Approximation Theory and its Applications

, Volume 11, Issue 4, pp 78–89

# Asymptotically sharp error bounds for a quadrature rule for Cauchy principal value integrals based on piecewise linear interpolation

• Kai Diethelm
Article

## Abstract

For the numerical evaluation of Cauchy principal value integrals of the form$$\int_{ - 1}^1 {f\left( x \right)\left( {x - \lambda } \right)^{ - 1} dx}$$ with λ∈ (−1,1) and f∈C’[−1’,1], we investigate the quadrature formula Q n+1 Spl1 [·;λ] obtained by replacing the integrand function f by its piecewise linear interpolant at an equidistant set of nodes as proposed by Rabinowitz (Math. Comp., 51:741–747,1988). We give upper bounds for the Peano—type error constants
$$\rho _s \left( {R_{n + 1}^{Spl1} \left[ { \bullet ;\lambda } \right]} \right): = sup\left\{ {\left| {\int_{ - 1}^1 {\frac{{f\left( x \right)}}{{x - \lambda }}dx - Q_{n + 1}^{Spl1} \left[ {f;\lambda } \right]} } \right|:f \in C'\left[ { - 1,1} \right],\left\| {f^{\left( s \right)} } \right\|_\infty \leqslant 1} \right\}$$
for s∈{1,2}. These are the best possible constants in inequalities of the type
$$\left| {\int_{ - 1}^1 {\frac{{f\left( x \right)}}{{x - \lambda }}dx - Q_{n + 1}^{Spl1} \left[ {f;\lambda } \right]} } \right|: \leqslant c_{1,n + 1} \left( \lambda \right)\left\| {f^{\left( 1 \right)} } \right\|_\infty$$
. Furthermore, we prove that our upper bounds are asymptotically sharp.

## Keywords

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