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 MbyN matrices, (A, D), (B, E) and (C, F) are given matrix pairs of size MbyM, NbyN and MbyN, 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 transpose 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), (B, E)] = sigma_min(Z) using reverse communicaton with DLACON.
DTGSY2 also (IJOB >= 1) contributes to the computation in DTGSYL of an upper bound on the separation between to matrix pairs. Then the input (A, D), (B, E) are subpencils of the matrix pair in DTGSYL. See DTGSYL for details.
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. (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) 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 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) 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 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 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 Difestimate under computation by DTGSYL, 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 DTGSY2 is called by DTGSYL.
 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 DTGSY2 is called by DTGSYL.
 IWORK (workspace) INTEGER array, dimension (M+N+2)
 PQ (output) INTEGER
 On exit, the number of subsystems (of size 2by2, 4by4 and 8by8) solved by this routine.
 INFO (output) INTEGER

On exit, if INFO is set to
=0: Successful exit
<0: If INFO = i, the ith 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 byBo Kagstrom and Peter Poromaa, Department of Computing Science,
Umea University, S901 87 Umea, Sweden.