A Novel Class of Symmetric and Nonsymmetric Periodizing Variable Transformations for Numerical Integration

Variable transformations for numerical integration have been used for improving the accuracy of the trapezoidal rule. Specifically, one first transforms the integral ${I[f]=\int^1_0f(x) dx}$ via a variable transformation ${x=\phi(t)}$ that maps [0,1] to itself, and then approximates the resulting transformed integral ${I[f]= \int^1_0 f\big(\phi(t)\big)\phi'(t) dt}$ by the trapezoidal rule. In this work, we propose a new class of symmetric and nonsymmetric variable transformations which we denote ${\mathcal{T}_{r,s}}$ , where r and s are positive scalars assigned by the user. A simple representative of this class is ${\phi(t)=(sin\frac{\pi}{2}t)^r/[(sin\frac{\pi}{2}t)^r+(\cos\frac{\pi}{2}t)^s]}$ . We show that, in case ${f\in C^\infty[0,1]}$ , or ${\in C^\infty(0,1)}$ 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 ${f\in C^\infty[0,1]}$ and we employ ${\phi\in{\mathcal T}_{r,r}}$ , 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.