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TRED1(NM,N,A,D,E,E2)

       SUBROUTINE TRED1(NM,N,A,D,E,E2)
 C
       INTEGER I,J,K,L,N,II,NM,JP1
       REAL A(NM,N),D(N),E(N),E2(N)
       REAL F,G,H,SCALE
 C
 C     THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE TRED1,
 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 REDUCES A REAL SYMMETRIC MATRIX
 C     TO A SYMMETRIC TRIDIAGONAL MATRIX USING
 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        A CONTAINS THE REAL SYMMETRIC INPUT MATRIX.  ONLY THE
 C          LOWER TRIANGLE OF THE MATRIX NEED BE SUPPLIED.
 C
 C     ON OUTPUT
 C
 C        A CONTAINS INFORMATION ABOUT THE ORTHOGONAL TRANS-
 C          FORMATIONS USED IN THE REDUCTION IN ITS STRICT LOWER
 C          TRIANGLE.  THE FULL UPPER TRIANGLE OF A IS UNALTERED.
 C
 C        D CONTAINS THE DIAGONAL ELEMENTS OF THE TRIDIAGONAL MATRIX.
 C
 C        E CONTAINS THE SUBDIAGONAL ELEMENTS OF THE TRIDIAGONAL
 C          MATRIX IN ITS LAST N-1 POSITIONS.  E(1) IS SET TO ZERO.
 C
 C        E2 CONTAINS THE SQUARES OF THE CORRESPONDING ELEMENTS OF E.
 C          E2 MAY COINCIDE WITH E IF THE SQUARES ARE NOT NEEDED.
 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