function [ x, dp2, p1 ] = hermite_root ( x, order )

%% HERMITE_ROOT improves an approximate root of a Hermite polynomial.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    19 February 2008
%
%  Author:
%
%    FORTRAN77 original version by Arthur Stroud, Don Secrest
%    MATLAB version by John Burkardt
%
%  Reference:
%
%    Arthur Stroud, Don Secrest,
%    Gaussian Quadrature Formulas,
%    Prentice Hall, 1966,
%    LC: QA299.4G3S7.
%
%  Parameters:
%
%    Input, real X, the approximate root.
%
%    Input, integer ORDER, the order of the Hermite polynomial.
%
%    Output, real X, an improved estimate of the approximate root.
%
%    Output, real DP2, the value of H'(ORDER)(X).
%
%    Output, real P1, the value of H(ORDER-1)(X).
%
  eps = r8_epsilon ( x );
  maxstep = 10;

  for i = 1 : maxstep

    [ p2, dp2, p1 ] = hermite_recur ( x, order );

    d = p2 / dp2;
    x = x - d;

    if ( abs ( d ) <= eps * ( abs ( x ) + 1.0 ) )
      return
    end

  end