SLAGV2
June 2010
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.
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
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA