DLAGV2  - compute the Generalized Schur factorization of a
       real 2-by-2 matrix pencil (A,B) where B is upper  triangu­
       lar


SYNOPSIS

       SUBROUTINE DLAGV2( A,  LDA,  B, LDB, ALPHAR, ALPHAI, BETA,
                          CSL, SNL, CSR, SNR )

           INTEGER        LDA, LDB

           DOUBLE         PRECISION CSL, CSR, SNL, SNR

           DOUBLE         PRECISION A( LDA, *  ),  ALPHAI(  2  ),
                          ALPHAR( 2 ), B( LDB, * ), BETA( 2 )


PURPOSE

       DLAGV2  computes  the Generalized Schur factorization of a
       real 2-by-2 matrix pencil (A,B) where B is upper  triangu­
       lar.  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) DOUBLE PRECISION array, dimension
               (LDA, 2)
               On entry, the 2 x 2 matrix A.  On exit, A is over­
               written by the ``A-part'' of the generalized Schur
               form.

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

       B       (input/output) 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     (input) 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 pencil (A,B),  k=1,2,  i  =
               sqrt(-1).  Note that BETA(k) may be zero.

       CSL     (output) DOUBLE PRECISION
               The cosine of the left rotation matrix.

       SNL     (output) DOUBLE PRECISION
               The sine of the left rotation matrix.

       CSR     (output) DOUBLE PRECISION
               The cosine of the right rotation matrix.

       SNR     (output) DOUBLE PRECISION
               The sine of the right rotation matrix.


FURTHER DETAILS

       Based on contributions by
          Mark  Fahey,  Department  of Mathematics, Univ. of Ken­
       tucky, USA


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