ZGEGS (3lapack)

compute for a pair of N-by-N complex nonsymmetric matrices A,

SYNOPSIS

  SUBROUTINE ZGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHA, BETA, VSL,
                    LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, INFO )

      CHARACTER     JOBVSL, JOBVSR

      INTEGER       INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N

      DOUBLE        PRECISION RWORK( * )

      COMPLEX*16    A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VSL(
                    LDVSL, * ), VSR( LDVSR, * ), WORK( * )

PURPOSE

  DGEGS computes for a pair of N-by-N complex nonsymmetric matrices A, B:
  the generalized eigenvalues (alpha, beta), the complex Schur form (A, B),
  and optionally left and/or right Schur vectors (VSL and VSR).

  (If only the generalized eigenvalues are needed, use the driver ZGEGV
  instead.)

  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)

  The (generalized) Schur form of a pair of matrices is the result of
  multiplying both matrices on the left by one unitary matrix and both on the
  right by another unitary matrix, these two unitary matrices being chosen so
  as to bring the pair of matrices into upper triangular form with the
  diagonal elements of B being non-negative real numbers (this is also called
  complex Schur form.)

  The left and right Schur vectors are the columns of VSL and VSR,
  respectively, where VSL and VSR are the unitary matrices
  which reduce A and B to Schur form:

  Schur form of (A,B) = ( (VSL)**H A (VSR), (VSL)**H B (VSR) )

ARGUMENTS

  JOBVSL   (input) CHARACTER*1
           = 'N':  do not compute the left Schur vectors;
           = 'V':  compute the left Schur vectors.

  JOBVSR   (input) CHARACTER*1
           = 'N':  do not compute the right Schur vectors;
           = 'V':  compute the right Schur vectors.

  N       (input) INTEGER
          The order of the matrices A, B, VSL, and VSR.  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) Schur vectors are to be computed.  On
          exit, the generalized Schur form of A.

  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) Schur vectors are to be computed.  On
          exit, the generalized Schur form of B.

  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.
          ALPHA(j), j=1,...,N  and  BETA(j), j=1,...,N  are the diagonals of
          the complex Schur form (A,B) output by ZGEGS.  The  BETA(j) will be
          non-negative real.

          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).

  VSL     (output) COMPLEX*16 array, dimension (LDVSL,N)
          If JOBVSL = 'V', VSL will contain the left Schur vectors.  (See
          "Purpose", above.) Not referenced if JOBVSL = 'N'.

  LDVSL   (input) INTEGER
          The leading dimension of the matrix VSL. LDVSL >= 1, and if JOBVSL
          = 'V', LDVSL >= N.

  VSR     (output) COMPLEX*16 array, dimension (LDVSR,N)
          If JOBVSR = 'V', VSR will contain the right Schur vectors.  (See
          "Purpose", above.) Not referenced if JOBVSR = 'N'.

  LDVSR   (input) INTEGER
          The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR
          = 'V', LDVSR >= 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 N*(NB+1).

  RWORK   (workspace) DOUBLE PRECISION array, dimension (3*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.  (A,B) are not in Schur form,
          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 ZGGBAK (computing VSL)
          =N+8: error return from ZGGBAK (computing VSR)
          =N+9: error return from ZLASCL (various places)