Difference between revisions of "Chebyshev quadrature formula"
m (label) |
|||
| Line 2: | Line 2: | ||
An interpolation quadrature formula with equal coefficients: | An interpolation quadrature formula with equal coefficients: | ||
| − | $$\int\limits_{-1}^1f(x)dx\cong C\sum_{k=1}^Nf(x_k).\tag{*}$$ | + | $$\int\limits_{-1}^1f(x)\,dx\cong C\sum_{k=1}^Nf(x_k).\label{*}\tag{*}$$ |
| − | The weight function is equal to one, and the integration interval is finite and is taken to coincide with $[-1,1]$. The number of parameters defining the quadrature formula \ | + | The weight function is equal to one, and the integration interval is finite and is taken to coincide with $[-1,1]$. The number of parameters defining the quadrature formula \eqref{*} is $N+1$ ($N$ nodes and the value of the coefficient $C$). The parameters are determined by the requirement that \eqref{*} is exact for all polynomials of degree $N$ or less, or equivalently, for the monomials $1,x,\ldots,x^N$. The parameter $C$ is obtained from the condition that the quadrature formula is exact for $f(x)=1$, and is equal to $2/N$. The nodes $x_1,\ldots,x_N$ turn out to be real only for $N=1,\ldots,7$ and $N=9$. For $N=1,\ldots,7$ the nodes were calculated by P.L. Chebyshev. For $N\geq10$ among the nodes of the Chebyshev quadrature formula there always are complex ones (cf. [[#References|[1]]]). The algebraic degree of precision of the Chebyshev quadrature formula is $N$ for odd $N$ and $N+1$ for even $N$. Formula \eqref{*} was proposed by Chebyshev in 1873. |
====References==== | ====References==== | ||
Latest revision as of 16:58, 14 February 2020
An interpolation quadrature formula with equal coefficients:
$$\int\limits_{-1}^1f(x)\,dx\cong C\sum_{k=1}^Nf(x_k).\label{*}\tag{*}$$
The weight function is equal to one, and the integration interval is finite and is taken to coincide with $[-1,1]$. The number of parameters defining the quadrature formula \eqref{*} is $N+1$ ($N$ nodes and the value of the coefficient $C$). The parameters are determined by the requirement that \eqref{*} is exact for all polynomials of degree $N$ or less, or equivalently, for the monomials $1,x,\ldots,x^N$. The parameter $C$ is obtained from the condition that the quadrature formula is exact for $f(x)=1$, and is equal to $2/N$. The nodes $x_1,\ldots,x_N$ turn out to be real only for $N=1,\ldots,7$ and $N=9$. For $N=1,\ldots,7$ the nodes were calculated by P.L. Chebyshev. For $N\geq10$ among the nodes of the Chebyshev quadrature formula there always are complex ones (cf. [1]). The algebraic degree of precision of the Chebyshev quadrature formula is $N$ for odd $N$ and $N+1$ for even $N$. Formula \eqref{*} was proposed by Chebyshev in 1873.
References
| [1] | N.M. Krylov, "Approximate calculation of integrals" , Macmillan (1962) (Translated from Russian) |
Comments
This formula is to be distinguished from the Gauss–Chebyshev quadrature formula (cf. Gauss quadrature formula), which is defined using a weight function $\neq1$.
The original reference for Chebyshev's quadrature formula is [a3]. S.N. Bernshtein [a2] has shown that the nodes are real only if $N\leq7$ or $N=9$. A detailed discussion of the formula can be found in [a4]. Tables of quadrature nodes are given in [a1].
References
| [a1] | A. Segun, M. Abramowitz, "Handbook of mathematical functions" , Appl. Math. Ser. , 55 , Nat. Bur. Standards (1970) |
| [a2] | S.N. Bernshtein, "Sur les formules quadratures de Cotes et Chebyshev" C.R. Acad. Sci. USSR , 14 pp. 323–326 |
| [a3] | P.L. Chebyshev, "Sur les quadratures" J. Math. Pures Appl. , 19 : 2 (1874) pp. 19–34 (Oeuvres, Vol. 2, pp. 165–180) |
| [a4] | F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1974) |
| [a5] | P.J. Davis, P. Rabinowitz, "Methods of numerical integration" , Acad. Press (1984) |
Chebyshev quadrature formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_quadrature_formula&oldid=32708