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.TH SLAGV2 l "15 June 2000" "LAPACK version 3.0" ")"
.SH NAME
SLAGV2 - compute the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular
.SH SYNOPSIS
.TP 19
SUBROUTINE SLAGV2(
A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL,
CSR, SNR )
.TP 19
.ti +4
INTEGER
LDA, LDB
.TP 19
.ti +4
REAL
CSL, CSR, SNL, SNR
.TP 19
.ti +4
REAL
A( LDA, * ), ALPHAI( 2 ), ALPHAR( 2 ),
B( LDB, * ), BETA( 2 )
.SH 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
.br
1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0
types), then
.br
[ 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
.br
[ 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.
.br
.SH ARGUMENTS
.TP 8
A (input/output) REAL array, dimension (LDA, 2)
On entry, the 2 x 2 matrix A.
On exit, A is overwritten by the ``A-part'' of the
generalized Schur form.
.TP 8
LDA (input) INTEGER
THe leading dimension of the array A. LDA >= 2.
.TP 8
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.
.TP 8
LDB (input) INTEGER
THe leading dimension of the array B. LDB >= 2.
.TP 8
ALPHAR (output) REAL array, dimension (2)
ALPHAI (output) REAL array, dimension (2)
BETA (output) REAL array, dimension (2)
(ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the
pencil (A,B), k=1,2, i = sqrt(-1). Note that BETA(k) may
be zero.
.TP 8
CSL (output) REAL
The cosine of the left rotation matrix.
.TP 8
SNL (output) REAL
The sine of the left rotation matrix.
.TP 8
CSR (output) REAL
The cosine of the right rotation matrix.
.TP 8
SNR (output) REAL
The sine of the right rotation matrix.
.SH FURTHER DETAILS
Based on contributions by
.br
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
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