# tred3.f

          SUBROUTINE TRED3 (N, NV, A, D, E, E2)
    C***BEGIN PROLOGUE  TRED3
    C***PURPOSE  Reduce a real symmetric matrix stored in packed form to
    C            symmetric tridiagonal matrix using orthogonal
    C            transformations.
    C***LIBRARY   SLATEC (EISPACK)
    C***CATEGORY  D4C1B1
    C***TYPE      SINGLE PRECISION (TRED3-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 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 A.  N is an INTEGER variable.
    C
    C        NV is an INTEGER variable set equal to the dimension of the
    C          array A as specified in the calling program.  NV must not
    C          be less than  N*(N+1)/2.
    C
    C        A contains the lower triangle, stored row-wise, of the real
    C          symmetric packed matrix.  A is a one-dimensional REAL
    C          array, dimensioned A(NV).
    C
    C     On Output
    C
    C        A contains information about the orthogonal transformations
    C          used in the reduction in its first N*(N+1)/2 positions.
    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  TRED3