# orthes.f

          SUBROUTINE ORTHES (NM, N, LOW, IGH, A, ORT)
    C***BEGIN PROLOGUE  ORTHES
    C***PURPOSE  Reduce a real general matrix to upper Hessenberg form
    C            using orthogonal similarity transformations.
    C***LIBRARY   SLATEC (EISPACK)
    C***CATEGORY  D4C1B2
    C***TYPE      SINGLE PRECISION (ORTHES-S, CORTH-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 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 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 general matrix to be reduced to upper
    C          Hessenberg form.  A is a two-dimensional REAL array,
    C          dimensioned A(NM,N).
    C
    C     On OUTPUT
    C
    C        A contains the upper Hessenberg matrix.  Some information about
    C          the orthogonal transformations used in the reduction
    C          is stored in the remaining triangle under the Hessenberg
    C          matrix.
    C
    C        ORT contains further information about the orthogonal trans-
    C          formations used in the reduction.  Only elements LOW+1
    C          through IGH are used.  ORT is a one-dimensional REAL array,
    C          dimensioned ORT(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  ORTHES