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dgegv


 NAME
      DGEGV - a pair of N-by-N real nonsymmetric matrices A, B

 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
      For a pair of N-by-N real nonsymmetric matrices A, B:

         compute the generalized eigenvalues (alphar +/- alphai*i,
      beta)
         compute the left and/or right generalized eigenvectors
                 (VL and VR)

      The second action is optional -- see the description of
      JOBVL and JOBVR below.

      A generalized eigenvalue for a pair of matrices (A,B) is,
      roughly speaking, a scalar w or a ratio  alpha/beta = w,
      such that  A - w*B is singular.  It is usually represented
      as the pair (alpha,beta), as there is a reasonable interpre-
      tation for beta=0, and even for both being zero.  A good
      beginning reference is the book, "Matrix Computations", by
      G. Golub & C. van Loan (Johns Hopkins U. Press)

      A right generalized eigenvector corresponding to a general-
      ized eigenvalue  w  for a pair of matrices (A,B) is a vector
      r  such that  (A - w B) r = 0 .  A left generalized eigen-
      vector is a vector
                             H
      l  such that  (A - w B) l = 0 .

      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 eigen-
              vectors;
              = 'V':  compute the left generalized eigenvectors.

      JOBVR   (input) CHARACTER*1

              = 'N':  do not compute the right generalized eigen-
              vectors;
              = 'V':  compute the right generalized eigenvectors.

      N       (input) INTEGER
              The number of rows and columns in the matrices A, B,
              VL, and VR.  N >= 0.

 N)
      A       (input/workspace) DOUBLE PRECISION array, dimension (LDA,
              On entry, the first of the pair of matrices whose
              generalized eigenvalues and (optionally) generalized
              eigenvectors are to be computed.  On exit, the con-
              tents will have been destroyed.  (For a description
              of the contents of A on exit, see "Further Details",
              below.)

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

 N)
      B       (input/workspace) DOUBLE PRECISION array, dimension (LDB,
              On entry, the second of the pair of matrices whose
              generalized eigenvalues and (optionally) generalized
              eigenvectors are to be computed.  On exit, the con-
              tents will have been destroyed.  (For a description
              of the contents of B on exit, see "Further Details",
              below.)

      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, dimen-
              sion (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) nega-
              tive.

              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.  How-
              ever, 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 generalized eigenvectors.
              (See "Purpose", above.)  Real eigenvectors take one
              column, complex take two columns, the first for the
              real part and the second for the imaginary part.
              Complex eigenvectors correspond to an eigenvalue
              with positive imaginary part.  Each eigenvector will
              be scaled so the largest component will have
              abs(real part) + abs(imag. part) = 1, *except* that
              for eigenvalues with alpha=beta=0, a zero vector
              will be returned as the corresponding eigenvector.
              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 JOBVL = 'V', the right generalized eigenvectors.
              (See "Purpose", above.)  Real eigenvectors take one
              column, complex take two columns, the first for the
              real part and the second for the imaginary part.
              Complex eigenvectors correspond to an eigenvalue
              with positive imaginary part.  Each eigenvector will
              be scaled so the largest component will have
              abs(real part) + abs(imag. part) = 1, *except* that
              for eigenvalues with alpha=beta=0, a zero vector
              will be returned as the corresponding eigenvector.
              Not referenced if JOBVR = 'N'.

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

 (LWORK)
      WORK    (workspace/output) DOUBLE PRECISION array, dimension
              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 gen-
              erally 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) ).

      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 entries close to one
      (except for the entries that start out zero.)

      After the eigenvalues and eigenvectors of the balanced
      matrices have been computed, DGGBAK transforms the eigenvec-
      tors back to what they would have been (in perfect arith-
      metic) 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 Computa-
      tions",
          by Golub & van Loan, pub. by Johns Hopkins U. Press.