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.

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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

September 2012

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