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 TRANS
INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE,
LDF, LWORK, M, N
REAL DIF, SCALE
INTEGER IWORK( * )
REAL A( LDA, * ), B( LDB, * ), C( LDC, * ),
D( LDD, * ), E( LDE, * ), F( LDF, * ),
WORK( * )
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
If IJOB >= 1, STGSYL computes a Frobenius norm-based esti
mate 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) CHARACTER*1
= 'N', solve the generalized Sylvester equation
(1). = 'T', solve the 'transposed' system (3).
IJOB (input) INTEGER
Specifies what kind of functionality to be per
formed. =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 com
puted. (look ahead strategy IJOB = 1 is used).
=4: Only an estimate of Dif[(A,D), (B,E)] is com
puted. ( SGECON on sub-systems is used ). Not
referenced if TRANS = 'T'.
M (input) INTEGER
The order of the matrices A and D, and the row
dimension of the matrices C, F, R and L.
N (input) INTEGER
The order of the matrices B and E, and the column
dimension of the matrices C, F, R and L.
A (input) REAL array, dimension (LDA, M)
The upper quasi triangular matrix A.
LDA (input) INTEGER
The leading dimension of the array A. LDA >=
max(1, M).
B (input) REAL array, dimension (LDB, N)
The upper quasi triangular matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >=
max(1, N).
C (input/output) REAL array, dimension (LDC, N)
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
LDC (input) INTEGER
The leading dimension of the array C. LDC >=
max(1, M).
D (input) REAL array, dimension (LDD, M)
The upper triangular matrix D.
LDD (input) INTEGER
The leading dimension of the array D. LDD >=
max(1, M).
E (input) REAL array, dimension (LDE, N)
The upper triangular matrix E.
LDE (input) INTEGER
The leading dimension of the array E. LDE >=
max(1, N).
F (input/output) REAL array, dimension (LDF, N)
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 computa
tion of the Dif-estimate.
LDF (input) INTEGER
The leading dimension of the array F. LDF >=
max(1, M).
DIF (output) REAL
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.
SCALE (output) REAL
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 solutions
R and L, respectively, to the homogeneous system
with C = F = 0. Normally, SCALE = 1.
WORK (workspace/output) REAL array, dimension (LWORK)
If IJOB = 0, WORK is not referenced. Otherwise,
on exit, if INFO = 0, WORK(1) returns the optimal
LWORK.
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+6)
INFO (output) INTEGER
=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 eigen
values.
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 General
ized 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.
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