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TRBAK1(NM,N,A,E,M,Z)

       SUBROUTINE TRBAK1(NM,N,A,E,M,Z)
 C
       INTEGER I,J,K,L,M,N,NM
       DOUBLE PRECISION A(NM,N),E(N),Z(NM,M)
       DOUBLE PRECISION S
 C
 C     THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE TRBAK1,
 C     NUM. MATH. 11, 181-195(1968) BY MARTIN, REINSCH, AND WILKINSON.
 C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971).
 C
 C     THIS SUBROUTINE FORMS THE EIGENVECTORS OF A REAL SYMMETRIC
 C     MATRIX BY BACK TRANSFORMING THOSE OF THE CORRESPONDING
 C     SYMMETRIC TRIDIAGONAL MATRIX DETERMINED BY  TRED1.
 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        A CONTAINS INFORMATION ABOUT THE ORTHOGONAL TRANS-
 C          FORMATIONS USED IN THE REDUCTION BY  TRED1
 C          IN ITS STRICT LOWER TRIANGLE.
 C
 C        E CONTAINS THE SUBDIAGONAL ELEMENTS OF THE TRIDIAGONAL
 C          MATRIX IN ITS LAST N-1 POSITIONS.  E(1) IS ARBITRARY.
 C
 C        M IS THE NUMBER OF EIGENVECTORS TO BE BACK TRANSFORMED.
 C
 C        Z CONTAINS THE EIGENVECTORS TO BE BACK TRANSFORMED
 C          IN ITS FIRST M COLUMNS.
 C
 C     ON OUTPUT
 C
 C        Z CONTAINS THE TRANSFORMED EIGENVECTORS
 C          IN ITS FIRST M COLUMNS.
 C
 C     NOTE THAT TRBAK1 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