DTGSY2(3) solves the generalized Sylvester equation

## 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 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, 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 sub-pencils 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 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) 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 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) 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 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 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 sub-system. 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 2-by-2, 4-by-4 and 8-by-8) solved by this routine.
INFO (output) INTEGER
On exit, if INFO is set to =0: Successful exit
<0: If INFO = -i, the i-th 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 by

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