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

• Authors: Avram Sidi
• Venue: Journal of Scientific Computing
• Year: 2007

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.