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BALBAK(NM,N,LOW,IGH,SCALE,M,Z)

       SUBROUTINE BALBAK(NM,N,LOW,IGH,SCALE,M,Z)
 C
       INTEGER I,J,K,M,N,II,NM,IGH,LOW
       DOUBLE PRECISION SCALE(N),Z(NM,M)
       DOUBLE PRECISION S
 C
 C     THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE BALBAK,
 C     NUM. MATH. 13, 293-304(1969) BY PARLETT AND REINSCH.
 C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 315-326(1971).
 C
 C     THIS SUBROUTINE FORMS THE EIGENVECTORS OF A REAL GENERAL
 C     MATRIX BY BACK TRANSFORMING THOSE OF THE CORRESPONDING
 C     BALANCED MATRIX DETERMINED BY  BALANC.
 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  BALANC.
 C
 C        SCALE CONTAINS INFORMATION DETERMINING THE PERMUTATIONS
 C          AND SCALING FACTORS USED BY  BALANC.
 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     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