ctgsy2(3)

NAME

CTGSY2 - solve 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,

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 * 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. 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 trans

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

trices 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 ma

trix pairs in CTGSYL.

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 Frobe

nius norm-based estimate of the separation between two matrix

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

bution 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 solu

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

= 1.

RDSUM (input/output) REAL

On entry, the sum of squares of computed contribu

tions to the Dif-estimate under computation by CTGSYL, 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 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 contri

butions 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 ille

gal.
>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

CTGSY2(3)