slagv2(3S)

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

SYNOPSIS
     SUBROUTINE SLAGV2( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL, CSR,
                        SNR )

         INTEGER        LDA, LDB

         REAL           CSL, CSR, SNL, SNR

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

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

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.
ARGUMENTS
     A       (input/output) 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     (input) INTEGER
             THe leading dimension of the array A.  LDA >= 2.

     B       (input/output) 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     (input) INTEGER
             THe leading dimension of the array B.  LDB >= 2.

     ALPHAR  (output) REAL array, dimension (2)
             ALPHAI  (output) REAL array, dimension (2) BETA    (output) 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     (output) REAL
             The cosine of the left rotation matrix.

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

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

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

FURTHER DETAILS
     Based on contributions by
        Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA


SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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