SLAGV2 - compute the Generalized Schur factorization of a
real 2-by-2 matrix pencil (A,B) where B is upper triangu
lar
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 triangu
lar. 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.
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))/BETA(k) are the eigenval
ues 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
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