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SUBROUTINE ORTHES(NM,N,LOW,IGH,A,ORT)
C
INTEGER I,J,M,N,II,JJ,LA,MP,NM,IGH,KP1,LOW
REAL A(NM,N),ORT(IGH)
REAL F,G,H,SCALE
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 BURTON S. GARBOW,
C MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C THIS VERSION DATED AUGUST 1983.
C
C ------------------------------------------------------------------
C