An overview of the Ortpol toolbox.
The goal of this toolbox is to provide Orthogonal Polynomials. This toolbox is released under the LGPL licence.
The following script plots the Chebyshev polynomial of degree 10.
x=linspace(-1,1,100); y=chebyshev_eval(x,10); h=scf(); plot(x,y,"r-") xtitle("Chebyshev Polynomial","x","T10(x)") | ![]() | ![]() |
The following script computes the roots of the Chebyshev polynomial of degree 10.
If required, the coefficients of the polynomial can
be obtained with the coeff
function.
T=chebyshev_poly(x,10); c=coeff(T) | ![]() | ![]() |
Let w be an integrable function on a real interval I, and let f be the associated distribution. Assume that X is a random variable associated with the distribution f. Consider the orthogonal polynomials associated with w.
Therefore,
Moreover,
for n greater or equal to 1.
This is because
for n greater or equal to 1.
This property explains why there is no *_expectation
function : the result is independent from the particular type of
orthogonal polynomial.
Copyright (C) 2015 - Michael Baudin
Jean-Marc Martinez
Numerical Mathematics, Series: Texts in Applied Mathematics, Vol. 37, Alfio Quarteroni, Riccardo Sacco, Fausto Saleri, 2nd ed. 2007, XVIII, 657 p. 135 illus.
Calculation of Gauss Quadrature Rules, Gene H. Golub and John H. Welsch, Mathematics of Computation, Vol. 23, No. 106 (Apr., 1969), pp. 221-230