Error estimation in automatic quadrature routines

Abstract

A new algorithm for estimating the error in quadrature approximations is presented. Based on the same integrand evaluations that we need for approximating the integral, one may, for many quadrature rules, compute a sequence of null rule approximations. These null rule approximations are then used to produce an estimate of the local error. The algorithm allows us to take advantage of the degree of precision of the basic quadrature rule. In the experiments we show that the algorithm works satisfactorily for a selection of different quadrature rules on all test families of integrals.

Nine-stage multi-derivative Runge-Kutta method of order 12

“…The stability region of HBT(12)9 has a remarkably good shape. The stepsize is controlled by a formula which uses y (4) n and y (6) n . On the basis of CPU time versus the maximum global error, and the number of steps versus the maximum global energy error, HBT(12)9 wins over DP(8,7)13M and T12 in solving several well-known test problems.…”

Adaptive Cubatures on Triangle

“…A null rule has the degree d if it integrates to zero all basic monomials of degree ≤ d and fails to do so for a monomial of degree d + 1. [2,3,9] for additional information).…”

An efficient and reliable quadrature algorithm for integration with respect to binomial measures

“…The usual strategy is to adapt the calculation considering a local error index and derive some global error estimate. The algorithm can be adapted from the coteglob code presented in [11] and can be summarized as: Output: estimated integral I and estimated error E Routine utilized: local quadrature rule, local error estimator 6 If we try to perform an extrapolation based on GL 2 formulae we can gain at most one degree of exactness (if α = 1/2) and cannot obtain the interpolatory rule. 7 The notation H 4 is quite arbitrary, it comes from the observation that in the case α = 1/2 the formula coincides with NC 4 and can be also interpreted as an Hermite formula with the central node counted twice.…”

“…We will here present and generalize the best known local error estimate, based on the null rules. For the general theory and the proofs of the results we refer to [6,11,12].…”

“…We will have that asymptotically, λ i = O(2 −k ). Given these considerations, following [6,11], we will use the following estimate:…”