
ZTGSY2(l) ) ZTGSY2(l)
NAME
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 MbyN 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 MbyN matrices, (A, D),
(B, E) and (C, F) are given matrix pairs of size MbyM, NbyN and MbyN, 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 Kro
necker 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.
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 subpencils of two
matrix pairs in ZTGSYL.
ARGUMENTS
TRANS (input) CHARACTER
= 'N', solve the generalized Sylvester equation (1). = 'T': solve the 'trans
posed' 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 normbased estimate of the
separation between two matrix pairs is computed. (look ahead strategy is used).
=2: A contribution from this subsystem to a Frobenius normbased estimate of the
separation between two matrix pairs is computed. (DGECON on subsystems 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 righthandside 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) 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 righthandside 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 matri
ces 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 contributions to the Difestimate under
computation by ZTGSYL, where the scaling factor RDSCAL (see below) has been fac
tored out. On exit, the corresponding sum of squares updated with the contribu
tions from the current subsystem. 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.
FURTHER DETAILS
Based on contributions by
Bo Kagstrom and Peter Poromaa, Department of Computing Science,
Umea University, S901 87 Umea, Sweden.
LAPACK version 3.0 15 June 2000 ZTGSY2(l) 
