SYNOPSIS
 SUBROUTINE CTGSY2(
 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
 REAL RDSCAL, RDSUM, SCALE
 COMPLEX A( LDA, * ), B( LDB, * ), C( LDC, * ), D( LDD, * ), E( LDE, * ), F( LDF, * )
PURPOSE
CTGSY2 solves the generalized Sylvester equation (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 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 CLACON.
CTGSY2 also (IJOB >= 1) contributes to the computation in CTGSYL 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 CTGSYL.
ARGUMENTS
 TRANS (input) CHARACTER*1
 = '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 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. (SGECON 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 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 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 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 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 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 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) REAL
 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) REAL
 On entry, the sum of squares of computed contributions to the Difestimate under computation by CTGSYL, 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 subsystem. If TRANS = 'T' RDSUM is not touched. NOTE: RDSUM only makes sense when CTGSY2 is called by CTGSYL.
 RDSCAL (input/output) REAL
 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 CTGSY2 is called by CTGSYL.
 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 byBo Kagstrom and Peter Poromaa, Department of Computing Science,
Umea University, S901 87 Umea, Sweden.