STGSY2(3f)

NAME
STGSY2 - the generalized Sylvester equation

SYNOPSIS

SUBROUTINE STGSY2(
    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

REAL
    RDSCAL, RDSUM, SCALE

INTEGER
    IWORK( * )

REAL
    A( LDA, * ), B( LDB, * ), C( LDC, * ), D( LDD, * ), E( LDE, * ), F( LDF, * )

PURPOSE
STGSY2 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 SLACON.
STGSY2 also (IJOB >= 1) contributes to the computation in STGSYL 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 STGSYL. See STGSYL 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. (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) REAL 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) REAL 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) REAL 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) REAL 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) REAL 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) REAL 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 STGSYL, 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 STGSY2 is called by STGSYL.

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 STGSY2 is called by STGSYL.

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.