DTGSY2 - solve 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 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 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 trans
pose 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),
DLACON.
DTGSY2 also (IJOB >= 1) contributes to the computation in
STGSYL of an upper bound on the separation between to
matrix pairs. Then the input (A, D), (B, E) are sub-pen
cils of the matrix pair in DTGSYL. See STGSYL for details.
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 per
formed. = 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 com
puted. (DGECON on sub-systems is used.) Not ref
erenced if TRANS = 'T'.
M (input) INTEGER
On entry, M specifies the order of A and D, and
the row dimension 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 =
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 perturbed system but the
input matrices A, B, D and E have not been
changed. If SCALE = 0, R and L will hold the solu
tions to the homogeneous system with C = F = 0.
Normally, SCALE = 1.
RDSUM (input/output) DOUBLE PRECISION
On entry, the sum of squares of computed contribu
tions to the Dif-estimate under computation by
DTGSYL, where the scaling factor RDSCAL (see
below) has been factored out. On exit, the corre
sponding sum of squares updated with the contribu
tions from the current sub-system. If TRANS = 'T'
RDSUM is not touched. NOTE: RDSUM only makes
sense when DTGSY2 is called by STGSYL.
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
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.
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