DLAGV2(1) LAPACK auxiliary routine (version 3.2) DLAGV2(1)NAME
DLAGV2 - computes the Generalized Schur factorization of a real 2-by-2
matrix pencil (A,B) where B is upper triangular
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 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) DOUBLE PRECISION 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.
LDA (input) INTEGER
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 pen‐
cil (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 Kentucky, USA
LAPACK auxiliary routine (versioNovember 2008 DLAGV2(1)
