A fast verified automatic integration algorithm using double exponential formula

Abstract

A fast verified automatic integration algorithm is proposed for calculating univariate integrals over finite intervals. This algorithm is based on the double exponential formula proposed by Takahasi and Mori. The double exponential formula uses a certain trapezoidal rule. This trapezoidal rule is determined by fixing two parameters, the width h of a subdivision of a finite interval and the number n of subdivision points of this subdivision. A theorem is presented for calculating h and n as a function of a given tolerance of the verified numerical integration of a definite integral. An efficient a priori method is also proposed for evaluating function calculation errors including rounding errors of floating point calculations. Combining these, a fast algorithm is proposed for verified automatic integration. Numerical examples are presented for illustrating effectiveness of the proposed algorithm.