Purpose
To solve the generalized complex Sylvester equation
A * R - L * B = scale * C, (1)
D * R - L * E = scale * F,
using Level 3 BLAS, where R and L are unknown M-by-N matrices, and
(A, D), (B, E) and (C, F) are given matrix pairs of size M-by-M,
N-by-N and M-by-N, respectively. A, B, D and E are complex upper
triangular (i.e., (A,D) and (B,E) are in generalized Schur form).
The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an
output scaling factor chosen to avoid overflow.
This routine is intended to be called only by SLICOT Library
routine MB04RZ. For efficiency purposes, the computations are
aborted when the absolute value of an element of R or L is greater
than a given value PMAX.
Specification
SUBROUTINE MB04RW( M, N, PMAX, A, LDA, B, LDB, C, LDC, D, LDD, E,
$ LDE, F, LDF, SCALE, IWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N
DOUBLE PRECISION PMAX, SCALE
C .. Array Arguments ..
INTEGER IWORK( * )
COMPLEX*16 A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), E( LDE, * ), F( LDF, * )
Arguments
Input/Output Parameters
M (input) INTEGER
The order of the matrices A and D, and the row dimension
of C, F, R and L. M >= 0.
N (input) INTEGER
The order of the matrices B and E, and the column
dimension of C, F, R and L. N >= 0.
PMAX (input) DOUBLE PRECISION
An upper bound for the "absolute value" of the elements of
the solution (R, L). (See FURTHER COMMENTS.)
PMAX >= 1.0D0.
A (input) COMPLEX*16 array, dimension (LDA, M)
On entry, the leading M-by-M upper triangular part of this
array must contain the matrix A in the generalized complex
Schur form, as returned by LAPACK routine ZGGES.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1, M).
B (input) COMPLEX*16 array, dimension (LDB, N)
On entry, the leading N-by-N upper triangular part of this
array must contain the matrix B in the generalized complex
Schur form.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1, N).
C (input/output) COMPLEX*16 array, dimension (LDC, N)
On entry, the leading M-by-N part of this array must
contain the right-hand-side of the first matrix equation
in (1).
On exit, if INFO = 0, the leading M-by-N part of this
array contains the solution R.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1, M).
D (input) COMPLEX*16 array, dimension (LDD, M)
On entry, the leading M-by-M upper triangular part of this
array must contain the matrix D in the generalized complex
Schur form. The diagonal elements are non-negative real.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1, M).
E (input) COMPLEX*16 array, dimension (LDE, N)
On entry, the leading N-by-N upper triangular part of this
array must contain the matrix E in the generalized complex
Schur form. The diagonal elements are non-negative real.
LDE INTEGER
The leading dimension of the array E. LDE >= max(1, N).
F (input/output) COMPLEX*16 array, dimension (LDF, N)
On entry, the leading M-by-N part of this array must
contain the right-hand-side of the second matrix equation
in (1).
On exit, if INFO = 0, the leading M-by-N part of this
array contains the solution L.
LDF INTEGER
The leading dimension of the array F. LDF >= max(1, M).
SCALE (output) DOUBLE PRECISION
On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions
R and L (C and F on entry) will hold the solutions 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 homogeneous system with C = 0
and F = 0. Normally, SCALE = 1.
Workspace
IWORK INTEGER array, dimension (M+N+2)Error Indicator
INFO INTEGER
= 0: successful exit;
= 1: an element of R or L had the absolute value greater
than the given value PMAX.
= 2: the matrix pairs (A, D) and (B, E) have common or
very close eigenvalues. The matrix Z in section
METHOD is (almost) singular.
Method
The routine uses an adaptation of the method for solving
generalized Sylvester equations [1], which controls the magnitude
of the individual elements of the computed solution [2].
In matrix notation, solving equation (1) corresponds to solve
Zx = scale * b, where Z is defined as
Z = [ kron(In, A) -kron(B', Im) ] (2)
[ kron(In, D) -kron(E', Im) ],
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.
References
[1] Kagstrom, B. and Westin, L.
Generalized Schur Methods with Condition Estimators for
Solving the Generalized Sylvester Equation.
IEEE Trans. Auto. Contr., 34, pp. 745-751, 1989.
[2] Kagstrom, B. and Westin, L.
GSYLV - Fortran Routines for the Generalized Schur Method with
Dif Estimators for Solving the Generalized Sylvester Equation.
Report UMINF-132.86, Institute of Information Processing,
Univ. of Umea, Sweden, July 1987.
[3] Kagstrom, B.
A Perturbation Analysis of the Generalized Sylvester Equation
(AR - LB, DR - LE ) = (C, F).
SIAM J. Matrix Anal. Appl., 15(4), pp. 1045-1060, 1994.
[4] Kagstrom, B. and Poromaa, P.
LAPACK-Style Algorithms and Software for Solving the
Generalized Sylvester Equation and Estimating the Separation
between Regular Matrix Pairs.
ACM Trans. on Math. Software, 22(1), pp. 78103, 1996.
Further Comments
For efficiency reasons, the "absolute value" of a complex number x is computed as |real(x)| + |imag(x)|.Example
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