A New Variable Transformation for Numerical Integration

Abstract

Presently, variable transformations are used to enhance the performance of lattice rules for multidimensional integration. The transformations that are in the literature so far are of either polynomial or exponential nature. Following a short survey of some of the transformations that have been found to be effective, we propose a new transformation, denoted the sin^m-transformation, that is neither polynomial nor exponential, but trigonometric, in nature. This transformation is also a representative of a general class of variable transformations that we denote S _m . We analyze the effect of transformations in S _m within the framework of one-dimensional integration, and show that they have some very interesting and useful properties. Present results indicate that transformations in S _m can be more advantageous than known polynomial transformations, and have less underflow and overflow problems than exponential ones. Indeed, the various numerical tests performed with the sin^m-transformation support this. We end the paper with numerical examples through which some of the theory is verified.