SLAGV2(3f)

NAME
SLAGV2 - the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular

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 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) 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.

LDA     (input) INTEGER
    THe leading dimension of the array A. LDA >= 2.

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.

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 eigenvalues 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 Kentucky, USA