NAME

stgsyl - solve the generalized Sylvester equation

SYNOPSIS

  SUBROUTINE STGSYL( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, 
 *      LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO)
  CHARACTER * 1 TRANS
  INTEGER IJOB, M, N, LDA, LDB, LDC, LDD, LDE, LDF, LWORK, INFO
  INTEGER IWORK(*)
  REAL SCALE, DIF
  REAL A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), E(LDE,*), F(LDF,*), WORK(*)

  SUBROUTINE STGSYL_64( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, 
 *      LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO)
  CHARACTER * 1 TRANS
  INTEGER*8 IJOB, M, N, LDA, LDB, LDC, LDD, LDE, LDF, LWORK, INFO
  INTEGER*8 IWORK(*)
  REAL SCALE, DIF
  REAL A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), E(LDE,*), F(LDF,*), WORK(*)

F95 INTERFACE

  SUBROUTINE TGSYL( TRANS, IJOB, [M], [N], A, [LDA], B, [LDB], C, [LDC], 
 *       D, [LDD], E, [LDE], F, [LDF], SCALE, DIF, [WORK], [LWORK], [IWORK], 
 *       [INFO])
  CHARACTER(LEN=1) :: TRANS
  INTEGER :: IJOB, M, N, LDA, LDB, LDC, LDD, LDE, LDF, LWORK, INFO
  INTEGER, DIMENSION(:) :: IWORK
  REAL :: SCALE, DIF
  REAL, DIMENSION(:) :: WORK
  REAL, DIMENSION(:,:) :: A, B, C, D, E, F

  SUBROUTINE TGSYL_64( TRANS, IJOB, [M], [N], A, [LDA], B, [LDB], C, 
 *       [LDC], D, [LDD], E, [LDE], F, [LDF], SCALE, DIF, [WORK], [LWORK], 
 *       [IWORK], [INFO])
  CHARACTER(LEN=1) :: TRANS
  INTEGER(8) :: IJOB, M, N, LDA, LDB, LDC, LDD, LDE, LDF, LWORK, INFO
  INTEGER(8), DIMENSION(:) :: IWORK
  REAL :: SCALE, DIF
  REAL, DIMENSION(:) :: WORK
  REAL, DIMENSION(:,:) :: A, B, C, D, E, F

C INTERFACE

#include <sunperf.h>

void stgsyl(char trans, int ijob, int m, int n, float *a, int lda, float *b, int ldb, float *c, int ldc, float *d, int ldd, float *e, int lde, float *f, int ldf, float *scale, float *dif, int *info);

void stgsyl_64(char trans, long ijob, long m, long n, float *a, long lda, float *b, long ldb, float *c, long ldc, float *d, long ldd, float *e, long lde, float *f, long ldf, float *scale, float *dif, long *info);

PURPOSE

stgsyl solves the generalized Sylvester equation:

            A * R - L * B = scale * C                 (1)
            D * R - L * E = scale * F

where R and L are unknown m-by-n matrices, (A, D), (B, E) and (C, F) are given matrix pairs of size m-by-m, n-by-n and m-by-n, respectively, with real entries. (A, D) and (B, E) must be in generalized (real) Schur canonical form, i.e. A, B are upper quasi triangular and D, E are upper triangular.

The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output scaling factor chosen to avoid overflow.

In matrix notation (1) is equivalent to solve Zx = scale b, where Z is defined as

           Z = [ kron(In, A)  -kron(B', Im) ]         (2)
               [ kron(In, D)  -kron(E', Im) ].

Here Ik is the identity matrix of size k and X' is the transpose of X. kron(X, Y) is the Kronecker product between the matrices X and Y.

If TRANS = 'T', STGSYL solves the transposed system Z'*y = scale*b, which is equivalent to solve for R and L in

            A' * R  + D' * L   = scale *  C           (3)
            R  * B' + L  * E'  = scale * (-F)

This case (TRANS = 'T') is used to compute an one-norm-based estimate of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D) and (B,E), using SLACON.

If IJOB >= 1, STGSYL computes a Frobenius norm-based estimate of Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the reciprocal of the smallest singular value of Z. See [1-2] for more information.

This is a level 3 BLAS algorithm.

ARGUMENTS

TRANS (input)

 = 'N', solve the generalized Sylvester equation (1).
 = 'T', solve the 'transposed' system (3).

IJOB (input)
Specifies what kind of functionality to be performed. =0: solve (1) only.

 =1: The functionality of 0 and 3.

 =2: The functionality of 0 and 4.

 =3: Only an estimate of Dif[(A,D), (B,E)] is computed.
(look ahead strategy IJOB   = 1 is used).
 =4: Only an estimate of Dif[(A,D), (B,E)] is computed.
( SGECON on sub-systems is used ).
Not referenced if TRANS  = 'T'.

M (input)
The order of the matrices A and D, and the row dimension of the matrices C, F, R and L.
N (input)
The order of the matrices B and E, and the column dimension of the matrices C, F, R and L.
A (input)
The upper quasi triangular matrix A.
LDA (input)
The leading dimension of the array A. LDA > = max(1, M).
B (input)
The upper quasi triangular matrix B.
LDB (input)
The leading dimension of the array B. LDB > = max(1, N).
C (input/output)
On entry, C contains the right-hand-side of the first matrix equation in (1) or (3). On exit, if IJOB = 0, 1 or 2, C has been overwritten by the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R, the solution achieved during the computation of the Dif-estimate.
LDC (input)
The leading dimension of the array C. LDC > = max(1, M).
D (input)
The upper triangular matrix D.
LDD (input)
The leading dimension of the array D. LDD > = max(1, M).
E (input)
The upper triangular matrix E.
LDE (input)
The leading dimension of the array E. LDE > = max(1, N).
F (input/output)
On entry, F contains the right-hand-side of the second matrix equation in (1) or (3). On exit, if IJOB = 0, 1 or 2, F has been overwritten by the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L, the solution achieved during the computation of the Dif-estimate.
LDF (input)
The leading dimension of the array F. LDF > = max(1, M).
SCALE (output)
On exit DIF is the reciprocal of a lower bound of the reciprocal of the Dif-function, i.e. DIF is an upper bound of Dif[(A,D), (B,E)] = sigma_min(Z), where Z as in (2). If IJOB = 0 or TRANS = 'T', DIF is not touched.
DIF (output)
On exit DIF is the reciprocal of a lower bound of the reciprocal of the Dif-function, i.e. DIF is an upper bound of Dif[(A,D), (B,E)] = sigma_min(Z), where Z as in (2). If IJOB = 0 or TRANS = 'T', DIF is not touched.
WORK (workspace)
If IJOB = 0, WORK is not referenced. Otherwise, on exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
The dimension of the array WORK. LWORK > = 1. If IJOB = 1 or 2 and TRANS = 'N', LWORK > = 2*M*N.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
IWORK (workspace)
dimension(M+N+2)

INFO (output)

 =0: successful exit

 <0: If INFO  = -i, the i-th argument had an illegal value.

 >0: (A, D) and (B, E) have common or close eigenvalues.

FURTHER DETAILS

Based on contributions by

   Bo Kagstrom and Peter Poromaa, Department of Computing Science,
   Umea University, S-901 87 Umea, Sweden.

[1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK Working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.

[2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal. Appl., 15(4):1045-1060, 1994

[3] B. Kagstrom and L. Westin, Generalized Schur Methods with Condition Estimators for Solving the Generalized Sylvester Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.