SLATEC Routines --- TRED3 ---

*DECK TRED3
      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