Sharp Error Bounds for the Trapezoidal Rule and Simpson's Rule

Volume 3, Issue 4, 2002

Article 39

SHARP ERROR BOUNDS FOR THE TRAPEZOIDAL RULE AND SIMPSON'S RULE

DEPARTMENT OF MATHEMATICS
TRINITY COLLEGE
HARTFORD, CT 06106-3100, USA.
E-Mail: david.cruzuribe@mail.trincoll.edu

DEPARTMENT OF MATHEMATICS
PURDUE UNIVERSITY
WEST LAFAYETTE,
N 47907-1395, USA.
E-Mail: neug@math.purdue.edu

Received 04 April, 2002; Accepted 01 May, 2002.
Communicated by: A. Fiorenza

ABSTRACT. We give error bounds for the trapezoidal rule and Simpson's rule for "rough" continuous functions---for instance, functions which are Hölder continuous, of bounded variation, or which are absolutely continuous and whose derivative is in L^p. These differ considerably from the classical results, which require the functions to have continuous higher derivatives. Further, we show that our results are sharp, and in many cases precisely characterize the functions for which equality holds. One consequence of these results is that for rough functions, the error estimates for the trapezoidal rule are better (that is, have smaller constants) than those for Simpson's rule.

Key words:
Numerical integration, Trapezoidal rule, Simpson's rule

2000 Mathematics Subject Classification:
26A42, 41A55.

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