zggev(3S)

NAME
     ZGGEV - compute for a pair of N-by-N complex nonsymmetric matrices (A,B),
     the generalized eigenvalues, and optionally, the left and/or right
     generalized eigenvectors

SYNOPSIS
     SUBROUTINE ZGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL, LDVL,
                       VR, LDVR, WORK, LWORK, RWORK, INFO )

         CHARACTER     JOBVL, JOBVR

         INTEGER       INFO, LDA, LDB, LDVL, LDVR, LWORK, N

         DOUBLE        PRECISION RWORK( * )

         COMPLEX*16    A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VL(
                       LDVL, * ), VR( LDVR, * ), WORK( * )

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
     ZGGEV computes for a pair of N-by-N complex nonsymmetric matrices (A,B),
     the generalized eigenvalues, and optionally, the left and/or right
     generalized eigenvectors. A generalized eigenvalue for a pair of matrices
     (A,B) is a scalar lambda or a ratio alpha/beta = lambda, such that A -
     lambda*B is singular. It is usually represented as the pair (alpha,beta),
     as there is a reasonable interpretation for beta=0, and even for both
     being zero.

     The right generalized eigenvector v(j) corresponding to the generalized
     eigenvalue lambda(j) of (A,B) satisfies

                  A * v(j) = lambda(j) * B * v(j).

     The left generalized eigenvector u(j) corresponding to the generalized
     eigenvalues lambda(j) of (A,B) satisfies

                  u(j)**H * A = lambda(j) * u(j)**H * B

     where u(j)**H is the conjugate-transpose of u(j).

ARGUMENTS
     JOBVL   (input) CHARACTER*1
             = 'N':  do not compute the left generalized eigenvectors;
             = 'V':  compute the left generalized eigenvectors.

     JOBVR   (input) CHARACTER*1
             = 'N':  do not compute the right generalized eigenvectors;
             = 'V':  compute the right generalized eigenvectors.

     N       (input) INTEGER
             The order of the matrices A, B, VL, and VR.  N >= 0.

     A       (input/output) COMPLEX*16 array, dimension (LDA, N)
             On entry, the matrix A in the pair (A,B).  On exit, A has been
             overwritten.

     LDA     (input) INTEGER
             The leading dimension of A.  LDA >= max(1,N).

     B       (input/output) COMPLEX*16 array, dimension (LDB, N)
             On entry, the matrix B in the pair (A,B).  On exit, B has been
             overwritten.

     LDB     (input) INTEGER
             The leading dimension of B.  LDB >= max(1,N).

     ALPHA   (output) COMPLEX*16 array, dimension (N)
             BETA    (output) COMPLEX*16 array, dimension (N) On exit,
             ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenvalues.

             Note: the quotients ALPHA(j)/BETA(j) may easily over- or
             underflow, and BETA(j) may even be zero.  Thus, the user should
             avoid naively computing the ratio alpha/beta.  However, ALPHA
             will be always less than and usually comparable with norm(A) in
             magnitude, and BETA always less than and usually comparable with
             norm(B).

     VL      (output) COMPLEX*16 array, dimension (LDVL,N)
             If JOBVL = 'V', the left generalized eigenvectors u(j) are stored
             one after another in the columns of VL, in the same order as
             their eigenvalues.  Each eigenvector will be scaled so the
             largest component will have abs(real part) + abs(imag. part) = 1.
             Not referenced if JOBVL = 'N'.

     LDVL    (input) INTEGER
             The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL =
             'V', LDVL >= N.

     VR      (output) COMPLEX*16 array, dimension (LDVR,N)
             If JOBVR = 'V', the right generalized eigenvectors v(j) are
             stored one after another in the columns of VR, in the same order
             as their eigenvalues.  Each eigenvector will be scaled so the
             largest component will have abs(real part) + abs(imag. part) = 1.
             Not referenced if JOBVR = 'N'.

     LDVR    (input) INTEGER
             The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR =
             'V', LDVR >= N.

     WORK    (workspace/output) COMPLEX*16 array, dimension (LWORK)
             On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

     LWORK   (input) INTEGER
             The dimension of the array WORK.  LWORK >= max(1,2*N).  For good
             performance, LWORK must generally be larger.

             If LWORK = -1, then a workspace query is assumed; the routine
             only calculates the optimal size of the WORK array, returns this
             value as the first entry of the WORK array, and no error message
             related to LWORK is issued by XERBLA.

     RWORK   (workspace/output) DOUBLE PRECISION array, dimension (8*N)

     INFO    (output) INTEGER
             = 0:  successful exit
             < 0:  if INFO = -i, the i-th argument had an illegal value.
             =1,...,N:  The QZ iteration failed.  No eigenvectors have been
             calculated, but ALPHA(j) and BETA(j) should be correct for
             j=INFO+1,...,N.  > N:  =N+1: other then QZ iteration failed in
             DHGEQZ,
             =N+2: error return from DTGEVC.

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

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