ztgsyl(3P) Sun Performance Library ztgsyl(3P)NAMEztgsyl - solve the generalized Sylvester equation
SYNOPSIS
SUBROUTINE ZTGSYL(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
DOUBLE COMPLEX A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), E(LDE,*),
F(LDF,*), WORK(*)
INTEGER IJOB, M, N, LDA, LDB, LDC, LDD, LDE, LDF, LWORK, INFO
INTEGER IWORK(*)
DOUBLE PRECISION SCALE, DIF
SUBROUTINE ZTGSYL_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
DOUBLE COMPLEX A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), E(LDE,*),
F(LDF,*), WORK(*)
INTEGER*8 IJOB, M, N, LDA, LDB, LDC, LDD, LDE, LDF, LWORK, INFO
INTEGER*8 IWORK(*)
DOUBLE PRECISION SCALE, DIF
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
COMPLEX(8), DIMENSION(:) :: WORK
COMPLEX(8), DIMENSION(:,:) :: A, B, C, D, E, F
INTEGER :: IJOB, M, N, LDA, LDB, LDC, LDD, LDE, LDF, LWORK, INFO
INTEGER, DIMENSION(:) :: IWORK
REAL(8) :: SCALE, DIF
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
COMPLEX(8), DIMENSION(:) :: WORK
COMPLEX(8), DIMENSION(:,:) :: A, B, C, D, E, F
INTEGER(8) :: IJOB, M, N, LDA, LDB, LDC, LDD, LDE, LDF, LWORK, INFO
INTEGER(8), DIMENSION(:) :: IWORK
REAL(8) :: SCALE, DIF
C INTERFACE
#include <sunperf.h>
void ztgsyl(char trans, int ijob, int m, int n, doublecomplex *a, int
lda, doublecomplex *b, int ldb, doublecomplex *c, int ldc,
doublecomplex *d, int ldd, doublecomplex *e, int lde, double‐
complex *f, int ldf, double *scale, double *dif, int *info);
void ztgsyl_64(char trans, long ijob, long m, long n, doublecomplex *a,
long lda, doublecomplex *b, long ldb, doublecomplex *c, long
ldc, doublecomplex *d, long ldd, doublecomplex *e, long lde,
doublecomplex *f, long ldf, double *scale, double *dif, long
*info);
PURPOSEztgsyl 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 complex entries. A, B, D and E are upper triangular (i.e., (A,D)
and (B,E) in generalized Schur form).
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 Ix is the identity matrix of size x and X' is the conjugate trans‐
pose of X. Kron(X, Y) is the Kronecker product between the matrices X
and Y.
If TRANS = 'C', y in the conjugate transposed system Z'*y = scale*b is
solved for, 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 = 'C') 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 ZLACON.
If IJOB >= 1, ZTGSYL 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.
This is a level-3 BLAS algorithm.
ARGUMENTS
TRANS (input)
= 'N': solve the generalized sylvester equation (1).
= 'C': solve the "conjugate 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 is used). =4: Only an estimate of Dif[(A,D),
(B,E)] is computed. (ZGECON on sub-systems is used). Not
referenced if TRANS = 'C'.
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 triangular matrix A.
LDA (input)
The leading dimension of the array A. LDA >= max(1, M).
B (input) The upper 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, R and L will hold the
solutions to the homogenious system with C = F = 0.
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 = 'C', DIF is not referenced.
WORK (workspace)
If IJOB = 0, WORK is not referenced. Otherwise, on exit, if
INFO=0 then 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) INTEGER array, 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 very 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 ztgsyl(3P)
