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. [1]). The algebraic degree of precision of the Chebyshev quadrature formula is for odd and for even . Formula (*) 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 .

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].

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)