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Orthogonal Polynomials >> Orthogonal Polynomials > Overview

Overview

An overview of the Ortpol toolbox.

Purpose

The goal of this toolbox is to provide Orthogonal Polynomials. This toolbox is released under the LGPL licence.

Example :

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.

T=chebyshev_poly(10);
x=roots(T)
scf();
plot(x,zeros(x),"bo")
xtitle("Chebyshev Roots","x","")

If required, the coefficients of the polynomial can be obtained with the coeff function.

T=chebyshev_poly(x,10);
c=coeff(T)

Expectation of polynomials

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.

Authors

Copyright (C) 2015 - Michael Baudin

Acknowledgments

Jean-Marc Martinez

Bibliography

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


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