SUBROUTINE DORTHE(NM,N,LOW,IGH,A,ORT)
C
      INTEGER I,J,M,N,II,JJ,LA,MP,NM,IGH,KP1,LOW
      DOUBLE PRECISION A(NM,N),ORT(IGH)
      DOUBLE PRECISION F,G,H,SCALE
      DOUBLE PRECISION DSQRT
C
C     THIS IS A DOUBLE-PRECISION VERSION OF THE
C     EISPACK SINGLE-PRECISION ROUTINE ORTHES.
C     IT WAS ADAPTED BY PHYLLIS FOX, MAY 28, 1975.
C
C     ORTHES 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 (DOUBLE PRECISION) 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
C     ------------------------------------------------------------------
C
      LA = IGH - 1
      KP1 = LOW + 1
      IF (LA .LT. KP1) GO TO 200
C
      DO 180 M = KP1, LA
         H = 0.0D0
         ORT(M) = 0.0D0
         SCALE = 0.0D0
C     ********** SCALE COLUMN (ALGOL TOL THEN NOT NEEDED) **********
         DO 90 I = M, IGH
   90    SCALE = SCALE + DABS(A(I,M-1))
C
         IF (SCALE .EQ. 0.0D0) GO TO 180
         MP = M + IGH
C     ********** FOR I=IGH STEP -1 UNTIL M DO -- **********
         DO 100 II = M, IGH
            I = MP - II
            ORT(I) = A(I,M-1) / SCALE
            H = H + ORT(I) * ORT(I)
  100    CONTINUE
C
         G = -DSIGN(DSQRT(H),ORT(M))
         H = H - ORT(M) * G
         ORT(M) = ORT(M) - G
C     ********** FORM (I-(U*UT)/H) * A **********
         DO 130 J = M, N
            F = 0.0D0
C     ********** FOR I=IGH STEP -1 UNTIL M DO -- **********
            DO 110 II = M, IGH
               I = MP - II
               F = F + ORT(I) * A(I,J)
  110       CONTINUE
C
            F = F / H
C
            DO 120 I = M, IGH
  120       A(I,J) = A(I,J) - F * ORT(I)
C
  130    CONTINUE
C     ********** FORM (I-(U*UT)/H)*A*(I-(U*UT)/H) **********
         DO 160 I = 1, IGH
            F = 0.0D0
C     ********** FOR J=IGH STEP -1 UNTIL M DO -- **********
            DO 140 JJ = M, IGH
               J = MP - JJ
               F = F + ORT(J) * A(I,J)
  140       CONTINUE
C
            F = F / H
C
            DO 150 J = M, IGH
  150       A(I,J) = A(I,J) - F * ORT(J)
C
  160    CONTINUE
C
         ORT(M) = SCALE * ORT(M)
         A(M,M-1) = SCALE * G
  180 CONTINUE
C
  200 RETURN
C     ********** LAST CARD OF DORTHE **********
      END

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