dhgeqz(3F)

NAME
     DHGEQZ - implement a single-/double-shift version of the QZ method for
     finding the generalized eigenvalues  w(j)=(ALPHAR(j) +
     i*ALPHAI(j))/BETAR(j) of the equation   det( A - w(i) B ) = 0  In
     addition, the pair A,B may be reduced to generalized Schur form

SYNOPSIS
     SUBROUTINE DHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB,
                        ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
                        INFO )

         CHARACTER      COMPQ, COMPZ, JOB

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

         DOUBLE         PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B(
                        LDB, * ), BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, *
                        )

PURPOSE
     DHGEQZ implements a single-/double-shift version of the QZ method for
     finding the generalized eigenvalues B is upper triangular, and A is block
     upper triangular, where the diagonal blocks are either 1-by-1 or 2-by-2,
     the 2-by-2 blocks having complex generalized eigenvalues (see the
     description of the argument JOB.)

     If JOB='S', then the pair (A,B) is simultaneously reduced to Schur form
     by applying one orthogonal tranformation (usually called Q) on the left
     and another (usually called Z) on the right.  The 2-by-2 upper-triangular
     diagonal blocks of B corresponding to 2-by-2 blocks of A will be reduced
     to positive diagonal matrices.  (I.e., if A(j+1,j) is non-zero, then
     B(j+1,j)=B(j,j+1)=0 and B(j,j) and B(j+1,j+1) will be positive.)

     If JOB='E', then at each iteration, the same transformations are
     computed, but they are only applied to those parts of A and B which are
     needed to compute ALPHAR, ALPHAI, and BETAR.

     If JOB='S' and COMPQ and COMPZ are 'V' or 'I', then the orthogonal
     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 ALPHAR, ALPHAI, 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 ALPHAR,
             ALPHAI, and BETA.

     COMPQ   (input) CHARACTER*1
             = 'N': do not modify Q.
             = 'V': multiply the array Q on the right by the transpose of the
             orthogonal 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 orthogonal
             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) DOUBLE PRECISION 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 generalized Schur form.
             If JOB='E', then on exit A will have been destroyed.  The
             diagonal blocks will be correct, but the off-diagonal portion
             will be meaningless.

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

     B       (input/output) DOUBLE PRECISION array, dimension (LDB, N)
             On entry, the N-by-N upper triangular matrix B.  Elements below
             the diagonal must be zero.  2-by-2 blocks in B corresponding to
             2-by-2 blocks in A will be reduced to positive diagonal form.
             (I.e., if A(j+1,j) is non-zero, then B(j+1,j)=B(j,j+1)=0 and
             B(j,j) and B(j+1,j+1) will be positive.)  If JOB='S', then on
             exit A and B will have been simultaneously reduced to Schur form.
             If JOB='E', then on exit B will have been destroyed.  Elements
             corresponding to diagonal blocks of A will be correct, but the
             off-diagonal portion will be meaningless.

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

     ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
             ALPHAR(1:N) will be set to real parts of the diagonal elements of
             A that would result from reducing A and B to Schur form and then
             further reducing them both to triangular form using unitary
             transformations s.t. the diagonal of B was non-negative real.
             Thus, if A(j,j) is in a 1-by-1 block (i.e., A(j+1,j)=A(j,j+1)=0),
             then ALPHAR(j)=A(j,j).  Note that the (real or complex) values
             (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the generalized
             eigenvalues of the matrix pencil A - wB.

     ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
             ALPHAI(1:N) will be set to imaginary parts of the diagonal
             elements of A that would result from reducing A and B to Schur
             form and then further reducing them both to triangular form using
             unitary transformations s.t. the diagonal of B was non-negative
             real.  Thus, if A(j,j) is in a 1-by-1 block (i.e.,
             A(j+1,j)=A(j,j+1)=0), then ALPHAR(j)=0.  Note that the (real or
             complex) values (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are
             the generalized eigenvalues of the matrix pencil A - wB.

     BETA    (output) DOUBLE PRECISION array, dimension (N)
             BETA(1:N) will be set to the (real) diagonal elements of B that
             would result from reducing A and B to Schur form and then further
             reducing them both to triangular form using unitary
             transformations s.t. the diagonal of B was non-negative real.
             Thus, if A(j,j) is in a 1-by-1 block (i.e., A(j+1,j)=A(j,j+1)=0),
             then BETA(j)=B(j,j).  Note that the (real or complex) values
             (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the generalized
             eigenvalues of the matrix pencil A - wB.  (Note that BETA(1:N)
             will always be non-negative, and no BETAI is necessary.)

     Q       (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
             If COMPQ='N', then Q will not be referenced.  If COMPQ='V' or
             'I', then the transpose of the orthogonal 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) DOUBLE PRECISION array, dimension (LDZ, N)
             If COMPZ='N', then Z will not be referenced.  If COMPZ='V' or
             'I', then the orthogonal 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) DOUBLE PRECISION 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).

     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 ALPHAR(i), ALPHAI(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 ALPHAR(i), ALPHAI(i), and
             BETA(i), i=INFO-N+1,...,N should be correct.  > 2*N:     various
             "impossible" errors.

FURTHER DETAILS
     Iteration counters:

     JITER  -- counts iterations.
     IITER  -- counts iterations run since ILAST was last
               changed.  This is therefore reset only when a 1-by-1 or
               2-by-2 block deflates off the bottom.