dtgsyl(3)

NAME

DTGSYL - solve the generalized Sylvester equation

SYNOPSIS

SUBROUTINE DTGSYL( TRANS, IJOB, M, N, A, LDA, B,  LDB,  C,
LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO
)
    CHARACTER      TRANS
    INTEGER        IJOB, INFO, LDA, LDB,  LDC,  LDD,  LDE,
LDF, LWORK, M, N
    DOUBLE         PRECISION DIF, SCALE
    INTEGER        IWORK( * )
    DOUBLE          PRECISION A( LDA, * ), B( LDB, * ), C(
LDC, * ), D( LDD, * ), E( LDE, * ), F( LDF, * ), WORK( * )

PURPOSE

DTGSYL solves the generalized Sylvester equation:

A * R - L * B = scale * C (1)
D * R - L * E = scale * F

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 (real) 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 (1) is equivalent to solve Zx = scale

b, where Z is defined as

Z = [ kron(In, A) -kron(B', Im) ] (2)

[ kron(In, D) -kron(E', Im) ].

Here 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 = 'T', DTGSYL solves the transposed system Z'*y =

scale*b, 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 (TRANS = 'T') is used to compute an one-norm

based estimate of Dif[(A,D), (B,E)], the separation between the

matrix pairs (A,D) and (B,E), using DLACON.

If IJOB >= 1, DTGSYL computes a Frobenius norm-based esti

mate of Dif[(A,D),(B,E)]. That is, the reciprocal of a lower

bound on the reciprocal of the smallest singular value of Z. See

[1-2] for more information.

This is a level 3 BLAS algorithm.

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 per

formed. =0: solve (1) only.
=1: The functionality of 0 and 3.
=2: The functionality of 0 and 4.
=3: Only an estimate of Dif[(A,D), (B,E)] is com

puted. (look ahead strategy IJOB = 1 is used). =4: Only an es

timate of Dif[(A,D), (B,E)] is computed. ( DGECON on sub-systems

is used ). Not referenced if TRANS = 'T'.

M (input) INTEGER

The order of the matrices A and D, and the row di

mension of the matrices C, F, R and L.

N (input) INTEGER

The order of the matrices B and E, and the column

dimension of the matrices C, F, R and L.

A (input) DOUBLE PRECISION array, dimension (LDA, M)

The upper quasi triangular matrix A.

LDA (input) INTEGER

The leading dimension of the array A. LDA >=

max(1, M).

B (input) DOUBLE PRECISION array, dimension (LDB, N)

The upper quasi triangular matrix B.

LDB (input) INTEGER

The leading dimension of the array 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) or (3). On exit, if IJOB = 0, 1 or

2, C has been overwritten by the solution R. If IJOB = 3 or 4 and

TRANS = 'N', C holds R, the solution achieved during the computa

tion of the Dif-estimate.

LDC (input) INTEGER

The leading dimension of the array C. LDC >=

max(1, M).

D (input) DOUBLE PRECISION array, dimension (LDD, M)

The upper triangular matrix D.

LDD (input) INTEGER

The leading dimension of the array D. LDD >=

max(1, M).

E (input) DOUBLE PRECISION array, dimension (LDE, N)

The upper triangular matrix E.

LDE (input) INTEGER

The leading dimension of the array 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) or (3). On exit, if IJOB = 0, 1 or

2, F has been overwritten by the solution L. If IJOB = 3 or 4 and

TRANS = 'N', F holds L, the solution achieved during the computa

tion of the Dif-estimate.

LDF (input) INTEGER

The leading dimension of the array F. LDF >=

max(1, M).

DIF (output) DOUBLE PRECISION

On exit DIF is the reciprocal of a lower bound of

the reciprocal of the Dif-function, i.e. DIF is an upper bound of

Dif[(A,D), (B,E)] = sigma_min(Z), where Z as in (2). IF IJOB = 0

or TRANS = 'T', DIF is not touched.

SCALE (output) DOUBLE PRECISION

On exit SCALE is the scaling factor in (1) or (3).

If 0 < SCALE < 1, C and F hold the solutions R and L, resp., to a

slightly perturbed system but the input matrices A, B, D and E

have not been changed. If SCALE = 0, C and F hold the solutions R

and L, respectively, to the homogeneous system with C = F = 0.

Normally, SCALE = 1.

WORK (workspace/output) DOUBLE PRECISION array, dimen

sion (LWORK)

If IJOB = 0, WORK is not referenced. Otherwise,

on exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK (input) INTEGER

The dimension of the array WORK. LWORK > = 1. If

IJOB = 1 or 2 and TRANS = 'N', LWORK >= 2*M*N.

If LWORK = -1, then a workspace query is assumed;

the routine only calculates the optimal size of the WORK array,

returns this value as the first entry of the WORK array, and no

error message related to LWORK is issued by XERBLA.

IWORK (workspace) INTEGER array, dimension (M+N+6)

INFO (output) INTEGER

=0: successful exit
<0: If INFO = -i, the i-th argument had an illegal

value.
>0: (A, D) and (B, E) have common or close eigen

values.

FURTHER DETAILS

Based on contributions by

Bo Kagstrom and Peter Poromaa, Department of Computing

Science,
Umea University, S-901 87 Umea, Sweden.

[1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms

and Software

for Solving the Generalized Sylvester Equation and Es

timating the
Separation between Regular Matrix Pairs, Report UMINF

- 93.23,
Department of Computing Science, Umea University,

S-901 87 Umea,
Sweden, December 1993, Revised April 1994, Also as LA

PACK Working
Note 75. To appear in ACM Trans. on Math. Software,

Vol 22,
No 1, 1996.

[2] B. Kagstrom, A Perturbation Analysis of the General

ized Sylvester

Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix

Anal.
Appl., 15(4):1045-1060, 1994

[3] B. Kagstrom and L. Westin, Generalized Schur Methods

with

Condition Estimators for Solving the Generalized

Sylvester
Equation, IEEE Transactions on Automatic Control, Vol.

34, No. 7,
July 1989, pp 745-751.

LAPACK version 3.0 15 June 2000

DTGSYL(3)