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shgeqz


 NAME
      SHGEQZ - 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 SHGEQZ( 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

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

 PURPOSE
      SHGEQZ 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 diag-
      onal blocks are either 1x1 or 2x2, the 2x2 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 using one orthogonal transformation (usually
      called Q) on the left and another (usually called Z) on the
      right.  The 2x2 upper-triangular diagonal blocks of B
      corresponding to 2x2 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 posi-
      tive.)

      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 accumu-
      lated 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 General-
      ized 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 transformation 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 transformation 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 ini-
              tialized to the identity first.

      N       (input) INTEGER
              The number of rows and columns in the matrices A, B,
              Q, and Z.  N must be at least 0.

      ILO     (input) INTEGER
              Columns 1 through ILO-1 of A and B are assumed to be
              in upper triangular form already, and will not be
              modified.  ILO must be at least 1.

      IHI     (input) INTEGER
              Rows IHI+1 through N of A and B are assumed to be in
              upper triangular form already, and will not be
              touched.  IHI may not be greater than N.

      A       (input/output) REAL array, dimension (LDA, N)
              On entry, the N x N upper Hessenberg matrix A.
              Entries below the subdiagonal must be zero.  If
              JOB='S', then on exit A and B will have been simul-
              taneously 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) REAL array, dimension (LDB, N)
              On entry, the N x N upper triangular matrix B.
              Entries below the diagonal must be zero.  2x2 blocks
              in B corresponding to 2x2 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.  Entries 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) REAL array, dimension (N)
              ALPHAR(1:N) will be set to real parts of the diago-
              nal 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 transforma-
              tions s.t. the diagonal of B was non-negative real.
              Thus, if A(j,j) is in a 1x1 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) REAL array, dimension (N)
              ALPHAI(1:N) will be set to imaginary parts of the
              diagonal elements of A that would result from reduc-
              ing A and B to Schur form and then further reducing
              them both to triangular form using unitary transfor-
              mations s.t. the diagonal of B was non-negative
              real.  Thus, if A(j,j) is in a 1x1 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) REAL array, dimension (N)
              BETA(1:N) will be set to the (real) diagonal ele-
              ments 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 1x1 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) REAL array, dimension (LDQ, N)
              If COMPQ='N', then Q will not be referenced.  If
              COMPQ='V' or 'I', then the transpose of the orthogo-
              nal 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 must be
              at least 1.  If COMPQ='V' or 'I', then LDQ must also
              be at least N.

      Z       (input/output) REAL array, dimension (LDZ, N)
              If COMPZ='N', then Z will not be referenced.  If
              COMPZ='V' or 'I', then the orthogonal transforma-
              tions 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 must be
              at least 1.  If COMPZ='V' or 'I', then LDZ must also
              be at least N.

      WORK    (workspace) REAL array, dimension (LWORK)
              On exit, if INFO is not negative, WORK(1) will be
              set to the optimal size of the array WORK.

      LWORK   (input) INTEGER
              The number of elements in WORK.  It must be at least
              max( 1, N ).

      INFO    (output) INTEGER
              < 0: if INFO = -i, the i-th argument had an illegal
              value
              = 0: successful exit.
              = 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 1x1

      or
                2x2 block deflates off the bottom.