SUBROUTINE ORTHES(NM,N,LOW,IGH,A,ORT)
C***BEGIN PROLOGUE  ORTHES
C***DATE WRITTEN   760101   (YYMMDD)
C***REVISION DATE  830518   (YYMMDD)
C***CATEGORY NO.  D4C1B2
C***KEYWORDS  EIGENVALUES,EIGENVECTORS,EISPACK
C***AUTHOR  SMITH, B. T., ET AL.
C***PURPOSE  Reduces real general matrix to upper Hessenberg form
C            orthogonal similarity transformations.
C***DESCRIPTION
C
C     This subroutine is a translation of the ALGOL procedure ORTHES,
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     orthogonal similarity transformations.
C
C     On INPUT
C
C        NM must be set to the row dimension of two-dimensional
C          array parameters as declared in the calling program
C          dimension statement.
C
C        N is the order of the matrix.
C
C        LOW and IGH are integers determined by the balancing
C          subroutine  BALANC.  If  BALANC  has not been used,
C          set LOW=1, IGH=N.
C
C        A contains the input matrix.
C
C     On OUTPUT
C
C        A contains the Hessenberg matrix.  Information about
C          the orthogonal transformations used in the reduction
C          is stored in the remaining triangle under the
C          Hessenberg matrix.
C
C        ORT contains further information about the transformations.
C          only elements LOW through IGH are used.
C
C     Questions and comments should be directed to B. S. Garbow,
C     APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
C     ------------------------------------------------------------------
C***REFERENCES  B. T. SMITH, J. M. BOYLE, J. J. DONGARRA, B. S. GARBOW,
C                 Y. IKEBE, V. C. KLEMA, C. B. MOLER, *MATRIX EIGEN-
C                 SYSTEM ROUTINES - EISPACK GUIDE*, SPRINGER-VERLAG,
C                 1976.
C***ROUTINES CALLED  (NONE)
C***END PROLOGUE  ORTHES