*DECK TRED1
      SUBROUTINE TRED1 (NM, N, A, D, E, E2)
C***BEGIN PROLOGUE  TRED1
C***PURPOSE  Reduce a real symmetric matrix to symmetric tridiagonal
C            matrix using orthogonal similarity transformations.
C***LIBRARY   SLATEC (EISPACK)
C***CATEGORY  D4C1B1
C***TYPE      SINGLE PRECISION (TRED1-S)
C***KEYWORDS  EIGENVALUES, EIGENVECTORS, EISPACK
C***AUTHOR  Smith, B. T., et al.
C***DESCRIPTION
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 the two-dimensional
C          array parameter, A, as declared in the calling program
C          dimension statement.  NM is an INTEGER variable.
C
C        N is the order of the matrix A.  N is an INTEGER variable.
C          N must be less than or equal to NM.
C
C        A contains the real symmetric input matrix.  Only the lower
C          triangle of the matrix need be supplied.  A is a two-
C          dimensional REAL array, dimensioned A(NM,N).
C
C     On Output
C
C        A contains information about the orthogonal transformations
C          used in the reduction in its strict lower triangle.  The
C          full upper triangle of A is unaltered.
C
C        D contains the diagonal elements of the symmetric tridiagonal
C          matrix.  D is a one-dimensional REAL array, dimensioned D(N).
C
C        E contains the subdiagonal elements of the symmetric
C          tridiagonal matrix in its last N-1 positions.  E(1) is set
C          to zero.  E is a one-dimensional REAL array, dimensioned
C          E(N).
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          E2 is a one-dimensional REAL array, dimensioned E2(N).
C
C     Questions and comments should be directed to B. S. Garbow,
C     APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
C     ------------------------------------------------------------------
C
C***REFERENCES  B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow,
C                 Y. Ikebe, V. C. Klema and C. B. Moler, Matrix Eigen-
C                 system Routines - EISPACK Guide, Springer-Verlag,
C                 1976.
C***ROUTINES CALLED  (NONE)
C***REVISION HISTORY  (YYMMDD)
C   760101  DATE WRITTEN
C   890831  Modified array declarations.  (WRB)
C   890831  REVISION DATE from Version 3.2
C   891214  Prologue converted to Version 4.0 format.  (BAB)
C   920501  Reformatted the REFERENCES section.  (WRB)
C***END PROLOGUE  TRED1
C
      INTEGER I,J,K,L,N,II,NM,JP1
      REAL A(NM,*),D(*),E(*),E2(*)
      REAL F,G,H,SCALE
C
C***FIRST EXECUTABLE STATEMENT  TRED1
      DO 100 I = 1, N
  100 D(I) = A(I,I)
C     .......... FOR I=N STEP -1 UNTIL 1 DO -- ..........
      DO 300 II = 1, N
         I = N + 1 - II
         L = I - 1
         H = 0.0E0
         SCALE = 0.0E0
         IF (L .LT. 1) GO TO 130
C     .......... SCALE ROW (ALGOL TOL THEN NOT NEEDED) ..........
         DO 120 K = 1, L
  120    SCALE = SCALE + ABS(A(I,K))
C
         IF (SCALE .NE. 0.0E0) GO TO 140
  130    E(I) = 0.0E0
         E2(I) = 0.0E0
         GO TO 290
C
  140    DO 150 K = 1, L
            A(I,K) = A(I,K) / SCALE
            H = H + A(I,K) * A(I,K)
  150    CONTINUE
C
         E2(I) = SCALE * SCALE * H
         F = A(I,L)
         G = -SIGN(SQRT(H),F)
         E(I) = SCALE * G
         H = H - F * G
         A(I,L) = F - G
         IF (L .EQ. 1) GO TO 270
         F = 0.0E0
C
         DO 240 J = 1, L
            G = 0.0E0
C     .......... FORM ELEMENT OF A*U ..........
            DO 180 K = 1, J
  180       G = G + A(J,K) * A(I,K)
C
            JP1 = J + 1
            IF (L .LT. JP1) GO TO 220
C
            DO 200 K = JP1, L
  200       G = G + A(K,J) * A(I,K)
C     .......... FORM ELEMENT OF P ..........
  220       E(J) = G / H
            F = F + E(J) * A(I,J)
  240    CONTINUE
C
         H = F / (H + H)
C     .......... FORM REDUCED A ..........
         DO 260 J = 1, L
            F = A(I,J)
            G = E(J) - H * F
            E(J) = G
C
            DO 260 K = 1, J
               A(J,K) = A(J,K) - F * E(K) - G * A(I,K)
  260    CONTINUE
C
  270    DO 280 K = 1, L
  280    A(I,K) = SCALE * A(I,K)
C
  290    H = D(I)
         D(I) = A(I,I)
         A(I,I) = H
  300 CONTINUE
C
      RETURN
      END
