CTGSY2(l) ) CTGSY2(l)NAME
CTGSY2 - solve the generalized Sylvester equation A * R - L * B =
scale * C (1) D * R - L * E = scale * F using Level 1 and 2 BLAS,
where R and L are unknown M-by-N matrices,
SYNOPSIS
SUBROUTINE CTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD,
E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, INFO )
CHARACTER TRANS
INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N
REAL RDSCAL, RDSUM, SCALE
COMPLEX A( LDA, * ), B( LDB, * ), C( LDC, * ), D( LDD, * ),
E( LDE, * ), F( LDF, * )
PURPOSE
CTGSY2 solves the generalized Sylvester equation A * R - L * B = scale
* C (1) D * R - L * E = scale * F using Level 1 and 2 BLAS, 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. 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 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.
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 is used to compute an estimate of Dif[(A, D), (B, E)] = =
sigma_min(Z) using reverse communicaton with CLACON.
CTGSY2 also (IJOB >= 1) contributes to the computation in CTGSYL of an
upper bound on the separation between to matrix pairs. Then the input
(A, D), (B, E) are sub-pencils of two matrix pairs in CTGSYL.
ARGUMENTS
TRANS (input) CHARACTER
= 'N', solve the generalized Sylvester equation (1). = 'T':
solve the 'transposed' system (3).
IJOB (input) INTEGER
Specifies what kind of functionality to be performed. =0:
solve (1) only.
=1: A contribution from this subsystem to a Frobenius norm-
based estimate of the separation between two matrix pairs is
computed. (look ahead strategy is used). =2: A contribution
from this subsystem to a Frobenius norm-based estimate of the
separation between two matrix pairs is computed. (SGECON on
sub-systems is used.) Not referenced if TRANS = 'T'.
M (input) INTEGER
On entry, M specifies the order of A and D, and the row dimen‐
sion of C, F, R and L.
N (input) INTEGER
On entry, N specifies the order of B and E, and the column
dimension of C, F, R and L.
A (input) COMPLEX array, dimension (LDA, M)
On entry, A contains an upper triangular matrix.
LDA (input) INTEGER
The leading dimension of the matrix A. LDA >= max(1, M).
B (input) COMPLEX array, dimension (LDB, N)
On entry, B contains an upper triangular matrix.
LDB (input) INTEGER
The leading dimension of the matrix B. LDB >= max(1, N).
C (input/ output) COMPLEX array, dimension (LDC, N)
On entry, C contains the right-hand-side of the first matrix
equation in (1). On exit, if IJOB = 0, C has been overwritten
by the solution R.
LDC (input) INTEGER
The leading dimension of the matrix C. LDC >= max(1, M).
D (input) COMPLEX array, dimension (LDD, M)
On entry, D contains an upper triangular matrix.
LDD (input) INTEGER
The leading dimension of the matrix D. LDD >= max(1, M).
E (input) COMPLEX array, dimension (LDE, N)
On entry, E contains an upper triangular matrix.
LDE (input) INTEGER
The leading dimension of the matrix E. LDE >= max(1, N).
F (input/ output) COMPLEX array, dimension (LDF, N)
On entry, F contains the right-hand-side of the second matrix
equation in (1). On exit, if IJOB = 0, F has been overwritten
by the solution L.
LDF (input) INTEGER
The leading dimension of the matrix F. LDF >= max(1, M).
SCALE (output) REAL
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 per‐
turbed 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 = F = 0. Normally, SCALE = 1.
RDSUM (input/output) REAL
On entry, the sum of squares of computed contributions to the
Dif-estimate under computation by CTGSYL, where the scaling
factor RDSCAL (see below) has been factored out. On exit, the
corresponding sum of squares updated with the contributions
from the current sub-system. If TRANS = 'T' RDSUM is not
touched. NOTE: RDSUM only makes sense when CTGSY2 is called by
CTGSYL.
RDSCAL (input/output) REAL
On entry, scaling factor used to prevent overflow in RDSUM. On
exit, RDSCAL is updated w.r.t. the current contributions in
RDSUM. If TRANS = 'T', RDSCAL is not touched. NOTE: RDSCAL
only makes sense when CTGSY2 is called by CTGSYL.
INFO (output) INTEGER
On exit, if INFO is set to =0: Successful exit
<0: If INFO = -i, input argument number i is illegal.
>0: The matrix pairs (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.
LAPACK version 3.0 15 June 2000 CTGSY2(l)
