CTGSY2(3) L * E = scale * F using Level 1 and 2 BLAS, where R and L are unknown M-by-N matrices,

## 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 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
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 sub-pencils 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 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. (SGECON 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 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 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).
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 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) 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 Dif-estimate 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 sub-system. 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 by

Bo Kagstrom and Peter Poromaa, Department of Computing Science,
Umea University, S-901 87 Umea, Sweden.