**gram_polynomial**,
a MATLAB code which
evaluates the Gram polynomials.

The Gram polynomial P(n,m,x) can be evaluated at a point x by:

P(0,m,x) = 1 P(1,m,x) = x P(n+1,m,x) = x * P(n,m,x) - beta(n,m) * P(n-1,m,x)where beta(n,m) = (m^2-n^2)*n^s/m^2/(4*n^2-1).

The polynomials are orthogonal with respect to a discrete inner product

(f,g) = sum ( 1 <= i <= m ) f(x(i)) * g(x(i))where

x(i) = - 1 + ( 2*i-1)/m, 1 <= i <= m.

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

**gram_polynomial** is available in
a MATLAB version.

bernstein_polynomial, a MATLAB code which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

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- Germund Dahlquist, Ake Bjorck, Numerical Methods in Scientific Computing, Volume 1, SIAM, 2008, ISBN: 978-0-898716-44-3, LC: QA297.D335 2008.

- gram_abscissas.m, returns the coordinates of the Gram polynomial abscissas.
- gram_beta.m, evaluates a factor used in evaluating Gram polynomials.
- gram_polynomial_coefficients.m, returns the coefficients of a Gram polynomial.
- gram_polynomial_evaluate.m, evaluates a Gram polynomial.
- gram_polynomial_inner_product.m, inner product of two Gram polynomials.
- gram_projection.m, projection of function onto Gram polynomial gj.
- r8poly_print.m, prints a polynomial given its coefficients.