A Novel Class of Symmetric and Nonsymmetric Periodizing Variable Transformations for Numerical Integration |
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Authors: | Avram Sidi |
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Affiliation: | (1) Computer Science Department, Technion – Israel Institute of Technology, Haifa, 32000, Israel |
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Abstract: | Variable transformations for numerical integration have been used for improving the accuracy of the trapezoidal rule. Specifically, one first transforms the integral via a variable transformation that maps [0,1] to itself, and then approximates the resulting transformed integral by the trapezoidal rule. In this work, we propose a new class of symmetric and nonsymmetric variable transformations which we denote , where r and s are positive scalars assigned by the user. A simple representative of this class is . We show that, in case , or but has algebraic (endpoint) singularities at x = 0 and/or x = 1, the trapezoidal rule on the transformed integral produces exceptionally high accuracies for special values of r and s. In particular, when and we employ , the error in the approximation is (i) O(h r ) for arbitrary r and (ii) O(h 2r ) if r is a positive odd integer at least 3, h being the integration step. We illustrate the use of these transformations and the accompanying theory with numerical examples. |
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Keywords: | Numerical integration variable transformations sin m -transformation Euler– Maclaurin expansions asymptotic expansions trapezoidal rule |
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