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Stochastic properties of quadrature formulas. (English) Zbl 0655.65041

Stochastic properties of several quadrature formulas are discussed. Let \(F\subset L_ 1([0,1]\beta d,\lambda \beta d)\) be a given set of functions and let \(S(f)=\int_{[0,1]\beta d} f(x)dx\) and \(\tilde S(f)=\phi(a_ 1,a_ 2,...,a_ n)\) be a quadrature method only usingthe function values at n knots \(a_ 1,a_ 2,...,a_ n\in [0,1]\beta d\). The maximal error of a method \(\tilde S\) is given by \(\Delta_{\max}(S)=\sup \| S(f)-S(f)\|,\) while the average error is given by \(\Delta_{\mu}(S)=\int_{F}| S(f)-\tilde S(f)| d\mu(f),\) where \(\mu\) is a Borel probability measure on \((F,\| \|_{\infty})\). It is shown that stochastic error bounds, i.e., the error bounds of the average error, for quadratic formulas are much smaller than deterministic ones (bound for the maximal error) in many cases, though it dependson the class of F on functions to be considered. Moreover, nonlinear methods, adaptive methods or even methods with varying cardinality are shown not be significantly better than the simple linear method with respect to certain stochastic error bounds.
Reviewer: K.Uosaki

MSC:

65D32 Numerical quadrature and cubature formulas
65C05 Monte Carlo methods
41A55 Approximate quadratures
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References:

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