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TRED3(N,NV,A,D,E,E2)

       SUBROUTINE TRED3(N,NV,A,D,E,E2)
 C
       INTEGER I,J,K,L,N,II,IZ,JK,NV,JM1
       REAL A(NV),D(N),E(N),E2(N)
       REAL F,G,H,HH,SCALE
 C
 C     THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE TRED3,
 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, STORED AS
 C     A ONE-DIMENSIONAL ARRAY, TO A SYMMETRIC TRIDIAGONAL MATRIX
 C     USING ORTHOGONAL SIMILARITY TRANSFORMATIONS.
 C
 C     ON INPUT
 C
 C        N IS THE ORDER OF THE MATRIX.
 C
 C        NV MUST BE SET TO THE DIMENSION OF THE ARRAY PARAMETER A
 C          AS DECLARED IN THE CALLING PROGRAM DIMENSION STATEMENT.
 C
 C        A CONTAINS THE LOWER TRIANGLE OF THE REAL SYMMETRIC
 C          INPUT MATRIX, STORED ROW-WISE AS A ONE-DIMENSIONAL
 C          ARRAY, IN ITS FIRST N*(N+1)/2 POSITIONS.
 C
 C     ON OUTPUT
 C
 C        A CONTAINS INFORMATION ABOUT THE ORTHOGONAL
 C          TRANSFORMATIONS USED IN THE REDUCTION.
 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
 C