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 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 Kentucky, USA