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, an -point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree or less by a suitable choice of the nodes and weights for . Th ...
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 orthogonal 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: ...
*
Chebyshev nodes In numerical analysis, Chebyshev nodes (also called Chebyshev points or a Chebyshev grid) are a set of specific algebraic numbers used as nodes for polynomial interpolation and numerical integration. They are the Projection (linear algebra), pr ...


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 Dig ...

Gauss–Chebyshev type 1 quadrature
an

free software Free software, libre software, libreware sometimes known as freedom-respecting software is computer software distributed open-source license, under terms that allow users to run the software for any purpose as well as to study, change, distribut ...
in C++, Fortran, and Matlab. {{DEFAULTSORT:Chebyshev-Gauss quadrature Numerical integration