lagv2(3) LAPACK lagv2(3)

lagv2 - lagv2: 2x2 generalized Schur factor


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.

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

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

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

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

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