Performance Evaluation of an Adaptive Quadrature and a Double Exponential Formula Using Arbitrary-Precision Floating-Point Arithmetic

Abstract

Using arbitrary-precision arithmetic provided by the GNU Multiple Precision Floating-Point Reliable Library, we implement AQE11D---that is, Ninomiya's adaptive 9-point Newton--Cotes rule extended with a sequence of higher-order rules---and Takahasi and Moris' double exponential (DE) formula. We evaluate them for Kahaner's 21 test problems. For both absolute tolerances $10^{-50}$ and $10^{-100}$, AQE11D attains target accuracy on all 21 problems; however, for strong endpoint singularity such as $1/\sqrt{x}$, it requires about $5.4\times10^{7}$ function evaluations at $10^{-100}$, roughly $7\times10^{4}$ times as many as the DE formula. The formula converges on 18 problems at both tolerances, demonstrating its strength against endpoint singularities but also its failure, as it stands, on problems with a singularity inside the integration interval.