# elmhes.f

          SUBROUTINE ELMHES (NM, N, LOW, IGH, A, INT)
    C***BEGIN PROLOGUE  ELMHES
    C***PURPOSE  Reduce a real general matrix to upper Hessenberg form
    C            using stabilized elementary similarity transformations.
    C***LIBRARY   SLATEC (EISPACK)
    C***CATEGORY  D4C1B2
    C***TYPE      SINGLE PRECISION (ELMHES-S, COMHES-C)
    C***KEYWORDS  EIGENVALUES, EIGENVECTORS, EISPACK
    C***AUTHOR  Smith, B. T., et al.
    C***DESCRIPTION
    C
    C     This subroutine is a translation of the ALGOL procedure ELMHES,
    C     NUM. MATH. 12, 349-368(1968) by Martin and Wilkinson.
    C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 339-358(1971).
    C
    C     Given a REAL GENERAL matrix, this subroutine
    C     reduces a submatrix situated in rows and columns
    C     LOW through IGH to upper Hessenberg form by
    C     stabilized elementary similarity transformations.
    C
    C     On INPUT
    C
    C        NM must be set to the row dimension of the two-dimensional
    C          array parameter, A, as declared in the calling program
    C          dimension statement.  NM is an INTEGER variable.
    C
    C        N is the order of the matrix, A.  N is an INTEGER variable.
    C          N must be less than or equal to NM.
    C
    C        LOW and IGH are two INTEGER variables determined by the
    C          balancing subroutine  BALANC.  If  BALANC  has not been
    C          used, set LOW=1 and IGH equal to the order of the matrix, N.
    C
    C        A contains the input matrix.  A is a two-dimensional REAL
    C          array, dimensioned A(NM,N).
    C
    C     On OUTPUT
    C
    C        A contains the upper Hessenberg matrix.  The multipliers which
    C          were used in the reduction are stored in the remaining
    C          triangle under the Hessenberg matrix.
    C
    C        INT contains information on the rows and columns interchanged
    C          in the reduction.  Only elements LOW through IGH are used.
    C          INT is a one-dimensional INTEGER array, dimensioned INT(IGH).
    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  ELMHES