SLATEC Routines --- TRED1 ---

*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