Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3 - routine i deprecated and has been replaced by routine DGGEV

SYNOPSIS

       SUBROUTINE DGEGV( 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( * )

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       This routine is deprecated and has been replaced by routine DGGEV.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 -- see
        "Further Details", below.

ARGUMENTS

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

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

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

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

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

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

        WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,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.
                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) 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 driver routine (version 3.2)        April 2011                             DGEGV(3lapack)