Chebyshev–Gauss Quadrature
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In
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
Chebyshev–Gauss quadrature is an extension of
Gaussian quadrature In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for more ...
method for approximating the value of integrals of the following kind: :\int_^ \frac \,dx and :\int_^ \sqrt g(x)\,dx. In the first case :\int_^ \frac \,dx \approx \sum_^n w_i f(x_i) where :x_i = \cos \left( \frac \pi \right) and the weight :w_i = \frac .Abramowitz, M & Stegun, I A, ''Handbook of Mathematical Functions'', 10th printing with corrections (1972), Dover, . Equation 25.4.38. In the second case :\int_^ \sqrt g(x)\,dx \approx \sum_^n w_i g(x_i) where :x_i = \cos \left( \frac \pi \right) and the weight : w_i = \frac \sin^2 \left( \frac \pi \right). \,Abramowitz, M & Stegun, I A, ''Handbook of Mathematical Functions'', 10th printing with corrections (1972), Dover, . Equation 25.4.40.


See also

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Chebyshev polynomials The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshe ...
*
Chebyshev nodes In numerical analysis, Chebyshev nodes are specific real algebraic numbers, namely the roots of the Chebyshev polynomials of the first kind. They are often used as nodes in polynomial interpolation because the resulting interpolation polynomial ...


References


External links


Chebyshev-Gauss Quadrature
from
Wolfram MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science D ...

Gauss–Chebyshev type 1 quadrature
an

free software Free software or libre software is computer software distributed under terms that allow users to run the software for any purpose as well as to study, change, and distribute it and any adapted versions. Free software is a matter of liberty, no ...
in C++, Fortran, and Matlab. {{DEFAULTSORT:Chebyshev-Gauss quadrature Numerical integration (quadrature)