SLATEC Routines --- QZIT ---

*DECK QZIT
      SUBROUTINE QZIT (NM, N, A, B, EPS1, MATZ, Z, IERR)
C***BEGIN PROLOGUE  QZIT
C***PURPOSE  The second step of the QZ algorithm for generalized
C            eigenproblems.  Accepts an upper Hessenberg and an upper
C            triangular matrix and reduces the former to
C            quasi-triangular form while preserving the form of the
C            latter.  Usually preceded by QZHES and followed by QZVAL
C            and QZVEC.
C***LIBRARY   SLATEC (EISPACK)
C***CATEGORY  D4C1B3
C***TYPE      SINGLE PRECISION (QZIT-S)
C***KEYWORDS  EIGENVALUES, EIGENVECTORS, EISPACK
C***AUTHOR  Smith, B. T., et al.
C***DESCRIPTION
C
C     This subroutine is the second step of the QZ algorithm
C     for solving generalized matrix eigenvalue problems,
C     SIAM J. NUMER. ANAL. 10, 241-256(1973) by MOLER and STEWART,
C     as modified in technical note NASA TN D-7305(1973) by WARD.
C
C     This subroutine accepts a pair of REAL matrices, one of them
C     in upper Hessenberg form and the other in upper triangular form.
C     It reduces the Hessenberg matrix to quasi-triangular form using
C     orthogonal transformations while maintaining the triangular form
C     of the other matrix.  It is usually preceded by  QZHES  and
C     followed by  QZVAL  and, possibly,  QZVEC.
C
C     On Input
C
C        NM must be set to the row dimension of the two-dimensional
C          array parameters, A, B, and Z, as declared in the calling
C          program dimension statement.  NM is an INTEGER variable.
C
C        N is the order of the matrices A and B.  N is an INTEGER
C          variable.  N must be less than or equal to NM.
C
C        A contains a real upper Hessenberg matrix.  A is a two-
C          dimensional REAL array, dimensioned A(NM,N).
C
C        B contains a real upper triangular matrix.  B is a two-
C          dimensional REAL array, dimensioned B(NM,N).
C
C        EPS1 is a tolerance used to determine negligible elements.
C          EPS1 = 0.0 (or negative) may be input, in which case an
C          element will be neglected only if it is less than roundoff
C          error times the norm of its matrix.  If the input EPS1 is
C          positive, then an element will be considered negligible
C          if it is less than EPS1 times the norm of its matrix.  A
C          positive value of EPS1 may result in faster execution,
C          but less accurate results.  EPS1 is a REAL variable.
C
C        MATZ should be set to .TRUE. if the right hand transformations
C          are to be accumulated for later use in computing
C          eigenvectors, and to .FALSE. otherwise.  MATZ is a LOGICAL
C          variable.
C
C        Z contains, if MATZ has been set to .TRUE., the transformation
C          matrix produced in the reduction by  QZHES, if performed, or
C          else the identity matrix.  If MATZ has been set to .FALSE.,
C          Z is not referenced.  Z is a two-dimensional REAL array,
C          dimensioned Z(NM,N).
C
C     On Output
C
C        A has been reduced to quasi-triangular form.  The elements
C          below the first subdiagonal are still zero, and no two
C          consecutive subdiagonal elements are nonzero.
C
C        B is still in upper triangular form, although its elements
C          have been altered.  The location B(N,1) is used to store
C          EPS1 times the norm of B for later use by  QZVAL  and  QZVEC.
C
C        Z contains the product of the right hand transformations
C          (for both steps) if MATZ has been set to .TRUE.
C
C        IERR is an INTEGER flag set to
C          Zero       for normal return,
C          J          if neither A(J,J-1) nor A(J-1,J-2) has become
C                     zero after a total of 30*N iterations.
C
C     Questions and comments should be directed to B. S. Garbow,
C     APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
C     ------------------------------------------------------------------
C
C***REFERENCES  B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow,
C                 Y. Ikebe, V. C. Klema and C. B. Moler, Matrix Eigen-
C                 system Routines - EISPACK Guide, Springer-Verlag,
C                 1976.
C***ROUTINES CALLED  (NONE)
C***REVISION HISTORY  (YYMMDD)
C   760101  DATE WRITTEN
C   890531  Changed all specific intrinsics to generic.  (WRB)
C   890831  Modified array declarations.  (WRB)
C   890831  REVISION DATE from Version 3.2
C   891214  Prologue converted to Version 4.0 format.  (BAB)
C   920501  Reformatted the REFERENCES section.  (WRB)
C***END PROLOGUE  QZIT