Abstract
Many algorithms that have been proposed for the numerical evaluation of Cauchy principal value integrals are numerically unstable. In this work we present some formulae to evaluate the known Gaussian quadrature rules for finite part integrals , and extend Clenshow's algorithm to evaluate these integrals in a stable way.
Highlights
In the theory of the numerical approximation of Cauchy principal value integrals, basically two kinds of Gaussian quadrature formulae have been investigated
Many algorithms that have been proposed for the numerical evaluation of Cauchy principal value integrals are numerically unstable
For the modified Gaussian formula, we refer to 1–7 and the literature cited therein
Summary
In the theory of the numerical approximation of Cauchy principal value integrals, basically two kinds of Gaussian quadrature formulae have been investigated. Many algorithms that have been proposed for the numerical evaluation of Cauchy principal value integrals are numerically unstable. This is true for the standard algorithm proposed in 3, Section 3 for the evaluation of the modified Gaussian formula 15, Section 4. We present some formulae to evaluate the known Gaussian quadrature rules for finite part integrals. The appendix, with closed form solutions for class of these integrals, supplements the paper
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