Automatic integration using asymptotically optimal adaptive Simpson quadrature

Automatic integration using asymptotically optimal adaptive Simpson quadrature

We present a novel theoretical approach to the analysis of adaptive quadratures and adaptive Simpson quadratures in particular which leads to the construction of a new algorithm for automatic integration. For a given function [Formula: see text] with [Formula: see text] and possible endpoint singularities the algorithm produces an approximation to [Formula: see text] within a given [Formula: se...

Asymptotically Optimal Weighted Numerical Integration

We study numerical integration of H older{type functions with respect to weights on the real line. Our study extends previous work by F. Curbera, [2] and relies on a connection between this problem and the approximation of distribution functions by empirical ones. The analysis is based on a lemma which is important within the theory of optimal designs for approximating stochastic processes. As...

Advances in Bayesian Automatic Adaptive Quadrature

If the condition E < τ is not fulfilled, the value of Q is refined, and the error estimate E is lowered, until (2) is satisfied. One way to reach this goal is the aaq. Essentially, this builds a subrange binary tree the root of which is the input integration domain [a, b] and the evolution of which is controlled by an associated priority queue. Over each subrange [α, β] ⊆ [a, b] of the subrange...

Self-Adaptive Quadrature and Numerical Path Integration

Handling accuracy in Bayesian automatic adaptive quadrature

New Bayesian inferences which significantly extend the coverage of the Bayesian automatic adaptive quadrature (BAAQ) approach to the solution of Riemann integrals are reported. The scrutiny of the possible floating-point machine number approximations of abscissa values inside an integration domain unveiled the occurrence of five classes of integration domains entering the quadrature problems: z...