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ORTBAK(NM,LOW,IGH,A,ORT,M,Z)

       SUBROUTINE ORTBAK(NM,LOW,IGH,A,ORT,M,Z)
 C
       INTEGER I,J,M,LA,MM,MP,NM,IGH,KP1,LOW,MP1
       REAL A(NM,IGH),ORT(IGH),Z(NM,M)
       REAL G
 C
 C     THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE ORTBAK,
 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     THIS SUBROUTINE FORMS THE EIGENVECTORS OF A REAL GENERAL
 C     MATRIX BY BACK TRANSFORMING THOSE OF THE CORRESPONDING
 C     UPPER HESSENBERG MATRIX DETERMINED BY  ORTHES.
 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        LOW AND IGH ARE INTEGERS DETERMINED BY THE BALANCING
 C          SUBROUTINE  BALANC.  IF  BALANC  HAS NOT BEEN USED,
 C          SET LOW=1 AND IGH EQUAL TO THE ORDER OF THE MATRIX.
 C
 C        A CONTAINS INFORMATION ABOUT THE ORTHOGONAL TRANS-
 C          FORMATIONS USED IN THE REDUCTION BY  ORTHES
 C          IN ITS STRICT LOWER TRIANGLE.
 C
 C        ORT CONTAINS FURTHER INFORMATION ABOUT THE TRANS-
 C          FORMATIONS USED IN THE REDUCTION BY  ORTHES.
 C          ONLY ELEMENTS LOW THROUGH IGH ARE USED.
 C
 C        M IS THE NUMBER OF COLUMNS OF Z TO BE BACK TRANSFORMED.
 C
 C        Z CONTAINS THE REAL AND IMAGINARY PARTS OF THE EIGEN-
 C          VECTORS TO BE BACK TRANSFORMED IN ITS FIRST M COLUMNS.
 C
 C     ON OUTPUT
 C
 C        Z CONTAINS THE REAL AND IMAGINARY PARTS OF THE
 C          TRANSFORMED EIGENVECTORS IN ITS FIRST M COLUMNS.
 C
 C        ORT HAS BEEN ALTERED.
 C
 C     NOTE THAT ORTBAK PRESERVES VECTOR EUCLIDEAN NORMS.
 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