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



NAME
       DGEGV

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



PURPOSE
            This routine is deprecated and has been replaced by routine DGGEV.

            DGEGV computes the eigenvalues and, optionally, the left and/or right
            eigenvectors of a real matrix pair (A,B).
            Given two square matrices A and B,
            the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
            eigenvalues lambda and corresponding (non-zero) eigenvectors x such
            that

               A*x = lambda*B*x.

            An alternate form is to find the eigenvalues mu and corresponding
            eigenvectors y such that

               mu*A*y = B*y.

            These two forms are equivalent with mu = 1/lambda and x = y if
            neither lambda nor mu is zero.  In order to deal with the case that
            lambda or mu is zero or small, two values alpha and beta are returned
            for each eigenvalue, such that lambda = alpha/beta and
            mu = beta/alpha.

            The vectors x and y in the above equations are right eigenvectors of
            the matrix pair (A,B).  Vectors u and v satisfying

               u**H*A = lambda*u**H*B  or  mu*v**H*A = v**H*B

            are left eigenvectors of (A,B).

            Note: this routine performs "full balancing" on A and B




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

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

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

           A         (input/output)
                     A is DOUBLE PRECISION array, dimension (LDA, N)
                     On entry, the matrix A.
                     If JOBVL = 'V' or JOBVR = 'V', then on exit A
                     contains the real Schur form of A from the generalized Schur
                     factorization of the pair (A,B) after balancing.
                     If no eigenvectors were computed, then only the diagonal
                     blocks from the Schur form will be correct.  See DGGHRD and
                     DHGEQZ for details.

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

           B         (input/output)
                     B is DOUBLE PRECISION array, dimension (LDB, N)
                     On entry, the matrix B.
                     If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
                     upper triangular matrix obtained from B in the generalized
                     Schur factorization of the pair (A,B) after balancing.
                     If no eigenvectors were computed, then only those elements of
                     B corresponding to the diagonal blocks from the Schur form of
                     A will be correct.  See DGGHRD and DHGEQZ for details.

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

           ALPHAR    (output)
                     ALPHAR is DOUBLE PRECISION array, dimension (N)
                     The real parts of each scalar alpha defining an eigenvalue of
                     GNEP.

           ALPHAI    (output)
                     ALPHAI is DOUBLE PRECISION array, dimension (N)
                     The imaginary parts of each scalar alpha defining an
                     eigenvalue of GNEP.  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) = -ALPHAI(j).

           BETA      (output)
                     BETA is DOUBLE PRECISION array, dimension (N)
                     The scalars beta that define the eigenvalues of GNEP.

                     Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
                     beta = BETA(j) represent the j-th eigenvalue of the matrix
                     pair (A,B), in one of the forms lambda = alpha/beta or
                     mu = beta/alpha.  Since either lambda or mu may overflow,
                     they should not, in general, be computed.

           VL        (output)
                     VL is DOUBLE PRECISION array, dimension (LDVL,N)
                     If JOBVL = 'V', the left eigenvectors u(j) are stored
                     in the columns of VL, in the same order as their eigenvalues.
                     If the j-th eigenvalue is real, then u(j) = VL(:,j).
                     If the j-th and (j+1)-st 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 is scaled so that its largest component has
                     abs(real part) + abs(imag. part) = 1, except for eigenvectors
                     corresponding to an eigenvalue with alpha = beta = 0, which
                     are set to zero.
                     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 DOUBLE PRECISION array, dimension (LDVR,N)
                     If JOBVR = 'V', the right eigenvectors x(j) are stored
                     in the columns of VR, in the same order as their eigenvalues.
                     If the j-th eigenvalue is real, then x(j) = VR(:,j).
                     If the j-th and (j+1)-st eigenvalues form a complex conjugate
                     pair, then
                       x(j) = VR(:,j) + i*VR(:,j+1)
                     and
                       x(j+1) = VR(:,j) - i*VR(:,j+1).

                     Each eigenvector is scaled so that its largest component has
                     abs(real part) + abs(imag. part) = 1, except for eigenvalues
                     corresponding to an eigenvalue with alpha = beta = 0, which
                     are set to zero.
                     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 DOUBLE PRECISION 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(1,8*N).
                     For good performance, LWORK must generally be larger.
                     To compute the optimal value of LWORK, call ILAENV to get
                     blocksizes (for DGEQRF, DORMQR, and DORGQR.)  Then compute:
                     NB  -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR;
                     The optimal LWORK is:
                         2*N + MAX( 6*N, N*(NB+1) ).

                     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)
                     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 ALPHAR(j), ALPHAI(j), and BETA(j)
                           should be correct for j=INFO+1,...,N.
                     > N:  errors that usually indicate LAPACK problems:
                           =N+1: error return from DGGBAL
                           =N+2: error return from DGEQRF
                           =N+3: error return from DORMQR
                           =N+4: error return from DORGQR
                           =N+5: error return from DGGHRD
                           =N+6: error return from DHGEQZ (other than failed
                                                           iteration)
                           =N+7: error return from DTGEVC
                           =N+8: error return from DGGBAK (computing VL)
                           =N+9: error return from DGGBAK (computing VR)
                           =N+10: error return from DLASCL (various calls)






FURTHER DETAILS
             Balancing
             ---------

             This driver calls DGGBAL to both permute and scale rows and columns
             of A and B.  The permutations PL and PR are chosen so that PL*A*PR
             and PL*B*R will be upper triangular except for the diagonal blocks
             A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
             possible.  The diagonal scaling matrices DL and DR are chosen so
             that the pair  DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
             one (except for the elements that start out zero.)

             After the eigenvalues and eigenvectors of the balanced matrices
             have been computed, DGGBAK transforms the eigenvectors back to what
             they would have been (in perfect arithmetic) if they had not been
             balanced.

             Contents of A and B on Exit
             -------- -- - --- - -- ----

             If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
             both), then on exit the arrays A and B will contain the real Schur
             form[*] of the "balanced" versions of A and B.  If no eigenvectors
             are computed, then only the diagonal blocks will be correct.

             [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations",
                 by Golub & van Loan, pub. by Johns Hopkins U. Press.



LAPACK routine                  31 October 2017                       DGEGV(3)