ZTGSY2(3S)ZTGSY2(3S)NAMEZTGSY2 - 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, * )
IMPLEMENTATION
These routines are part of the SCSL Scientific Library and can be loaded
using either the -lscs or the -lscs_mp option. The -lscs_mp option
directs the linker to use the multi-processor version of the library.
When linking to SCSL with -lscs or -lscs_mp, the default integer size is
4 bytes (32 bits). Another version of SCSL is available in which integers
are 8 bytes (64 bits). This version allows the user access to larger
memory sizes and helps when porting legacy Cray codes. It can be loaded
by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
only one of the two versions; 4-byte integer and 8-byte integer library
calls cannot be mixed.
PURPOSEZTGSY2 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
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ZTGSY2(3S)ZTGSY2(3S)
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.
ZTGSY2 also (IJOB >= 1) contributes to the computation in ZTGSYL 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 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 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 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.
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ZTGSY2(3S)ZTGSY2(3S)
LDC (input) INTEGER
The leading dimension of the matrix C. LDC >= max(1, M).
D (input) COMPLEX*16 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*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 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 ZTGSYL, 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 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 illegal.
>0: The matrix pairs (A, D) and (B, E) have common or very close
eigenvalues.
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ZTGSY2(3S)ZTGSY2(3S)FURTHER DETAILS
Based on contributions by
Bo Kagstrom and Peter Poromaa, Department of Computing Science,
Umea University, S-901 87 Umea, Sweden.
SEE ALSOINTRO_LAPACK(3S), INTRO_SCSL(3S)
This man page is available only online.
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