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Periodic Schur factorization of a real 2-by-2 matrix pair (A,B) with B upper triangular

**[Specification]**
**[Arguments]**
**[Method]**
**[References]**
**[Comments]**
**[Example]**
**Purpose**

To compute the periodic Schur factorization of a real 2-by-2
matrix pair (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 pair (A,B) has two real eigenvalues, then
[ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ]
[ 0 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ]
[ b11 b12 ] := [ CSR SNR ] [ b11 b12 ] [ CSL -SNL ]
[ 0 b22 ] [ -SNR CSR ] [ 0 b22 ] [ SNL CSL ],
2) if the pair (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 ] := [ CSR SNR ] [ b11 b12 ] [ CSL -SNL ]
[ 0 b22 ] [ -SNR CSR ] [ 0 b22 ] [ SNL CSL ].
This is a modified version of the LAPACK routine DLAGV2 for
computing the real, generalized Schur decomposition of a
two-by-two matrix pencil.

**Specification**
SUBROUTINE MB03YT( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL,
$ CSR, SNR )
C .. Scalar Arguments ..
INTEGER LDA, LDB
DOUBLE PRECISION CSL, CSR, SNL, SNR
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), ALPHAI(2), ALPHAR(2), B(LDB,*),
$ BETA(2)

**Arguments**
**Input/Output Parameters**

A (input/output) DOUBLE PRECISION array, dimension (LDA,2)
On entry, the leading 2-by-2 part of this array must
contain the matrix A.
On exit, the leading 2-by-2 part of this array contains
the matrix A of the pair in periodic Schur form.
LDA INTEGER
The leading dimension of the array A. LDA >= 2.
B (input/output) DOUBLE PRECISION array, dimension (LDB,2)
On entry, the leading 2-by-2 part of this array must
contain the upper triangular matrix B.
On exit, the leading 2-by-2 part of this array contains
the matrix B of the pair in periodic Schur form.
LDB 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
pair (A,B), k=1,2, i = sqrt(-1). ALPHAI(1) >= 0.
CSL (output) DOUBLE PRECISION
The cosine of the first rotation matrix.
SNL (output) DOUBLE PRECISION
The sine of the first rotation matrix.
CSR (output) DOUBLE PRECISION
The cosine of the second rotation matrix.
SNR (output) DOUBLE PRECISION
The sine of the second rotation matrix.

**References**
[1] Van Loan, C.
Generalized Singular Values with Algorithms and Applications.
Ph. D. Thesis, University of Michigan, 1973.

**Further Comments**
None

**Example**
**Program Text**

None

**Program Data**
None

**Program Results**
None

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