ZGGEV(3)       LAPACK routine of NEC Numeric Library Collection       ZGGEV(3)



NAME
       ZGGEV

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



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

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

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

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

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

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

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

           ALPHA     (output)
                     ALPHA is COMPLEX*16 array, dimension (N)

           BETA      (output)
                     BETA is 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)
                     VL is 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 is scaled so the largest component has
                     abs(real part) + abs(imag. part) = 1.
                     Not referenced if JOBVL = 'N'.

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

           VR        (output)
                     VR is 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 is scaled so the largest component has
                     abs(real part) + abs(imag. part) = 1.
                     Not referenced if JOBVR = 'N'.

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

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

           LWORK     (input)
                     LWORK is 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     (output)
                     RWORK is DOUBLE PRECISION array, dimension (8*N)

           INFO      (output)
                     INFO is 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.



LAPACK routine                  31 October 2017                       ZGGEV(3)