A probabilistic theory for error estimation in automatic integration
We show that probabilistic analysis provides a mathematical foundation for heuristic error estimation in automatic integration. Let \(\int^{b}_{a} f(t)dt\) be approximated by \(Q_ n (f) = \sum^{n}_{i=1} a_ i f(t_ i)\) and assume on a uniform mesh there exists a constant \(c>0\), such that \(| \int^{b}_{a} f(t)dt - Q_ n(f) | \leq c/n^ k \sup_{a\leq t\leq b} | f^{(k)}(t) |,\) for any \(f\in C^ k_ 0 [a,b]\). Consider the following heuristic error criterion: \(| \int^{b}_{a} f(t)dt -Q_ n(f) | \approx c/n^ k \max_{1\leq i\leq n} | [t_{i-k },\ldots,t_ i]f|,\) where \([t_{i-k},\ldots,t_ i]f\) is the \(k\)th divided difference of \(f\) at \(t_{i-k},\dots,t_ i\), \(k<n\), and by convention \(t_{-i} = a-t_ i\), \(f(t_ i) = 0\), \(i=1,\dots,n\). Our main result is the following theorem. There exist constants \(c_1\geq0\), \(c_2>0\), such that \[ \text{Average } \left| \int^{b}_{a} g(t)dt - Q_ n(f) \right|^2 \leq \frac{c_ 1}{n^{2k+2}} + c_2 n \left(\frac{c}{n^ k} \max_{1\leq i\leq n} | [t_{i-k},\dots,t_ i]f|\right)^2 \] \[ g \in C^ k_0[a,b] \qquad g(t_ i) = f(t_ i) \qquad i=1,\ldots,n \] for any \((f(t_ 1),\ldots,f(t_ n))^ T\in R^ n\). Here the average represen ts integration over \(C^ k_ 0[a,b]\) with respect to the conditional Wiener meas ure induced from the Wiener measure on \(C^ k_ 0[a,b]\) by the constraints \(g(t_ i)=f(t_ i)\), \(i=1,...,n\). The computational aspects of the theory are also discussed. In particular, we show that estimating the conditional average error is equivalent to estimating error with the spline interpolation as a test integrand, and that this process can be made implicit using error criteria with local error- checks.
scientific article; zbMATH DE number 4090680
scientific article