SLATEC Routines --- HTRIDI ---

*DECK HTRIDI
      SUBROUTINE HTRIDI (NM, N, AR, AI, D, E, E2, TAU)
C***BEGIN PROLOGUE  HTRIDI
C***PURPOSE  Reduce a complex Hermitian matrix to a real symmetric
C            tridiagonal matrix using unitary similarity
C            transformations.
C***LIBRARY   SLATEC (EISPACK)
C***CATEGORY  D4C1B1
C***TYPE      SINGLE PRECISION (HTRIDI-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 TRED1, 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
C     to a real symmetric tridiagonal matrix using
C     unitary similarity transformations.
C
C     On INPUT
C
C        NM must be set to the row dimension of the two-dimensional
C          array parameters, AR and AI, as declared in the calling
C          program dimension statement.  NM is an INTEGER variable.
C
C        N is the order of the matrix A=(AR,AI).  N is an INTEGER
C          variable. N must be less than or equal to NM.
C
C        AR and AI contain the real and imaginary parts, respectively,
C          of the complex Hermitian input matrix.  Only the lower
C          triangle of the matrix need be supplied.  AR and AI are two-
C          dimensional REAL arrays, dimensioned AR(NM,N) and AI(NM,N).
C
C     On OUTPUT
C
C        AR and AI contain some information about the unitary trans-
C          formations used in the reduction in the strict lower triangle
C          of AR and the full lower triangle of AI.  The rest of the
C          matrices are unaltered.
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  HTRIDI