DLAGV2 - compute the Generalized Schur factorization of a
real 2-by-2 matrix pencil (A,B) where B is upper triangu
lar
SYNOPSIS
SUBROUTINE DLAGV2( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA,
CSL, SNL, CSR, SNR )
INTEGER LDA, LDB
DOUBLE PRECISION CSL, CSR, SNL, SNR
DOUBLE PRECISION A( LDA, * ), ALPHAI( 2 ),
ALPHAR( 2 ), B( LDB, * ), BETA( 2 )
PURPOSE
DLAGV2 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) DOUBLE PRECISION 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.
THe leading dimension of the array A. LDA >= 2.
B (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (2)
ALPHAI (output) DOUBLE PRECISION array, dimension
(2) BETA (output) DOUBLE PRECISION 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.
CSL (output) DOUBLE PRECISION
The cosine of the left rotation matrix.
SNL (output) DOUBLE PRECISION
The sine of the left rotation matrix.
CSR (output) DOUBLE PRECISION
The cosine of the right rotation matrix.
SNR (output) DOUBLE PRECISION
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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