function [ wval, dwdx ] = r8poly_lagrange_factor ( npol, xpol, xval )
%*****************************************************************************80
%
%% R8POLY_LAGRANGE_FACTOR evaluates the polynomial Lagrange factor at a point.
%
% Discussion:
%
% The power sum form of a polynomial is:
%
% p(x) = c(0) + c(1) * x + ... + c(n-1) * x^(n-1) + c(n) * x^(n)
%
% Because MATLAB doesn't allow 0 indexing, we use an adjusted formula:
%
% p(x) = c(1) + c(2) * x + ... + c(n) * x^(n-1) + c(n+1) * x^(n)
%
% W(X) = Product ( 1 <= I <= NPOL ) ( X - XPOL(I) )
%
% Suppose F(X) is at least N times continuously differentiable in the
% interval [A,B]. Pick NPOL distinct points XPOL(I) in [A,B] and compute
% the interpolating polynomial P(X) of order NPOL ( and degree NPOL-1)
% which passes through all the points ( XPOL(I), F(XPOL(I)) ).
% Then in the interval [A,B], the maximum error
%
% abs ( F(X) - P(X) )
%
% is bounded by:
%
% C * FNMAX * W(X)
%
% where
%
% C is a constant,
% FNMAX is the maximum value of the NPOL-th derivative of F in [A,B],
% W(X) is the Lagrange factor.
%
% Thus, the value of W(X) is useful as part of an estimated bound
% for the interpolation error.
%
% Note that the Chebyshev abscissas have the property that they minimize
% the value of W(X) over the interval [A,B]. Hence, if the abscissas may
% be chosen arbitrarily, the Chebyshev abscissas have this advantage over
% other choices.
%
% For a set of points XPOL(I), 1 <= I <= NPOL, the IPOL-th Lagrange basis
% polynomial L(IPOL)(X), has the property:
%
% L(IPOL)( XPOL(J) ) = delta ( IPOL, J )
%
% and may be expressed as:
%
% L(IPOL)(X) = W(X) / ( ( X - XPOL(IPOL) ) * W'(XPOL(IPOL)) )
%
% Licensing:
%
% This code is distributed under the GNU LGPL license.
%
% Modified:
%
% 26 April 2005
%
% Author:
%
% John Burkardt
%
% Input:
%
% integer NPOL, the number of abscissas.
% NPOL must be at least 1.
%
% real XPOL(NPOL), the abscissas, which should
% be distinct.
%
% real XVAL, the evaluation point.
%
% Output:
%
% real WVAL, the value of the Lagrange factor at XVAL.
%
% real DWDX, the derivative of W with respect to XVAL.
%
wval = prod ( xval - xpol(1:npol) );
dwdx = 0.0;
for i = 1 : npol
term = 1.0;
for j = 1 : npol
if ( i ~= j )
term = term * ( xval - xpol(j) );
end
end
dwdx = dwdx + term;
end
return
end