dtgsyl(3P) Sun Performance Library dtgsyl(3P)NAMEdtgsyl - solve the generalized Sylvester equation
SYNOPSIS
SUBROUTINE DTGSYL(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(*)
DOUBLE PRECISION SCALE, DIF
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), E(LDE,*),
F(LDF,*), WORK(*)
SUBROUTINE DTGSYL_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(*)
DOUBLE PRECISION SCALE, DIF
DOUBLE PRECISION 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(8) :: SCALE, DIF
REAL(8), DIMENSION(:) :: WORK
REAL(8), 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(8) :: SCALE, DIF
REAL(8), DIMENSION(:) :: WORK
REAL(8), DIMENSION(:,:) :: A, B, C, D, E, F
C INTERFACE
#include <sunperf.h>
void dtgsyl(char trans, int ijob, int m, int n, double *a, int lda,
double *b, int ldb, double *c, int ldc, double *d, int ldd,
double *e, int lde, double *f, int ldf, double *scale, double
*dif, int *info);
void dtgsyl_64(char trans, long ijob, long m, long n, double *a, long
lda, double *b, long ldb, double *c, long ldc, double *d,
long ldd, double *e, long lde, double *f, long ldf, double
*scale, double *dif, long *info);
PURPOSEdtgsyl 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', DTGSYL 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 DLACON.
If IJOB >= 1, DTGSYL 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. ( DGECON 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 com‐
putation 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 com‐
putation of the Dif-estimate.
LDF (input)
The leading dimension of the array F. LDF >= max(1, M).
SCALE (output)
On exit SCALE is the scaling factor in (1) or (3). If 0 <
SCALE < 1, C and F hold the solutions R and L, resp., to a
slightly perturbed system but the input matrices A, B, D and
E have not been changed. If SCALE = 0, C and F hold the solu‐
tions R and L, respectively, to the homogeneous system with C
= F = 0. Normally, SCALE = 1.
DIF (output)
On exit DIF is the reciprocal of a lower bound of the recip‐
rocal 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+6)
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.
6 Mar 2009 dtgsyl(3P)
