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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Generalized noninterpolatory rules for Cauchy principal value integrals

Author: Philip Rabinowitz
Journal: Math. Comp. 54 (1990), 271-279
MSC: Primary 65D30
MathSciNet review: 990601
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Abstract: Consider the Cauchy principal value integral

$\displaystyle I(kf;\lambda ) = \oint k(x)\frac{{f(x)}}{{x - \lambda }}\,dx,\quad - 1 < \lambda < 1.$

If we approximate $ f(x)$ by $ \sum _{j = 0}^N\;{a_j}{p_j}(x;w)$ where $ \{ {p_j}\} $ is a sequence of orthonormal polynomials with respect to an admissible weight function w and $ {a_j} = (f,{p_j})$, then an approximation to $ I(kf;\lambda )$ is given by $ \sum _{j = 0}^N\;{a_j}I(k{p_j};\lambda )$. If, in turn, we approximate $ {a_j}$ by $ {a_{jm}} = \sum _{i = 1}^m\;{w_{im}}f({x_{im}}){p_j}({x_{im}})$, then we get a double sequence of approximations $ \{ Q_m^N(f;\lambda )\} $ to $ I(kf;\lambda )$. We study the convergence of this sequence by relating it to the sequence of approximations associated with $ I(wf;\lambda )$ which has been investigated previously.

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Keywords: Cauchy principal value integrals, numerical integration, noninterpolatory integration rules, orthogonal polynomials
Article copyright: © Copyright 1990 American Mathematical Society