slagv2(3)

NAME

SLAGV2 - compute the Generalized Schur factorization of a

real 2-by-2 matrix pencil (A,B) where B is upper triangular

SYNOPSIS

SUBROUTINE  SLAGV2(  A, LDA, B, LDB, ALPHAR, ALPHAI, BETA,
CSL, SNL, CSR, SNR )
    INTEGER        LDA, LDB
    REAL           CSL, CSR, SNL, SNR
    REAL           A( LDA, * ), ALPHAI( 2 ), ALPHAR( 2  ),
B( LDB, * ), BETA( 2 )

PURPOSE

SLAGV2 computes the Generalized Schur factorization of a

real 2-by-2 matrix pencil (A,B) where B is upper triangular. This

routine computes orthogonal (rotation) matrices given by CSL, SNL

and CSR, SNR such that

1) if the pencil (A,B) has two real eigenvalues (include

0/0 or 1/0

types), then

[ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ]
[ 0 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ]

[ b11 b12 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ]
[ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ],

2) if the pencil (A,B) has a pair of complex conjugate

eigenvalues,

then

[ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ]
[ a21 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ]

[ b11 0 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ]
[ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ]

where b11 >= b22 > 0.

ARGUMENTS

A (input/output) REAL array, dimension (LDA, 2)

On entry, the 2 x 2 matrix A. On exit, A is over

written by the ``A-part'' of the generalized Schur form.

LDA (input) INTEGER

THe leading dimension of the array A. LDA >= 2.

B (input/output) REAL array, dimension (LDB, 2)

On entry, the upper triangular 2 x 2 matrix B. On

exit, B is overwritten by the ``B-part'' of the generalized Schur

form.

LDB (input) INTEGER

THe leading dimension of the array B. LDB >= 2.

ALPHAR (output) REAL array, dimension (2)

ALPHAI (output) REAL array, dimension (2) BETA

(output) REAL array, dimension (2) (ALPHAR(k)+i*ALPHAI(k))/BE

TA(k) are the eigenvalues of the pencil (A,B), k=1,2, i =

sqrt(-1). Note that BETA(k) may be zero.

CSL (output) REAL

The cosine of the left rotation matrix.

SNL (output) REAL

The sine of the left rotation matrix.

CSR (output) REAL

The cosine of the right rotation matrix.

SNR (output) REAL

The sine of the right rotation matrix.

FURTHER DETAILS

Based on contributions by

Mark Fahey, Department of Mathematics, Univ. of Ken

tucky, USA

LAPACK version 3.0 15 June 2000

SLAGV2(3)