Chebyshev quadrature formula
An interpolation quadrature formula with equal coefficients:
The weight function is equal to one, and the integration interval is finite and is taken to coincide with . The number of parameters defining the quadrature formula (*) is ( nodes and the value of the coefficient ). The parameters are determined by the requirement that (*) is exact for all polynomials of degree or less, or equivalently, for the monomials . The parameter is obtained from the condition that the quadrature formula is exact for , and is equal to . The nodes turn out to be real only for and . For the nodes were calculated by P.L. Chebyshev. For among the nodes of the Chebyshev quadrature formula there always are complex ones (cf. ). The algebraic degree of precision of the Chebyshev quadrature formula is for odd and for even . Formula (*) was proposed by Chebyshev in 1873.
|||N.M. Krylov, "Approximate calculation of integrals" , Macmillan (1962) (Translated from Russian)|
The original reference for Chebyshev's quadrature formula is [a3]. S.N. Bernshtein [a2] has shown that the nodes are real only if or . A detailed discussion of the formula can be found in [a4]. Tables of quadrature nodes are given in [a1].
|[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=16279