DTGSY2(1) LAPACK auxiliary routine (version 3.2) DTGSY2(1)NAME
DTGSY2 - solves the generalized Sylvester equation
SYNOPSIS
SUBROUTINE DTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD,
E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, IWORK, PQ,
INFO )
CHARACTER TRANS
INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N, PQ
DOUBLE PRECISION RDSCAL, RDSUM, SCALE
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ), D(
LDD, * ), E( LDE, * ), F( LDF, * )
PURPOSE
DTGSY2 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, with real entries. (A, D) and (B, E) must be
in generalized Schur canonical form, i.e. A, B are upper quasi triangu‐
lar and D, E are upper triangular. The solution (R, L) overwrites (C,
F). 0 <= SCALE <= 1 is an output scaling factor chosen to avoid over‐
flow.
In matrix notation solving equation (1) corresponds to solve Z*x =
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. In
the process of solving (1), we solve a number of such systems where
Dim(In), Dim(In) = 1 or 2.
If TRANS = 'T', solve the transposed system Z'*y = scale*b for y, 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 DLACON.
DTGSY2 also (IJOB >= 1) contributes to the computation in DTGSYL of an
upper bound on the separation between to matrix pairs. Then the input
(A, D), (B, E) are sub-pencils of the matrix pair in DTGSYL. See DTGSYL
for details.
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 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. (DGECON 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) DOUBLE PRECISION array, dimension (LDA, M)
On entry, A contains an upper quasi triangular matrix.
LDA (input) INTEGER
The leading dimension of the matrix A. LDA >= max(1, M).
B (input) DOUBLE PRECISION array, dimension (LDB, N)
On entry, B contains an upper quasi triangular matrix.
LDB (input) INTEGER
The leading dimension of the matrix B. LDB >= max(1, N).
C (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) 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 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) DOUBLE PRECISION
On entry, the sum of squares of computed contributions to the
Dif-estimate under computation by DTGSYL, 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 DTGSY2 is called by
DTGSYL.
RDSCAL (input/output) DOUBLE PRECISION
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 DTGSY2 is called by DTGSYL.
IWORK (workspace) INTEGER array, dimension (M+N+2)
PQ (output) INTEGER
On exit, the number of subsystems (of size 2-by-2, 4-by-4 and
8-by-8) solved by this routine.
INFO (output) INTEGER
On exit, if INFO is set to =0: Successful exit
<0: If INFO = -i, the i-th argument had an illegal value.
>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 auxiliary routine (versioNovember 2008 DTGSY2(1)
