On the evaluation of one-dimensional Cauchy principal value integrals by rules based on cubic spline interpolation

In this note we consider the numerical evaluation of one dimensional Cauchy principal value integrals of the form $$\rlap{--} \smallint _a^b \frac{{k(x)f(x)}}{{x - \lambda }}dx, a< \lambda< b,$$ by rules obtained by “subtracting out” the singularity and then applying product quadratures based on cubic spline interpolation at equally spaced nodes. Convergence results are established for Hölder continuous functions of order, μ, 0<μ≤1, and asymptotic rates are obtained for functionsf≠C ^k [a, b],k=1, 2, 3 or 4. Some comparisons with other methods and numerical examples are also given.     

Error bounds for direct and composite cubature formulae—I

In this paper direct and composite two dimensional integration formulae are compared, and conditions are given under which certain error bounds turn out to be better for either one or the other approach.

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Sommario In questo articolo vengono confrontate formule dirette e formule composte di integrazione numerica in due dimensioni, determinando condizioni sotto cui risultano migliori certi limiti di errore per l'uno o per l'altro approccio.

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This work has been done within the GNAFA-CNR research activity.