.TH DTGSY2 1 "February 2007" " LAPACK auxiliary routine (version 3.1.1) " " LAPACK auxiliary routine (version 3.1.1) "
.SH NAME
DTGSY2 - the generalized Sylvester equation
.SH SYNOPSIS
.TP 19
SUBROUTINE DTGSY2(
TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL,
IWORK, PQ, INFO )
.TP 19
.ti +4
CHARACTER
TRANS
.TP 19
.ti +4
INTEGER
IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N,
PQ
.TP 19
.ti +4
DOUBLE
PRECISION RDSCAL, RDSUM, SCALE
.TP 19
.ti +4
INTEGER
IWORK( * )
.TP 19
.ti +4
DOUBLE
PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
D( LDD, * ), E( LDE, * ), F( LDF, * )
.SH PURPOSE
DTGSY2 solves the generalized Sylvester equation:
A * R - L * B = scale * C (1)
.br
D * R - L * E = scale * F,
.br
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 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.
.br
In matrix notation solving equation (1) corresponds to solve
Z*x = scale*b, where Z is defined as
.br
Z = [ kron(In, A) -kron(B\(aq, Im) ] (2)
[ kron(In, D) -kron(E\(aq, Im) ],
.br
Ik is the identity matrix of size k and X\(aq 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.
.br
If TRANS = \(aqT\(aq, solve the transposed system Z\(aq*y = scale*b for y,
which is equivalent to solve for R and L in
.br
A\(aq * R + D\(aq * L = scale * C (3)
R * B\(aq + L * E\(aq = scale * -F
.br
This case is used to compute an estimate of Dif[(A, D), (B, E)] =
sigma_min(Z) using reverse communicaton with DLACON.
.br
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.
.br
.SH ARGUMENTS
.TP 8
TRANS (input) CHARACTER*1
= \(aqN\(aq, solve the generalized Sylvester equation (1).
= \(aqT\(aq: solve the \(aqtransposed\(aq system (3).
.TP 8
IJOB (input) INTEGER
Specifies what kind of functionality to be performed.
= 0: solve (1) only.
.br
= 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 = \(aqT\(aq.
.TP 8
M (input) INTEGER
On entry, M specifies the order of A and D, and the row
dimension of C, F, R and L.
.TP 8
N (input) INTEGER
On entry, N specifies the order of B and E, and the column
dimension of C, F, R and L.
.TP 8
A (input) DOUBLE PRECISION array, dimension (LDA, M)
On entry, A contains an upper quasi triangular matrix.
.TP 8
LDA (input) INTEGER
The leading dimension of the matrix A. LDA >= max(1, M).
.TP 8
B (input) DOUBLE PRECISION array, dimension (LDB, N)
On entry, B contains an upper quasi triangular matrix.
.TP 8
LDB (input) INTEGER
The leading dimension of the matrix B. LDB >= max(1, N).
.TP 8
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.
.TP 8
LDC (input) INTEGER
The leading dimension of the matrix C. LDC >= max(1, M).
.TP 8
D (input) DOUBLE PRECISION array, dimension (LDD, M)
On entry, D contains an upper triangular matrix.
.TP 8
LDD (input) INTEGER
The leading dimension of the matrix D. LDD >= max(1, M).
.TP 8
E (input) DOUBLE PRECISION array, dimension (LDE, N)
On entry, E contains an upper triangular matrix.
.TP 8
LDE (input) INTEGER
The leading dimension of the matrix E. LDE >= max(1, N).
.TP 8
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.
.TP 8
LDF (input) INTEGER
The leading dimension of the matrix F. LDF >= max(1, M).
.TP 8
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 = F = 0. Normally,
SCALE = 1.
.TP 8
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 = \(aqT\(aq RDSUM is not touched.
NOTE: RDSUM only makes sense when DTGSY2 is called by DTGSYL.
.TP 8
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 = \(aqT\(aq, RDSCAL is not touched.
NOTE: RDSCAL only makes sense when DTGSY2 is called by
DTGSYL.
.TP 8
IWORK (workspace) INTEGER array, dimension (M+N+2)
.TP 8
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.
.TP 8
INFO (output) INTEGER
On exit, if INFO is set to
=0: Successful exit
.br
<0: If INFO = -i, the i-th argument had an illegal value.
.br
>0: The matrix pairs (A, D) and (B, E) have common or very
close eigenvalues.
.SH FURTHER DETAILS
Based on contributions by
.br
Bo Kagstrom and Peter Poromaa, Department of Computing Science,
Umea University, S-901 87 Umea, Sweden.
.br