zhgeqz (3lapack)

CXML

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.

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