# dlagv2 (l) - Linux Man Pages

## 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 ]
a22    -SNL  CSL a21 a22  SNR  CSR ]
b11 b12 :=  CSL  SNL b11 b12  CSR -SNR ]
b22    -SNL  CSL   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   :=  CSL  SNL b11 b12  CSR -SNR ]
b22    -SNL  CSL   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-partaqaq 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-partaqaq 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