zhgeqz(3F)

NAME
     ZHGEQZ - implement a single-shift version of the QZ method for finding
     the generalized eigenvalues w(i)=ALPHA(i)/BETA(i) of the equation   det(
     A - w(i) B ) = 0  If JOB='S', then the pair (A,B) is simultaneously
     reduced to Schur form (i.e., A and B are both upper triangular) by
     applying one unitary tranformation (usually called Q) on the left and
     another (usually called Z) on the right

SYNOPSIS
     SUBROUTINE ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, ALPHA,
                        BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO )

         CHARACTER      COMPQ, COMPZ, JOB

         INTEGER        IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, LWORK, N

         DOUBLE         PRECISION RWORK( * )

         COMPLEX*16     A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), Q(
                        LDQ, * ), WORK( * ), Z( LDZ, * )

PURPOSE
     ZHGEQZ implements a single-shift version of the QZ method for finding the
     generalized eigenvalues w(i)=ALPHA(i)/BETA(i) of the equation A are then
     ALPHA(1),...,ALPHA(N), and of B are BETA(1),...,BETA(N).

     If JOB='S' and COMPQ and COMPZ are 'V' or 'I', then the unitary
     transformations used to reduce (A,B) are accumulated into the arrays Q
     and Z s.t.:

          Q(in) A(in) Z(in)* = Q(out) A(out) Z(out)*
          Q(in) B(in) Z(in)* = Q(out) B(out) Z(out)*

     Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
          Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
          pp. 241--256.


ARGUMENTS
     JOB     (input) CHARACTER*1
             = 'E': compute only ALPHA and BETA.  A and B will not necessarily
             be put into generalized Schur form.  = 'S': put A and B into
             generalized Schur form, as well as computing ALPHA and BETA.

     COMPQ   (input) CHARACTER*1
             = 'N': do not modify Q.
             = 'V': multiply the array Q on the right by the conjugate
             transpose of the unitary tranformation that is applied to the
             left side of A and B to reduce them to Schur form.  = 'I': like
             COMPQ='V', except that Q will be initialized to the identity
             first.
     COMPZ   (input) CHARACTER*1
             = 'N': do not modify Z.
             = 'V': multiply the array Z on the right by the unitary
             tranformation that is applied to the right side of A and B to
             reduce them to Schur form.  = 'I': like COMPZ='V', except that Z
             will be initialized to the identity first.

     N       (input) INTEGER
             The order of the matrices A, B, Q, and Z.  N >= 0.

     ILO     (input) INTEGER
             IHI     (input) INTEGER It is assumed that A is already upper
             triangular in rows and columns 1:ILO-1 and IHI+1:N.  1 <= ILO <=
             IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

     A       (input/output) COMPLEX*16 array, dimension (LDA, N)
             On entry, the N-by-N upper Hessenberg matrix A.  Elements below
             the subdiagonal must be zero.  If JOB='S', then on exit A and B
             will have been simultaneously reduced to upper triangular form.
             If JOB='E', then on exit A will have been destroyed.

     LDA     (input) INTEGER
             The leading dimension of the array A.  LDA >= max( 1, N ).

     B       (input/output) COMPLEX*16 array, dimension (LDB, N)
             On entry, the N-by-N upper triangular matrix B.  Elements below
             the diagonal must be zero.  If JOB='S', then on exit A and B will
             have been simultaneously reduced to upper triangular form.  If
             JOB='E', then on exit B will have been destroyed.

     LDB     (input) INTEGER
             The leading dimension of the array B.  LDB >= max( 1, N ).

     ALPHA   (output) COMPLEX*16 array, dimension (N)
             The diagonal elements of A when the pair (A,B) has been reduced
             to Schur form.  ALPHA(i)/BETA(i) i=1,...,N are the generalized
             eigenvalues.

     BETA    (output) COMPLEX*16 array, dimension (N)
             The diagonal elements of B when the pair (A,B) has been reduced
             to Schur form.  ALPHA(i)/BETA(i) i=1,...,N are the generalized
             eigenvalues.  A and B are normalized so that BETA(1),...,BETA(N)
             are non-negative real numbers.

     Q       (input/output) COMPLEX*16 array, dimension (LDQ, N)
             If COMPQ='N', then Q will not be referenced.  If COMPQ='V' or
             'I', then the conjugate transpose of the unitary transformations
             which are applied to A and B on the left will be applied to the
             array Q on the right.

     LDQ     (input) INTEGER
             The leading dimension of the array Q.  LDQ >= 1.  If COMPQ='V' or
             'I', then LDQ >= N.

     Z       (input/output) COMPLEX*16 array, dimension (LDZ, N)
             If COMPZ='N', then Z will not be referenced.  If COMPZ='V' or
             'I', then the unitary transformations which are applied to A and
             B on the right will be applied to the array Z on the right.

     LDZ     (input) INTEGER
             The leading dimension of the array Z.  LDZ >= 1.  If COMPZ='V' or
             'I', then LDZ >= 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,N).

     RWORK   (workspace) DOUBLE PRECISION array, dimension (N)

     INFO    (output) INTEGER
             = 0: successful exit
             < 0: if INFO = -i, the i-th argument had an illegal value
             = 1,...,N: the QZ iteration did not converge.  (A,B) is not in
             Schur form, but ALPHA(i) and BETA(i), i=INFO+1,...,N should be
             correct.  = N+1,...,2*N: the shift calculation failed.  (A,B) is
             not in Schur form, but ALPHA(i) and BETA(i), i=INFO-N+1,...,N
             should be correct.  > 2*N:     various "impossible" errors.

FURTHER DETAILS
     We assume that complex ABS works as long as its value is less than
     overflow.