CXML
        	    
LAPACK version 3.0
zgegv(3)

PURPOSE
  ZGEGV - routine is deprecated and has been replaced by routine ZGGEV

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

      CHARACTER	    JOBVL, JOBVR

      INTEGER	    INFO, LDA, LDB, LDVL, LDVR, LWORK, N

      DOUBLE	    PRECISION RWORK( * )

      COMPLEX*16    A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VL(
		    LDVL, * ), VR( LDVR, * ), WORK( * )

DESCRIPTION
  This routine is deprecated and has been replaced by routine ZGGEV.  ZGEGV
  computes for a pair of N-by-N complex nonsymmetric matrices A and B, the
  generalized eigenvalues (alpha, 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) COMPLEX*16 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) COMPLEX*16 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).

  ALPHA	  (output) COMPLEX*16 array, dimension (N)
	  BETA	  (output) 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) COMPLEX*16 array, dimension (LDVL,N)
	  If JOBVL = 'V', the left generalized eigenvectors.  (See "Purpose",
	  above.) 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) COMPLEX*16 array, dimension (LDVR,N)
	  If JOBVR = 'V', the right generalized eigenvectors.  (See
	  "Purpose", above.) 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) COMPLEX*16 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(1,2*N).  For good
	  performance, LWORK must generally be larger.	To compute the
	  optimal value of LWORK, call ILAENV to get blocksizes (for ZGEQRF,
	  ZUNMQR, and CUNGQR.)	Then compute: NB  -- MAX of the blocksizes
	  for ZGEQRF, ZUNMQR, and CUNGQR; The optimal LWORK is	MAX( 2*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.

  RWORK	  (workspace/output) DOUBLE PRECISION array, dimension (8*N)

  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 ALPHA(j) and BETA(j) should be correct for
	  j=INFO+1,...,N.  > N:	 errors that usually indicate LAPACK
	  problems:
	  =N+1: error return from ZGGBAL
	  =N+2: error return from ZGEQRF
	  =N+3: error return from ZUNMQR
	  =N+4: error return from ZUNGQR
	  =N+5: error return from ZGGHRD
	  =N+6: error return from ZHGEQZ (other than failed iteration) =N+7:
	  error return from ZTGEVC
	  =N+8: error return from ZGGBAK (computing VL)
	  =N+9: error return from ZGGBAK (computing VR)
	  =N+10: error return from ZLASCL (various calls)

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

  This driver calls ZGGBAL 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, ZGGBAK 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 complex Schur form[*] of
  the "balanced" versions of A and B.  If no eigenvectors are computed, then
  only the diagonal blocks will be correct.

  [*] In other words, upper triangular form.