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Approximation Theory and its Applications

, Volume 11, Issue 4, pp 78–89 | Cite as

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

  • Kai Diethelm
Article
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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

Quadrature Formula Singular Integral Quadrature Rule Linear Quadrature Cauchy Principal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Braß H., Quadraturverfahren, Vandenhoeck & Ruprecht, Göttingen, 1977.MATHGoogle Scholar
  2. [2]
    Davis P. J., and Rabinowita P., Methods of Numerical Integration, Academic Press, Orlando, 2nd edition, 1984.MATHGoogle Scholar
  3. [3]
    Diethelm K., A Definiteness Criterion for Linear Functionals and its Application to Cauchy Principal Value Quadrature, J. Comput. Appl. Math., to appear.Google Scholar
  4. [4]
    Diethelm K., Error Estimates for a Quadrature Rule for Cauchy Principal Value Integrals. In W. Gautschi, editor, Mathematics of Computation 1943–1993: A Half-Century of Computational Mathematics, volume 48 of Proc. Symp. Appl. Math., 1994, pp. 287–291.Google Scholar
  5. [5]
    —, Modified Compound Quadrature Rules for Strongly Singular Integrals, Computing, 52(4):337–354, 1994.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Diethelm K., Peano Kernels and Bounds for the Error Constants of Gaussian and Related Quadrature Rules for Cauchy Principal Value Integrals. Numer. Math., to appear.Google Scholar
  7. [7]
    —, Uniform Convergence of Optimal Order Quadrature Rules for Cauchy Principal Value Integrals, J. Comput. Appl. Math., 56:321–329, 1994.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Gerasoulis A., Piecewise Polynomial Quadratures for Cauchy Singular Integrals, SIAM J. Numer. Anal., 23:891–902, 1986.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Köhler P., Asymptotically Sharp Error Estimates for Modified Compound Quadrature Formulae for Cauchy Principal Value Integrals, Computing, to appear.Google Scholar
  10. [10]
    Palamara, Orsi A., Spline Approximation for Cauchy Principal Value Integrals. J. Comput. Appl. Math., 30:191–201, 1990.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Rabinowitz, P., Convergence Results for Piecewise Linear Quadratures for Cauchy Prinipal Value Integrals, Math. Comp., 51:741–747, 1988.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Sard, A., Integral Representations of Remainders, Duke Math. J., 15:333–345, 1948.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Stewart, C.E., On the Numerical Evaluation of Singular Integrals of Cauchy Type, J. Soc. Indust. Appl. Math., 8:342–353, 1960.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Stolle, H. W., Strauß, R., On the Numerical Integration of Certain Singular Integrals, Computing, 48:177–189, 1992.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Strauß R., Eine Interpolationsquadratur für Cauchy-Hauptwertintegrale, Rostock, Math. Kolloq., 22:57–66, 1983.MATHGoogle Scholar

Copyright information

© Springer 1995

Authors and Affiliations

  • Kai Diethelm
    • 1
  1. 1.Institut für MathematikUniversität HildesheimHildesheimGermany

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