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

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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.

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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 Save time finding and organizing research with Mendeley