# lagv2 - Man Page

lagv2: 2x2 generalized Schur factor

## Synopsis

### Functions

subroutine dlagv2 (a, lda, b, ldb, alphar, alphai, beta, csl, snl, csr, snr)
DLAGV2 computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular.
subroutine slagv2 (a, lda, b, ldb, alphar, alphai, beta, csl, snl, csr, snr)
SLAGV2 computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular.

## Function Documentation

### subroutine dlagv2 (double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( 2 ) alphar, double precision, dimension( 2 ) alphai, double precision, dimension( 2 ) beta, double precision csl, double precision snl, double precision csr, double precision snr)

DLAGV2 computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular.

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 ]
[  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.```
Parameters

A

```          A is DOUBLE PRECISION 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

```          LDA is INTEGER
THe leading dimension of the array A.  LDA >= 2.```

B

```          B is DOUBLE PRECISION 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

```          LDB is INTEGER
THe leading dimension of the array B.  LDB >= 2.```

ALPHAR

`          ALPHAR is DOUBLE PRECISION array, dimension (2)`

ALPHAI

`          ALPHAI is DOUBLE PRECISION array, dimension (2)`

BETA

```          BETA is 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

```          CSL is DOUBLE PRECISION
The cosine of the left rotation matrix.```

SNL

```          SNL is DOUBLE PRECISION
The sine of the left rotation matrix.```

CSR

```          CSR is DOUBLE PRECISION
The cosine of the right rotation matrix.```

SNR

```          SNR is DOUBLE PRECISION
The sine of the right rotation matrix.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Definition at line 155 of file dlagv2.f.

### subroutine slagv2 (real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( 2 ) alphar, real, dimension( 2 ) alphai, real, dimension( 2 ) beta, real csl, real snl, real csr, real snr)

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

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.```
Parameters

A

```          A is 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

```          LDA is INTEGER
THe leading dimension of the array A.  LDA >= 2.```

B

```          B is 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

```          LDB is INTEGER
THe leading dimension of the array B.  LDB >= 2.```

ALPHAR

`          ALPHAR is REAL array, dimension (2)`

ALPHAI

`          ALPHAI is REAL array, dimension (2)`

BETA

```          BETA is 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

```          CSL is REAL
The cosine of the left rotation matrix.```

SNL

```          SNL is REAL
The sine of the left rotation matrix.```

CSR

```          CSR is REAL
The cosine of the right rotation matrix.```

SNR

```          SNR is REAL
The sine of the right rotation matrix.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Definition at line 155 of file slagv2.f.

## Author

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## Info

Tue Nov 28 2023 12:08:43 Version 3.12.0 LAPACK