ZGEGV (3lapack)

compute for a pair of N-by-N complex nonsymmetric matrices A and B, the generalized eigenvalues (alpha, beta), and optionally,

SYNOPSIS

  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( * )

PURPOSE

  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 JOBVL = '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) ).

  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.