dtgsy2(3)

NAME

DTGSY2 - solve 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 triangu

lar. 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 trans

pose of X. kron(X, Y) is the Kronecker product between the ma

trices 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

STGSYL of an upper bound on the separation between to matrix

pairs. Then the input (A, D), (B, E) are sub-pencils of the ma

trix pair in DTGSYL. See STGSYL for details.

ARGUMENTS

TRANS (input) CHARACTER

= 'N', solve the generalized Sylvester equation

(1). = 'T': solve the 'transposed' system (3).

IJOB (input) INTEGER

Specifies what kind of functionality to be per

formed. = 0: solve (1) only.
= 1: A contribution from this subsystem to a

Frobenius norm-based estimate of the separation between two ma

trix pairs is computed. (look ahead strategy is used). = 2: A

contribution from this subsystem to a Frobenius norm-based esti

mate 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 ma

trix.

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 ma

trix.

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 solu

tions to the homogeneous system with C = F = 0. Normally, SCALE =

1.

RDSUM (input/output) DOUBLE PRECISION

On entry, the sum of squares of computed contribu

tions to the Dif-estimate under computation by DTGSYL, where the

scaling factor RDSCAL (see below) has been factored out. On ex

it, the corresponding sum of squares updated with the contribu

tions from the current sub-system. If TRANS = 'T' RDSUM is not

touched. NOTE: RDSUM only makes sense when DTGSY2 is called by

STGSYL.

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 contri

butions 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.

LAPACK version 3.0 15 June 2000

DTGSY2(3)