function yval = bpab_approx ( n, a, b, ydata, xval )
%*****************************************************************************80
%
%% BPAB_APPROX evaluates the Bernstein polynomial for F(X) on [A,B].
%
% Discussion:
%
% The Bernstein polynomial BERN(F) for F(X) is an approximant, not an
% interpolant; in other words, its value is not guaranteed to equal
% that of F at any particular point. However, for a fixed interval
% [A,B], if we let N increase, the Bernstein polynomial converges
% uniformly to F everywhere in [A,B], provided only that F is continuous.
% Even if F is not continuous, but is bounded, the polynomial converges
% pointwise to F(X) at all points of continuity. On the other hand,
% the convergence is quite slow compared to other interpolation
% and approximation schemes.
%
% BERN(F)(X) = sum ( 0 <= I <= N ) F(X(I)) * B_BASE(I,X)
%
% where
%
% X(I) = ( ( N - I ) * A + I * B ) / N
% B_BASE(I,X) is the value of the I-th Bernstein basis polynomial at X.
%
% Licensing:
%
% This code is distributed under the GNU LGPL license.
%
% Modified:
%
% 06 June 2013
%
% Author:
%
% John Burkardt
%
% Reference:
%
% Kahaner, Moler, and Nash,
% Numerical Methods and Software,
% Prentice Hall, 1989.
%
% Parameters:
%
% Input, integer N, the degree of the Bernstein polynomial to be used.
%
% Input, real A, B, the endpoints of the interval on which the
% approximant is based. A and B should not be equal.
%
% Input, real YDATA(1:N+1), the data values at N+1 equally spaced points
% in [A,B]. If N = 0, then the evaluation point should be 0.5 * ( A + B).
% Otherwise, evaluation point I should be ( (N-I)*A + I*B ) / N ).
%
% Input, real XVAL, the point at which the Bernstein polynomial
% approximant is to be evaluated. XVAL does not have to lie in the
% interval [A,B].
%
% Output, real YVAL, the value of the Bernstein polynomial approximant
% for F, based in [A,B], evaluated at XVAL.
%
%
% Destroy all row vectors!
%
ydata = ydata ( : );
%
% Evaluate the Bernstein basis polynomials at XVAL.
%
bvec = bpab ( n, a, b, xval );
%
% Now compute the sum of YDATA(I) * BVEC(I).
%
yval = ydata(1:n+1)' * bvec(1:n+1);
return
end