer ldu, double precision, dimension(ldv, *) v, integer ldv, integer maxit, double precision tau, double precision, dimension(*) convlog, double precision, dimension(*) work, integer ldwork, integer info)

Iterative Refinement for the Coupled Generalized Sylvester Equations.

**Purpose:**

DLA_GGCSYLV_REFINE solves a coupled generalized Sylvester equation of the following forms op1(A) * R + SGN1 * L * op2(B) = E (1) op1(C) * R + SGN2 * L * op2(D) = F with iterative refinement, Thereby (A,C) is a M-by-M matrix pencil and (B,D) is a N-by-N matrix pencil. The right hand side (E,F) and the solution (R,L) are M-by-N matrices. The pencils (A,C) and (B,D) need to be given in the original form as well as in their generalized Schur decomposition since both are required in the iterative refinement procedure.

**Parameters***TRANSA**TRANSB**GUESS*GUESS is CHARACTER Specifies whether (R,L) contains an initial guess or nor not. = 'I': (R, L) contains an initial guess = 'N': No initial guess, (R,L) is set to zero at the begin of the iteration.

*SGN1*SGN1 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.

*SGN2*SGN2 is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the second equation.

*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 DOUBLE PRECISION array, dimension (LDA,M) The array A contains the original matrix A defining the equation.

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

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

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

*C*C is DOUBLE PRECISION array, dimension (LDC,M) The array C contains the original matrix C defining the equation.

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

*D*D is DOUBLE PRECISION array, dimension (LDD,N) The array D contains the original matrix D defining the equation.

*LDD*LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).

*R*R is DOUBLE PRECISION array, dimension (LDR,N) On input, the array R contains the initial guess R0 for the first solution matrix. On output, the array R contains the refine solution matrix R.

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

*L*L is DOUBLE PRECISION array, dimension (LDL,N) On input, the array L contains the initial guess for the second solution matrix. On output, the array L contains the solution L.

*LDL*LDL is INTEGER The leading dimension of the array L. LDF >= max(1,M).

*E*E is DOUBLE PRECISION array, dimension (LDE,N) On input, the array E contains the right hand side E.

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

*F*F is DOUBLE PRECISION array, dimension (LDF,N) On input, the array F contains the right hand side F.

*LDF*LDF is INTEGER The leading dimension of the array F. LDF >= 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,N) The array BS contains the generalized Schur decomposition of the B.

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

*CS*CS is DOUBLE PRECISION array, dimension (LDCS,M) The array CS contains the generalized Schur decomposition of the C.

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

*DS*DS is DOUBLE PRECISION array, dimension (LDDS,N) The array DS contains the generalized Schur decomposition of the D.

*LDDS*LDDS is INTEGER The leading dimension of the array DS. LDDS >= max(1,N).

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

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

*U*U is DOUBLE PRECISION array, dimension (LDU,N) The array U contains the left generalized Schur vectors for (B,D) as returned by DGGES.

*LDU*LDU is INTEGER The leading dimension of the array U. LDU >= max(1,N).

*V*V is DOUBLE PRECISION array, dimension (LDV,N) The array V contains the right generalized Schur vectors for (B,D) as returned by DGGES.

*LDV*LDV is INTEGER The leading dimension of the array V. LDV >= max(1,N).

*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 of both equations 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_TGCSYLV_DAG

DLA_TGCSYLV_LEVEL3

DLA_TGCSYLV_L3_2S

DLA_TGCSYLV_L2_UNOPT

DLA_TGCSYLV_L2

DLA_TGCSYLV_L2_REORDER

DLA_TGCSYLV_L2_LOCAL_COPY_32

DLA_TGCSYLV_L2_LOCAL_COPY_64

DLA_TGCSYLV_L2_LOCAL_COPY_96

DLA_TGCSYLV_L2_LOCAL_COPY_128

DLA_TGCSYLV_L2_LOCAL_COPY

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **381** of file **dla_ggcsylv_refine.f90**.

### subroutine dla_ggsylv (character, dimension(1) facta, character, dimension(1) factb, character, dimension(1) transa, character, dimension(1) transb, double precision sgn, integer m, integer n, double precision, dimension(lda,*) a, integer lda, double precision, dimension(ldb, *) b, integer ldb, double precision, dimension(ldc, *) c, integer ldc, double precision, dimension(ldd,*) d, integer ldd, double precision, dimension(ldqa, *) qa, integer ldqa, double precision, dimension(ldza, *) za, integer ldza, double precision, dimension(ldqb, *) qb, integer ldqb, double precision, dimension(ldzb, *) zb, integer ldzb, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer ldwork, integer info)

Frontend for the solution of Generalized Sylvester Equations.

**Purpose:**

DLA_GGSYLV solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + op1(C) * X * op2(D) = SCALE * Y (1) or op1(A) * X * op2(B) - op1(C) * X * op2(D) = SCALE * Y (2) where (A,C) is a M-by-M matrix pencil and (B,D) is a N-by-N matrix pencil. The right hand side Y and the solution X M-by-N matrices. The pencils (A,C) and (B,D) can be either given as general unreduced matrices, as generalized Hessenberg form, or in terms of their generalized Schur decomposition. If they are given as general matrices or as a generalized Hessenberg form their generalized Schur decomposition will be computed.

**Parameters***FACTA*FACTA is CHARACTER Specifies how the matrix pencil (A,C) is given. == 'N': The matrix pencil (A,C) is given as a general matrices and its Schur decomposition A = QA*S*ZA**T, C = QA*R*ZA**T will be computed. == 'F': The matrix pencil (A,C) is already in generalized Schur form and S, R, QA, and ZA are given. == 'H': The matrix pencil (A,C) is given in generalized Hessenberg form and its Schur decomposition A = QA*S*ZA**T, C = QA*R*ZA**T will be computed.

*FACTB*FACTB is CHARACTER Specifies how the matrix pencil (B,D) is given. == 'N': The matrix pencil (B,D) is given as a general matrices and its Schur decomposition B = QB*U*ZB**T, D = QB*V*ZB**T will be computed. == 'F': The matrix pencil (B,D) is already in generalized Schur form and U, V, QB, and ZB are given. == 'H': The matrix pencil (B,D) is given in generalized Hessenberg form and its Schur decomposition B = QB*U*ZB**T, D = QB*V*ZB**T will be computed.

*TRANSA**TRANSB**SGN*SGN is DOUBLE PRECISION, 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 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 factor S of the Schur decomposition of (A,C). If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix S of the Schur decomposition of (A,C). If FACT == 'H', the matrix A is an upper Hessenberg matrix of the generalized Hessenberg form (A,C) and it is overwritten with the (quasi-) upper triangular factor S of the Schur decomposition of (A,C).

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

*B*B is DOUBLE PRECISION array, dimension (LDB,N) If FACT == 'N', the matrix B is a general matrix and it is overwritten with the (quasi-) upper triangular factor U of the Schur decomposition of (B,D). If FACT == 'F', the matrix B contains its (quasi-) upper triangular matrix U of the Schur decomposition of (B,D). If FACT == 'H', the matrix B is an upper Hessenberg matrix of the generalized Hessenberg form (B,D) and it is overwritten with the (quasi-) upper triangular factor U of the Schur decomposition of (B,D).

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

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

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

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

*LDD*LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).

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

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

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

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

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

*LDQB*LDQB is INTEGER The leading dimension of the array QB. LDQB >= max(1,N).

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

*LDZB*LDZB is INTEGER The leading dimension of the array ZB. LDZB >= max(1,N).

*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 M-by-N 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: DHGGES 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_TGSYLV_DAG

DLA_TGSYLV_L3_COLWISE

DLA_TGSYLV_L3_2S

DLA_TGSYLV_L2_REORDER

DLA_TGSYLV_L2

DLA_TGSYLV_L2_COLWISE

DLA_TGSYLV_L2_LOCAL_COPY_32

DLA_TGSYLV_L2_LOCAL_COPY_64

DLA_TGSYLV_L2_LOCAL_COPY_96

DLA_TGSYLV_L2_LOCAL_COPY_128

DLA_TGSYLV_L2_LOCAL_COPY

DLA_TGSYLV_GARDINER_LAUB

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **322** of file **dla_ggsylv.f90**.

### subroutine dla_ggsylv_refine (character, dimension(1) transa, character, dimension(1) transb, character, dimension(1) guess, double precision sgn, integer m, integer n, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldb, *) b, integer ldb, double precision, dimension(ldc, *) c, integer ldc, double precision, dimension(ldd, *) d, integer ldd, 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(ldcs, *) cs, integer ldcs, double precision, dimension(ldds, *) ds, integer ldds, double precision, dimension(ldq, *) q, integer ldq, double precision, dimension(ldz, *) z, integer ldz, double precision, dimension(ldu, *) u, integer ldu, double precision, dimension(ldv, *) v, integer ldv, integer maxit, double precision tau, double precision, dimension(*) convlog, double precision, dimension(*) work, integer ldwork, integer info)

Iterative Refinement for the Generalized Sylvester Equations.

**Purpose:**

DLA_GGSYLV_REFINE solves a generalized Sylvester equation of the following forms op1(A) * X * op2(B) + SGN * op1(C) * X * op2(D) = Y (1) with iterative refinement, Thereby (A,C) is a M-by-M matrix pencil and (B,D) is a N-by-N matrix pencil. The right hand side Y and the solution X are M-by-N matrices. The pencils (A,C) and (B,D) need to be given in the original form as well as in their generalized Schur decomposition since both are required in the iterative refinement procedure.

**Parameters***TRANSA**TRANSB**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.

*SGN*SGN is DOUBLE PRECISION, allowed values: +/-1 Specifies the sign between in the first equation.

*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 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,N) 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,N).

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

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

*D*D is DOUBLE PRECISION array, dimension (LDD,N) The array D contains the original matrix D defining the eqaution.

*LDD*LDD is INTEGER The leading dimension of the array D. LDD >= max(1,N).

*X*X is DOUBLE PRECISION array, dimension (LDX,N) 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,N) 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,N) The array BS contains the generalized Schur decomposition of the B.

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

*CS*CS is DOUBLE PRECISION array, dimension (LDCS,M) The array CS contains the generalized Schur decomposition of the C.

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

*DS*DS is DOUBLE PRECISION array, dimension (LDDS,N) The array DS contains the generalized Schur decomposition of the D.

*LDDS*LDDS is INTEGER The leading dimension of the array DS. LDDS >= max(1,N).

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

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

*U*U is DOUBLE PRECISION array, dimension (LDU,N) The array U contains the left generalized Schur vectors for (B,D) as returned by DGGES.

*LDU*LDU is INTEGER The leading dimension of the array U. LDU >= max(1,N).

*V*V is DOUBLE PRECISION array, dimension (LDV,N) The array V contains the right generalized Schur vectors for (B,D) as returned by DGGES.

*LDV*LDV is INTEGER The leading dimension of the array V. LDV >= max(1,N).

*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**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_TGSYLV_DAG

DLA_TGSYLV_LEVEL3

DLA_TGSYLV_L3_2S

DLA_TGSYLV_L2_UNOPT

DLA_TGSYLV_L2

DLA_TGSYLV_L2_REORDER

DLA_TGSYLV_L2_LOCAL_COPY_32

DLA_TGSYLV_L2_LOCAL_COPY_64

DLA_TGSYLV_L2_LOCAL_COPY_96

DLA_TGSYLV_L2_LOCAL_COPY_128

DLA_TGSYLV_L2_LOCAL_COPY

DLA_TGSYLV_GARDINER_LAUB

**Author**Martin Koehler, MPI Magdeburg

**Date**January 2024

Definition at line **349** of file **dla_ggsylv_refine.f90**.

## Author

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