# cggsylv - Man Page

## Name

cggsylv — C-Interface

— C-Interface for generalized Sylvester equations.

## Synopsis

### Functions

void mepack_double_ggcsylv (const char *FACTA, const char *FACTB, const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *QA, int LDQA, double *ZA, int LDZA, double *QB, int LDQB, double *ZB, int LDZB, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, size_t LDWORK, int *INFO)
Frontend for the solution of Coupled Generalized Sylvester Equations.
void mepack_single_ggcsylv (const char *FACTA, const char *FACTB, const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *QA, int LDQA, float *ZA, int LDZA, float *QB, int LDQB, float *ZB, int LDZB, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, size_t LDWORK, int *INFO)
Frontend for the solution of Coupled Generalized Sylvester Equations.
void mepack_double_ggcsylv_dual (const char *FACTA, const char *FACTB, const char *TRANSA, const char *TRANSB, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *QA, int LDQA, double *ZA, int LDZA, double *QB, int LDQB, double *ZB, int LDZB, double *E, int LDE, double *F, int LDF, double *SCALE, double *WORK, size_t LDWORK, int *INFO)
Frontend for the solution of the dual Coupled Generalized Sylvester Equations.
void mepack_single_ggcsylv_dual (const char *FACTA, const char *FACTB, const char *TRANSA, const char *TRANSB, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *QA, int LDQA, float *ZA, int LDZA, float *QB, int LDQB, float *ZB, int LDZB, float *E, int LDE, float *F, int LDF, float *SCALE, float *WORK, size_t LDWORK, int *INFO)
Frontend for the solution of the dual Coupled Generalized Sylvester Equations.
void mepack_double_ggsylv (const char *FACTA, const char *FACTB, const char *TRANSA, const char *TRANSB, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *QA, int LDQA, double *ZA, int LDZA, double *QB, int LDQB, double *ZB, int LDZB, double *X, int LDX, double *SCALE, double *WORK, size_t LDWORK, int *INFO)
Frontend for the solution of Generalized Sylvester Equations.
void mepack_single_ggsylv (const char *FACTA, const char *FACTB, const char *TRANSA, const char *TRANSB, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *QA, int LDQA, float *ZA, int LDZA, float *QB, int LDQB, float *ZB, int LDZB, float *X, int LDX, float *SCALE, float *WORK, size_t LDWORK, int *INFO)
Frontend for the solution of Generalized Sylvester Equations.
void mepack_double_ggcsylv_refine (const char *TRANSA, const char *TRANSB, const char *GUESS, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *R, int LDR, double *L, int LDL, double *E, int LDE, double *F, int LDF, double *AS, int LDAS, double *BS, int LDBS, double *CS, int LDCS, double *DS, int LDDS, double *Q, int LDQ, double *Z, int LDZ, double *U, int LDU, double *V, int LDV, int *MAXIT, double *TAU, double *CONVLOG, double *WORK, size_t LDWORK, int *INFO)
Iterative Refinement for the Coupled Generalized Sylvester Equations.
void mepack_single_ggcsylv_refine (const char *TRANSA, const char *TRANSB, const char *GUESS, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *R, int LDR, float *L, int LDL, float *E, int LDE, float *F, int LDF, float *AS, int LDAS, float *BS, int LDBS, float *CS, int LDCS, float *DS, int LDDS, float *Q, int LDQ, float *Z, int LDZ, float *U, int LDU, float *V, int LDV, int *MAXIT, float *TAU, float *CONVLOG, float *WORK, size_t LDWORK, int *INFO)
Iterative Refinement for the Coupled Generalized Sylvester Equations.
void mepack_double_ggcsylv_dual_refine (const char *TRANSA, const char *TRANSB, const char *GUESS, double SGN1, double SGN2, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *R, int LDR, double *L, int LDL, double *E, int LDE, double *F, int LDF, double *AS, int LDAS, double *BS, int LDBS, double *CS, int LDCS, double *DS, int LDDS, double *Q, int LDQ, double *Z, int LDZ, double *U, int LDU, double *V, int LDV, int *MAXIT, double *TAU, double *CONVLOG, double *WORK, size_t LDWORK, int *INFO)
Iterative Refinement for the dual Coupled Generalized Sylvester Equations.
void mepack_single_ggcsylv_dual_refine (const char *TRANSA, const char *TRANSB, const char *GUESS, float SGN1, float SGN2, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *R, int LDR, float *L, int LDL, float *E, int LDE, float *F, int LDF, float *AS, int LDAS, float *BS, int LDBS, float *CS, int LDCS, float *DS, int LDDS, float *Q, int LDQ, float *Z, int LDZ, float *U, int LDU, float *V, int LDV, int *MAXIT, float *TAU, float *CONVLOG, float *WORK, size_t LDWORK, int *INFO)
Iterative Refinement for the dual Coupled Generalized Sylvester Equations.
void mepack_double_ggsylv_refine (const char *TRANSA, const char *TRANSB, const char *GUESS, double SGN, int M, int N, double *A, int LDA, double *B, int LDB, double *C, int LDC, double *D, int LDD, double *X, int LDX, double *Y, int LDY, double *AS, int LDAS, double *BS, int LDBS, double *CS, int LDCS, double *DS, int LDDS, double *Q, int LDQ, double *Z, int LDZ, double *U, int LDU, double *V, int LDV, int *MAXIT, double *TAU, double *CONVLOG, double *WORK, size_t LDWORK, int *INFO)
Iterative Refinement for the Generalized Sylvester Equations.
void mepack_single_ggsylv_refine (const char *TRANSA, const char *TRANSB, const char *GUESS, float SGN, int M, int N, float *A, int LDA, float *B, int LDB, float *C, int LDC, float *D, int LDD, float *X, int LDX, float *Y, int LDY, float *AS, int LDAS, float *BS, int LDBS, float *CS, int LDCS, float *DS, int LDDS, float *Q, int LDQ, float *Z, int LDZ, float *U, int LDU, float *V, int LDV, int *MAXIT, float *TAU, float *CONVLOG, float *WORK, size_t LDWORK, int *INFO)
Iterative Refinement for the Generalized Sylvester Equations.

## Detailed Description

C-Interface for generalized Sylvester equations.

The Fortran routines to solve the generalized Sylvester equation with arbitrary coefficients are wrapped in C to provide an easier access to them. All wrapper routines are direct wrappers to the corresponding Fortran subroutines without sanity checks. These are performed by the Fortran routines. The only difference is that the C interface does not allow LAPACK-like work_space queries. For this purpose the mepack_memory_frontend function needs to be used.

## Function Documentation

### void mepack_double_ggcsylv (const char * FACTA, const char * FACTB, const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * QA, int LDQA, double * ZA, int LDZA, double * QB, int LDQB, double * ZB, int LDZB, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, size_t LDWORK, int * INFO)

Frontend for the solution of Coupled Generalized Sylvester Equations.

Purpose:

mepack_double_ggcsylv solves a 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 matrix pencils (A,C)
and (B,D) can be either given as general unreduced matrices or in terms of
their generalized Schur decomposition. If they are given as general matrices
their generalized Schur decomposition will be computed..fi

Remarks
This function is a wrapper for dla_ggcsylv.

See also
dla_ggcsylv

Parameters
FACTA

FACTA is String
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.

FACTB

         FACTB is String
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.

TRANSA

         TRANSA is String
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 String
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).

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).

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).

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).

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).

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).

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).

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).

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 optimal workspace is given by \ref mepack_memory_frontend.

LDWORK

         LDWORK is INTEGER
Size of the workspace for the algorithm counted in floating point numbers of the actual precision.
The C interface does not support the workspace query by setting LDWORK == -1 on input. In this case,
the \ref mepack_memory_frontend function have to be used.

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
Attention

The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 295 of file cgsylv.c.

### void mepack_double_ggcsylv_dual (const char * FACTA, const char * FACTB, const char * TRANSA, const char * TRANSB, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * QA, int LDQA, double * ZA, int LDZA, double * QB, int LDQB, double * ZB, int LDZB, double * E, int LDE, double * F, int LDF, double * SCALE, double * WORK, size_t LDWORK, int * INFO)

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

Purpose:

mepack_double_ggcsylv_dual 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)..fi

Remarks
This function is a wrapper for dla_ggcsylv_dual.

See also
dla_ggcsylv_dual

Parameters
FACTA

FACTA is String
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.

FACTB

         FACTB is String
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.

TRANSA

         TRANSA is String
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 String
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).

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).

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).

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).

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).

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).

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).

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).

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 optimal workspace is given by \ref mepack_memory_frontend.

LDWORK

         LDWORK is INTEGER
Size of the workspace for the algorithm counted in floating point numbers of the actual precision.
The C interface does not support the workspace query by setting LDWORK == -1 on input. In this case,
the \ref mepack_memory_frontend function have to be used.

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
Attention

The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 313 of file cgsylv_dual.c.

### void mepack_double_ggcsylv_dual_refine (const char * TRANSA, const char * TRANSB, const char * GUESS, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * R, int LDR, double * L, int LDL, double * E, int LDE, double * F, int LDF, double * AS, int LDAS, double * BS, int LDBS, double * CS, int LDCS, double * DS, int LDDS, double * Q, int LDQ, double * Z, int LDZ, double * U, int LDU, double * V, int LDV, int * MAXIT, double * TAU, double * CONVLOG, double * WORK, size_t LDWORK, int * INFO)

Iterative Refinement for the dual Coupled Generalized Sylvester Equations.

Purpose:

mepack_double_ggcsylv_dual_refine 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)..fi

Remarks
This function is a wrapper for dla_ggcsylv_dual_refine.

See also
dla_ggcsylv_dual_refine

Parameters
TRANSA

TRANSA is String
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 String
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 String
Specifies whether (R,L) contains an initial guess on input or not.
= 'I': (R,L) contains an initial guess for the solution
== 'N': No initial guess is provided. (R,L) are set to zero.

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 for the first solution.
On output, the array R contains the solution 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.
On output, the array L contains the second solution.

LDL

         LDL is INTEGER
The leading dimension of the array L.  LDL >= 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 (LDC,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, 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.

LDWORK

         LDWORK is INTEGER
Size of the workspace for the algorithm. This can be determined by a call \ref mepack_memory_frontend.

INFO

         INFO is INTEGER
== 0:  Success
> 0:  Iteration failed in step INFO
< 0:  if INFO == -i, the i-Th argument had an illegal value
Attention

The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 374 of file cgsylv_dual.c.

### void mepack_double_ggcsylv_refine (const char * TRANSA, const char * TRANSB, const char * GUESS, double SGN1, double SGN2, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * R, int LDR, double * L, int LDL, double * E, int LDE, double * F, int LDF, double * AS, int LDAS, double * BS, int LDBS, double * CS, int LDCS, double * DS, int LDDS, double * Q, int LDQ, double * Z, int LDZ, double * U, int LDU, double * V, int LDV, int * MAXIT, double * TAU, double * CONVLOG, double * WORK, size_t LDWORK, int * INFO)

Iterative Refinement for the Coupled Generalized Sylvester Equations.

Purpose:

mepack_double_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 matrix 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.
Remarks

This function is a wrapper for dla_ggcsylv_refine

See also

dla_ggcsylv_refine

Parameters

TRANSA

         TRANSA is String
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 String
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 String
Specifies whether (R,L) contains an initial guess on input or not.
= 'I': (R,L) contains an initial guess for the solution
== 'N': No initial guess is provided. (R,L) are set to zero.

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 for the first solution.
On output, the array R contains the solution 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.
On output, the array L contains the second solution.

LDL

         LDL is INTEGER
The leading dimension of the array L.  LDL >= 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, 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.

LDWORK

         LDWORK is INTEGER
Size of the workspace for the algorithm. This can be determined by a call \ref mepack_memory_frontend.

INFO

         INFO is INTEGER
== 0:  Success
> 0:  Iteration failed in step INFO
< 0:  if INFO == -i, the i-Th argument had an illegal value
Attention

The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 353 of file cgsylv.c.

### void mepack_double_ggsylv (const char * FACTA, const char * FACTB, const char * TRANSA, const char * TRANSB, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * QA, int LDQA, double * ZA, int LDZA, double * QB, int LDQB, double * ZB, int LDZB, double * X, int LDX, double * SCALE, double * WORK, size_t LDWORK, int * INFO)

Frontend for the solution of Generalized Sylvester Equations.

Purpose:

mepack_double_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 matrix pencils (A,C)
and (B,D) can be either given as general unreduced matrices or in terms of
their generalized Schur decomposition. If they are given as general matrices
their generalized Schur decomposition will be computed..fi

Remarks
This function is a wrapper for dla_ggsylv.

See also
dla_ggsylv

Parameters
FACTA

FACTA is String
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.

FACTB

         FACTB is String
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.

TRANSA

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

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).

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).

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).

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).

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).

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).

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).

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).

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 optimal workspace is given by \ref mepack_memory_frontend.

LDWORK

         LDWORK is INTEGER
Size of the workspace for the algorithm counted in floating point numbers of the actual precision.
The C interface does not support the workspace query by setting LDWORK == -1 on input. In this case,
the \ref mepack_memory_frontend function have to be used.

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
Attention

The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 280 of file ggsylv.c.

### void mepack_double_ggsylv_refine (const char * TRANSA, const char * TRANSB, const char * GUESS, double SGN, int M, int N, double * A, int LDA, double * B, int LDB, double * C, int LDC, double * D, int LDD, double * X, int LDX, double * Y, int LDY, double * AS, int LDAS, double * BS, int LDBS, double * CS, int LDCS, double * DS, int LDDS, double * Q, int LDQ, double * Z, int LDZ, double * U, int LDU, double * V, int LDV, int * MAXIT, double * TAU, double * CONVLOG, double * WORK, size_t LDWORK, int * INFO)

Iterative Refinement for the Generalized Sylvester Equations.

Purpose:

mepack_double_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 matrix 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.
Remarks

This function is a wrapper for dla_ggsylv_refine

See also

dla_ggsylv_refine

Parameters

TRANSA

         TRANSA is String
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 String
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 String
Specifies whether X  contains an initial guess on input or not.
= 'I': X contains an initial guess for the solution
== 'N': No initial guess is provided. X is set to zero.

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 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 B.  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).

X

         X is DOUBLE PRECISION array, dimension (LDX,N)
On input, the array X contains the initial guess.
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.

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, 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.

LDWORK

         LDWORK is INTEGER
Size of the workspace for the algorithm. This can be determined by a call \ref mepack_memory_frontend.

INFO

         INFO is INTEGER
== 0:  Success
> 0:  Iteration failed in step INFO
< 0:  if INFO == -i, the i-Th argument had an illegal value
Attention

The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 324 of file ggsylv.c.

### void mepack_single_ggcsylv (const char * FACTA, const char * FACTB, const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * QA, int LDQA, float * ZA, int LDZA, float * QB, int LDQB, float * ZB, int LDZB, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, size_t LDWORK, int * INFO)

Frontend for the solution of Coupled Generalized Sylvester Equations.

Purpose:

mepack_single_ggcsylv solves a 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) M-by-N matrices. The matrix pencils (A,C)
and (B,D) can be either given as general unreduced matrices or in terms of
their generalized Schur decomposition. If they are given as general matrices
their generalized Schur decomposition will be computed..fi

Remarks
This function is a wrapper for sla_ggcsylv.

See also
sla_ggcsylv

Parameters
FACTA

FACTA is String
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.

FACTB

         FACTB is String
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.

TRANSA

         TRANSA is String
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 String
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).

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).

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).

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).

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).

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).

LDZA

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

QB

         QB is DOUBLE PRECISION array, dimension (LDQB,M)
If FACT == 'N', the matrix QB is an empty M-by-M 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).

LDQB

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

ZB

         ZB is DOUBLE PRECISION array, dimension (LDZB,M)
If FACT == 'N', the matrix ZB is an empty M-by-M 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).

LDZB

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

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 X.  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 X.  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 optimal workspace is given by \ref mepack_memory_frontend.

LDWORK

         LDWORK is INTEGER
Size of the workspace for the algorithm counted in floating point numbers of the actual precision.
The C interface does not support the workspace query by setting LDWORK == -1 on input. In this case,
the \ref mepack_memory_frontend function have to be used.

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
Attention

The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 598 of file cgsylv.c.

### void mepack_single_ggcsylv_dual (const char * FACTA, const char * FACTB, const char * TRANSA, const char * TRANSB, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * QA, int LDQA, float * ZA, int LDZA, float * QB, int LDQB, float * ZB, int LDZB, float * E, int LDE, float * F, int LDF, float * SCALE, float * WORK, size_t LDWORK, int * INFO)

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

Purpose:

mepack_single_ggcsylv_dual solves a generalized coupled  Sylvester equation of the following form

op1(A)**T * R  + op1(C)**T * R               = 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)..fi

Remarks
This function is a wrapper for sla_ggcsylv_dual.

See also
sla_ggcsylv_dual

Parameters
FACTA

FACTA is String
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.

FACTB

         FACTB is String
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.

TRANSA

         TRANSA is String
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 String
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 SINGLE PRECISION, allowed values: +/-1
Specifies the sign between in the first equation.

SGN2

         SGN2 is SINGLE 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 SINGLE 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).

LDA

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

B

         B is SINGLE 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).

LDB

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

C

         C is SINGLE 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).

LDC

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

D

         D is SINGLE 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).

LDD

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

QA

         QA is SINGLE 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).

LDQA

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

ZA

         ZA is SINGLE 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).

LDZA

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

QB

         QB is SINGLE PRECISION array, dimension (LDQB,M)
If FACT == 'N', the matrix QB is an empty M-by-M 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).

LDQB

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

ZB

         ZB is SINGLE PRECISION array, dimension (LDZB,M)
If FACT == 'N', the matrix ZB is an empty M-by-M 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).

LDZB

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

E

         E is SINGLE 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 X.  LDE >= max(1,M).

F

         F is SINGLE 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 X.  LDF >= max(1,M).

SCALE

         SCALE is SINGLE 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 SINGLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory_frontend.

LDWORK

         LDWORK is INTEGER
Size of the workspace for the algorithm counted in floating point numbers of the actual precision.
The C interface does not support the workspace query by setting LDWORK == -1 on input. In this case,
the \ref mepack_memory_frontend function have to be used.

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
Attention

The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 633 of file cgsylv_dual.c.

### void mepack_single_ggcsylv_dual_refine (const char * TRANSA, const char * TRANSB, const char * GUESS, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * R, int LDR, float * L, int LDL, float * E, int LDE, float * F, int LDF, float * AS, int LDAS, float * BS, int LDBS, float * CS, int LDCS, float * DS, int LDDS, float * Q, int LDQ, float * Z, int LDZ, float * U, int LDU, float * V, int LDV, int * MAXIT, float * TAU, float * CONVLOG, float * WORK, size_t LDWORK, int * INFO)

Iterative Refinement for the dual Coupled Generalized Sylvester Equations.

Purpose:

mepack_single_ggcsylv_dual_refine solves a generalized coupled  Sylvester equation of the following form

op1(A)**T * R  + op1(C)**T * R               = 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)..fi

Remarks
This function is a wrapper for sla_ggcsylv_dual_refine.

See also
sla_ggcsylv_dual_refine

Parameters
TRANSA

TRANSA is String
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 String
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 String
Specifies whether (R,L) contains an initial guess on input or not.
= 'I': (R,L) contains an initial guess for the solution
== 'N': No initial guess is provided. (R,L) are set to zero.

SGN1

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

SGN2

         SGN2 is SINGLE 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 SINGLE 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 SINGLE 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 SINGLE 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 SINGLE PRECISION array, dimension (LDA,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 SINGLE PRECISION array, dimension (LDR,N)
On input, the array R contains the initial guess for the first solution.
On output, the array R contains the solution R.

LDR

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

L

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

LDL

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

E

         E is SINGLE 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 SINGLE 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 SINGLE 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 SINGLE PRECISION array, dimension (LDBS,N)
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,N).

CS

         CS is SINGLE 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 SINGLE 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.  LDAS >= max(1,N).

Q

         Q is SINGLE 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 SINGLE 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 SINGLE 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 SINGLE 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, MAXIT <= 100
On exit, MAXIT contains the number of iteration steps taken by the algorithm.

TAU

         TAU is SINGLE 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 SINGLE 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 SINGLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm.

LDWORK

         LDWORK is INTEGER
Size of the workspace for the algorithm. This can be determined by a call \ref mepack_memory_frontend.

INFO

         INFO is INTEGER
== 0:  Success
> 0:  Iteration failed in step INFO
< 0:  if INFO == -i, the i-Th argument had an illegal value
Attention

The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 763 of file cgsylv_dual.c.

### void mepack_single_ggcsylv_refine (const char * TRANSA, const char * TRANSB, const char * GUESS, float SGN1, float SGN2, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * R, int LDR, float * L, int LDL, float * E, int LDE, float * F, int LDF, float * AS, int LDAS, float * BS, int LDBS, float * CS, int LDCS, float * DS, int LDDS, float * Q, int LDQ, float * Z, int LDZ, float * U, int LDU, float * V, int LDV, int * MAXIT, float * TAU, float * CONVLOG, float * WORK, size_t LDWORK, int * INFO)

Iterative Refinement for the Coupled Generalized Sylvester Equations.

Purpose:

mepack_single_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 matrix 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.
Remarks

This function is a wrapper for sla_ggcsylv_refine

See also

sla_ggcsylv_refine

Parameters

TRANSA

         TRANSA is String
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 String
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 String
Specifies whether (R,L) contains an initial guess on input or not.
= 'I': (R,L) contains an initial guess for the solution
== 'N': No initial guess is provided. (R,L) are set to zero.

SGN1

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

SGN2

         SGN2 is SINGLE 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 SINGLE 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 SINGLE 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 SINGLE 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 SINGLE PRECISION array, dimension (LDA,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 SINGLE PRECISION array, dimension (LDR,N)
On input, the array R contains the initial guess for the first solution.
On output, the array R contains the solution R.

LDR

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

L

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

LDL

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

E

         E is SINGLE 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 SINGLE 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 SINGLE 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 SINGLE PRECISION array, dimension (LDBS,N)
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,N).

CS

         CS is SINGLE 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 SINGLE 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.  LDAS >= max(1,N).

Q

         Q is SINGLE 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 SINGLE 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 SINGLE 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 SINGLE 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, MAXIT <= 100
On exit, MAXIT contains the number of iteration steps taken by the algorithm.

TAU

         TAU is SINGLE 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 SINGLE 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 SINGLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm.

LDWORK

         LDWORK is INTEGER
Size of the workspace for the algorithm. This can be determined by a call \ref mepack_memory_frontend.

INFO

         INFO is INTEGER
== 0:  Success
> 0:  Iteration failed in step INFO
< 0:  if INFO == -i, the i-Th argument had an illegal value
Attention

The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 722 of file cgsylv.c.

### void mepack_single_ggsylv (const char * FACTA, const char * FACTB, const char * TRANSA, const char * TRANSB, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * QA, int LDQA, float * ZA, int LDZA, float * QB, int LDQB, float * ZB, int LDZB, float * X, int LDX, float * SCALE, float * WORK, size_t LDWORK, int * INFO)

Frontend for the solution of Generalized Sylvester Equations.

Purpose:

mepack_single_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 matrix pencils (A,C)
and (B,D) can be either given as general unreduced matrices or in terms of
their generalized Schur decomposition. If they are given as general matrices
their generalized Schur decomposition will be computed..fi

Remarks
This function is a wrapper for sla_ggsylv.

See also
sla_ggsylv

Parameters
FACTA

FACTA is String
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.

FACTB

         FACTB is String
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.

TRANSA

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

SGN

         SGN is SINGLE 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 SINGLE 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).

LDA

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

B

         B is SINGLE 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).

LDB

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

C

         C is SINGLE 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).

LDC

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

D

         D is SINGLE 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).

LDD

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

QA

         QA is SINGLE 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).

LDQA

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

ZA

         ZA is SINGLE 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).

LDZA

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

QB

         QB is SINGLE PRECISION array, dimension (LDQB,M)
If FACT == 'N', the matrix QB is an empty M-by-M 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).

LDQB

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

ZB

         ZB is SINGLE PRECISION array, dimension (LDZB,M)
If FACT == 'N', the matrix ZB is an empty M-by-M 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).

LDZB

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

X

         X is SINGLE 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 SINGLE 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 SINGLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal workspace is given by \ref mepack_memory_frontend.

LDWORK

         LDWORK is INTEGER
Size of the workspace for the algorithm counted in floating point numbers of the actual precision.
The C interface does not support the workspace query by setting LDWORK == -1 on input. In this case,
the \ref mepack_memory_frontend function have to be used.

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
Attention

The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 568 of file ggsylv.c.

### void mepack_single_ggsylv_refine (const char * TRANSA, const char * TRANSB, const char * GUESS, float SGN, int M, int N, float * A, int LDA, float * B, int LDB, float * C, int LDC, float * D, int LDD, float * X, int LDX, float * Y, int LDY, float * AS, int LDAS, float * BS, int LDBS, float * CS, int LDCS, float * DS, int LDDS, float * Q, int LDQ, float * Z, int LDZ, float * U, int LDU, float * V, int LDV, int * MAXIT, float * TAU, float * CONVLOG, float * WORK, size_t LDWORK, int * INFO)

Iterative Refinement for the Generalized Sylvester Equations.

Purpose:

mepack_single_gesylv_refine solves a coupled 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 matrix 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.
Remarks

This function is a wrapper for sla_ggsylv_refine

See also

sla_ggsylv_refine

Parameters

TRANSA

         TRANSA is String
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 String
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 String
Specifies whether X  contains an initial guess on input or not.
= 'I': X contains an initial guess for the solution
== 'N': No initial guess is provided. X is set to zero.

SGN

         SGN is SINGLE 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 SINGLE 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 SINGLE 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 B.  LDB >= max(1,N).

C

         C is SINGLE 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 SINGLE PRECISION array, dimension (LDA,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).

X

         X is SINGLE PRECISION array, dimension (LDX,M)
On input, the array X contains the initial guess.
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 SINGLE PRECISION array, dimension (LDY,M)
On input, the array Y contains the right hand side.

LDY

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

AS

         AS is SINGLE 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 SINGLE PRECISION array, dimension (LDBS,N)
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,N).

CS

         CS is SINGLE 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 SINGLE 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.  LDAS >= max(1,N).

Q

         Q is SINGLE 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 SINGLE 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 SINGLE 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 SINGLE 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, MAXIT <= 100
On exit, MAXIT contains the number of iteration steps taken by the algorithm.

TAU

         TAU is SINGLE 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 SINGLE 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 SINGLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm.

LDWORK

         LDWORK is INTEGER
Size of the workspace for the algorithm. This can be determined by a call \ref mepack_memory_frontend.

INFO

         INFO is INTEGER
== 0:  Success
> 0:  Iteration failed in step INFO
< 0:  if INFO == -i, the i-Th argument had an illegal value
Attention

The Fortran/LAPACK-like workspace query with setting LDWORK=-1 on input will not work in the C interface. One have to use the mepack_memory_frontend function for this purpose.

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 660 of file ggsylv.c.

## Author

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## Info

Tue Mar 7 2023 Version 1.0.3 MEPACK