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Local spline approximation methods for singular product integration. (English) Zbl 0911.65017

The authors consider the numerical evaluation of integrals of the form \[ I(K,f)=\int_{-1}^{1} K(x)f(x)dx \] where \(K\) is singular, but absolutely integrable, and \(f\) is bounded in \([-1,1]\), or of the form \[ J(uf;\lambda)= \iint_{-1}^{1}u(x)\frac{f(x)}{x-\lambda}dx, \quad -1<\lambda<1 \] where \[ \iint_{-1}^{1} := lim_{\varepsilon \rightarrow 0} \left\{ \int_{-1}^{\lambda-\varepsilon} + \int_{\lambda+\varepsilon}^{1} \right\} \] is the Cauchy principal value of \(u \cdot f\), with \(u\) and \(f\) such that \(J(uf;\lambda)\) exists. The authors present quadrature formulas for (1) and (2), for which: i) the precision degree is the highest possible using spline approximation; ii) the nodes can be assumed equal to arbitrary points, where the integral function \(f\) is known and computable; iii) the number of the requested evaluations of \(f\) at the nodes is low; iv) a satisfactory convergence theory can be proved.

MSC:

65D32 Numerical quadrature and cubature formulas
41A55 Approximate quadratures
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