# qzhes.f

          SUBROUTINE QZHES (NM, N, A, B, MATZ, Z)
    C***BEGIN PROLOGUE  QZHES
    C***PURPOSE  The first step of the QZ algorithm for solving generalized
    C            matrix eigenproblems.  Accepts a pair of real general
    C            matrices and reduces one of them to upper Hessenberg
    C            and the other to upper triangular form using orthogonal
    C            transformations. Usually followed by QZIT, QZVAL, QZVEC.
    C***LIBRARY   SLATEC (EISPACK)
    C***CATEGORY  D4C1B3
    C***TYPE      SINGLE PRECISION (QZHES-S)
    C***KEYWORDS  EIGENVALUES, EIGENVECTORS, EISPACK
    C***AUTHOR  Smith, B. T., et al.
    C***DESCRIPTION
    C
    C     This subroutine is the first 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
    C     This subroutine accepts a pair of REAL GENERAL matrices and
    C     reduces one of them to upper Hessenberg form and the other
    C     to upper triangular form using orthogonal transformations.
    C     It is usually followed by  QZIT,  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 general matrix.  A is a two-dimensional
    C          REAL array, dimensioned A(NM,N).
    C
    C        B contains a real general matrix.  B is a two-dimensional
    C          REAL array, dimensioned B(NM,N).
    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     On Output
    C
    C        A has been reduced to upper Hessenberg form.  The elements
    C          below the first subdiagonal have been set to zero.
    C
    C        B has been reduced to upper triangular form.  The elements
    C          below the main diagonal have been set to zero.
    C
    C        Z contains the product of the right hand transformations if
    C          MATZ has been set to .TRUE.  Otherwise, Z is not referenced.
    C          Z is a two-dimensional REAL array, dimensioned Z(NM,N).
    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   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  QZHES