CXML
        	    
LAPACK version 3.0
dgegv(3)

PURPOSE
  DGEGV - routine is deprecated and has been replaced by routine DGGEV

SYNTAX
  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( * )

DESCRIPTION
  This routine is deprecated and has been replaced by routine DGGEV.  DGEGV
  computes for a pair of n-by-n real nonsymmetric matrices A and B, the
  generalized eigenvalues (alphar +/- alphai*i, beta), and optionally, the
  left and/or right generalized eigenvectors (VL and VR).

  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
  interpretation 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 generalized eigenvalue
  w  for a pair of matrices (A,B) is a vector  r  such that  (A - w B) r = 0
  .  A left generalized eigenvector is a vector l such that l**H * (A - w B)
  = 0, where l**H is the
  conjugate-transpose of l.

  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.

  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 first of the pair of matrices whose generalized
	  eigenvalues and (optionally) generalized eigenvectors are to be
	  computed.  On exit, the contents 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).

  B	  (input/output) DOUBLE PRECISION array, dimension (LDB, N)
	  On entry, the second of the pair of matrices whose generalized
	  eigenvalues and (optionally) generalized eigenvectors are to be
	  computed.  On exit, the contents 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, 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 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 JOBVR = '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.

  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.	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.