ZTGSY2 - 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 ZTGSY2( 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
DOUBLE PRECISION RDSCAL, RDSUM, SCALE
COMPLEX*16 A( LDA, * ), B( LDB, * ), C( LDC, * ),
D( LDD, * ), E( LDE, * ), F( LDF, * )
PURPOSE
ZTGSY2 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 trans
pose 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
ZLACON.
ZTGSYL of an upper bound on the separation between to
matrix pairs. Then the input (A, D), (B, E) are sub-pen
cils of two matrix pairs in ZTGSYL.
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 Frobe
nius 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 separa
tion 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 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) COMPLEX*16 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*16 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*16 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).
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*16 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*16 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
ZTGSYL, 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 ZTGSY2 is called by ZTGSYL.
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 ZTGSY2 is called by ZTGSYL.
INFO (output) INTEGER
On exit, if INFO is set to =0: Successful exit
<0: If INFO = -i, input argument number i is ille
gal.
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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