cgegs(3F)

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

SYNOPSIS
     SUBROUTINE CGEGS( 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

         REAL          RWORK( * )

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

PURPOSE
     SGEGS 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 CGEGV
     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 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 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 array, dimension (N)
             BETA    (output) COMPLEX 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 CGEGS.  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 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 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 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
             CGEQRF, CUNMQR, and CUNGQR.)  Then compute:  NB  -- MAX of the
             blocksizes for CGEQRF, CUNMQR, and CUNGQR; the optimal LWORK is
             N*(NB+1).

     RWORK   (workspace) REAL 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 CGGBAL
             =N+2: error return from CGEQRF
             =N+3: error return from CUNMQR
             =N+4: error return from CUNGQR
             =N+5: error return from CGGHRD
             =N+6: error return from CHGEQZ (other than failed iteration)
             =N+7: error return from CGGBAK (computing VSL)
             =N+8: error return from CGGBAK (computing VSR)
             =N+9: error return from CLASCL (various places)