Bivariate Product Cubature Using Peano Kernels for Local Error Estimates

Abstract

The error estimates of automatic integration by pure floating-point arithmetic are intrinsically embedded with uncertainty. This in critical cases can make the computation problematic. To avoid the problem, we use product rules to implement a self-validating subroutine for bivariate cubature over rectangular regions. Different from previous self-validating integrators for multiple variables (Storck in Scientific Computing with Automatic Result Verification, pp. 187---224, Academic Press, San Diego, [1993]; Wolfe in Appl. Math. Comput. 96:145---159, [1998]), which use derivatives of specific higher orders for the error estimates, we extend the ideas for univariate quadrature investigated in (Chen in Computing 78(1):81---99, [2006]) to our bivariate cubature to enable locally adaptive error estimates by full utilization of Peano kernels theorem. The mechanism for active recognition of unreachable error bounds is also set up. We demonstrate the effectiveness of our approach by comparing it with a conventional integrator.