dggev(3S)

NAME
     DGGEV - compute for a pair of N-by-N real nonsymmetric matrices (A,B)

SYNOPSIS
     SUBROUTINE DGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA,
                       VL, LDVL, VR, LDVR, WORK, LWORK, INFO )

         CHARACTER     JOBVL, JOBVR

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

         DOUBLE        PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), 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
     DGGEV computes for a pair of N-by-N real 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 eigenvector v(j) corresponding to the eigenvalue lambda(j) of
     (A,B) satisfies

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

     The left eigenvector u(j) corresponding to the eigenvalue 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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).

     ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
             ALPHAI  (output) DOUBLE PRECISION array, dimension (N) BETA
             (output) DOUBLE PRECISION array, dimension (N) On exit,
             (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the
             generalized eigenvalues.  If ALPHAI(j) is zero, then the j-th
             eigenvalue is real; if positive, then the j-th and (j+1)-st
             eigenvalues are a complex conjugate pair, with ALPHAI(j+1)
             negative.

             Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(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, ALPHAR and ALPHAI 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) DOUBLE PRECISION array, dimension (LDVL,N)
             If JOBVL = 'V', the left eigenvectors u(j) are stored one after
             another in the columns of VL, in the same order as their
             eigenvalues. If the j-th eigenvalue is real, then u(j) = VL(:,j),
             the j-th column of VL. If the j-th and (j+1)-th eigenvalues form
             a complex conjugate pair, then u(j) = VL(:,j)+i*VL(:,j+1) and
             u(j+1) = VL(:,j)-i*VL(:,j+1).  Each eigenvector will be scaled so
             the largest component 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) DOUBLE PRECISION array, dimension (LDVR,N)
             If JOBVR = 'V', the right eigenvectors v(j) are stored one after
             another in the columns of VR, in the same order as their
             eigenvalues. If the j-th eigenvalue is real, then v(j) = VR(:,j),
             the j-th column of VR. If the j-th and (j+1)-th eigenvalues form
             a complex conjugate pair, then v(j) = VR(:,j)+i*VR(:,j+1) and
             v(j+1) = VR(:,j)-i*VR(:,j+1).  Each eigenvector will be scaled so
             the largest component 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) DOUBLE PRECISION 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(1,8*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.

     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 ALPHAR(j), ALPHAI(j), and BETA(j) should be
             correct for j=INFO+1,...,N.  > N:  =N+1: other than QZ iteration
             failed in DHGEQZ.
             =N+2: error return from DTGEVC.

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

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