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