gram_polynomial, a Python code which evaluates the Gram polynomials, also known as the discrete Chebyshev polyomials.

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^2/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))
  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 MIT license


gram_polynomial is available in a MATLAB version and a Python version.

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

Source Code:

Last revised on 11 October 2022.