Gaussian quadrature formulae of the third kind for Cauchy principal value integrals: Basic properties and error estimates

22Citations
Citations of this article
5Readers
Mendeley users who have this article in their library.

Abstract

Let ∏n-1[f] be the polynomial of degree n-1 interpolating the function f at the points x1,x2, ...,xn with Pn(xi) = 0, i.e., at the nodes of the classical Gaussian quadrature formula. For the numerical approximation of the Cauchy principal value integral {cauchy integral}1-1 f{hook}(x)(x-λ)-1 dx with λ ∈ (-1,1) and f ∈ C1[-1,1], we present the quadrature formula Qn+1G3 given by Q n+1 G3[f{hook};λ]:= ∫ - 1 πn-1[f{hook}](x)-πn-1[f{hook}](λ) x-λdx+f{hook}(λ)1n 1-λ 1+λ. We show that this quadrature formula does not have the disadvantages of the other two well-known quadrature formulae based on the same set of nodes. In particular, we prove that the sequence ( based on the same set of nodes. In particular, we prove that the sequence (Qn + 1G3[f; λ]) converges to the true value of the integral uniformly for all λ ∈ (-1, 1). We give estimates for the error term. Furthermore, we state some relations connecting the present quadrature formula to the previously introduced formulae. © 1995.

Cite

CITATION STYLE

APA

Diethelm, K. (1995). Gaussian quadrature formulae of the third kind for Cauchy principal value integrals: Basic properties and error estimates. Journal of Computational and Applied Mathematics, 65(1–3), 97–114. https://doi.org/10.1016/0377-0427(95)00103-4

Register to see more suggestions

Mendeley helps you to discover research relevant for your work.

Already have an account?

Save time finding and organizing research with Mendeley

Sign up for free