# htrid3.f

          SUBROUTINE HTRID3 (NM, N, A, D, E, E2, TAU)
    C***BEGIN PROLOGUE  HTRID3
    C***PURPOSE  Reduce a complex Hermitian (packed) matrix to a real
    C            symmetric tridiagonal matrix by unitary similarity
    C            transformations.
    C***LIBRARY   SLATEC (EISPACK)
    C***CATEGORY  D4C1B1
    C***TYPE      SINGLE PRECISION (HTRID3-S)
    C***KEYWORDS  EIGENVALUES, EIGENVECTORS, EISPACK
    C***AUTHOR  Smith, B. T., et al.
    C***DESCRIPTION
    C
    C     This subroutine is a translation of a complex analogue of
    C     the ALGOL procedure TRED3, NUM. MATH. 11, 181-195(1968)
    C     by Martin, Reinsch, and Wilkinson.
    C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971).
    C
    C     This subroutine reduces a COMPLEX HERMITIAN matrix, stored as
    C     a single square array, to a real symmetric tridiagonal matrix
    C     using unitary 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.  N is an INTEGER variable.
    C          N must be less than or equal to NM.
    C
    C        A contains the lower triangle of the complex Hermitian input
    C          matrix.  The real parts of the matrix elements are stored
    C          in the full lower triangle of A, and the imaginary parts
    C          are stored in the transposed positions of the strict upper
    C          triangle of A.  No storage is required for the zero
    C          imaginary parts of the diagonal elements.  A is a two-
    C          dimensional REAL array, dimensioned A(NM,N).
    C
    C     On OUTPUT
    C
    C        A contains some information about the unitary transformations
    C          used in the reduction.
    C
    C        D contains the diagonal elements of the real symmetric
    C          tridiagonal matrix.  D is a one-dimensional REAL array,
    C          dimensioned D(N).
    C
    C        E contains the subdiagonal elements of the real tridiagonal
    C          matrix in its last N-1 positions.  E(1) is set to zero.
    C          E is a one-dimensional REAL array, dimensioned E(N).
    C
    C        E2 contains the squares of the corresponding elements of E.
    C          E2(1) is set to zero.  E2 may coincide with E if the squares
    C          are not needed.  E2 is a one-dimensional REAL array,
    C          dimensioned E2(N).
    C
    C        TAU contains further information about the transformations.
    C          TAU is a one-dimensional REAL array, dimensioned TAU(2,N).
    C
    C     Calls PYTHAG(A,B) for sqrt(A**2 + B**2).
    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  PYTHAG
    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  HTRID3