Abstract
One of the standard methods for computing Cauchy principal value integrals is to subtract the singularity, and then to apply a given quadrature formula. This results in a quadrature formula for the Cauchy principal value integral which is called a modified quadrature formula. Here, we consider the case that this given quadrature formula is a compound quadrature formula, and derive error estimates of the form |R[f]| ≤κ i ∥f (i)∥∞ (whereR[f] is the error of the modified quadrature formula). In contrast to previous estimates, the behaviour ofκ i when the number of quadrature nodes tends to infinity is determined exactly.
Zusammenfassung
Eine der Standardmethoden zur Berechnung Cauchyscher Hauptwerte ist es, zunächst die Singularität herauszuziehen und dann eine gegebene Quadraturformel anzuwenden. Die daraus resultierende Quadraturformel für das Cauchysche Hauptwert-Integral wird als modifizierte Quadraturformel bezeichnet. In dieser Arbeit wird der Fall betrachtet, daß die gegebene Quadraturformel eine zusammengesetzte Quadraturformel ist, und es werden Fehlerabschätzungen der Form |R[f]| ≤κ i ∥f (i)∥∞ hergeleitet (wobeiR[f] der Fehler der modifizierten Quadraturformel ist). Im Gegensatz zu früheren Abschätzungen wird das Verhalten vonκ i für den Fall, daß die Anzahl der Stützstellen gegen Unendlich geht, exakt bestimmt.
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Köhler, P. Asymptotically sharp error estimates for modified compound quadrature formulae for Cauchy principal value integrals. Computing 55, 255–269 (1995). https://doi.org/10.1007/BF02238435
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DOI: https://doi.org/10.1007/BF02238435