Anzeige
Erschienen in:
2015 | OriginalPaper | Buchkapitel
4. Quasi-Monte Carlo Methods
verfasst von : Harald Niederreiter, Arne Winterhof
Erschienen in: Applied Number Theory
Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.
Wählen Sie Textabschnitte aus um mit Künstlicher Intelligenz passenden Patente zu finden. powered by
Markieren Sie Textabschnitte, um KI-gestützt weitere passende Inhalte zu finden. powered by
Abstract
There are many scientific as well as real-world applications where we run into the problem of computing a definite integral. In calculus courses you are taught that a definite integral \(\int _{a}^{b}f(u)du\) is evaluated by the fundamental theorem of integral calculus which says that where the function F is an antiderivative of the integrand f. What you are often not told is that there are many cases where F cannot be expressed in finite terms by means of elementary functions, and in such situations the formula (4.1) is useless for computational purposes. Examples are \(\int _{0}^{1}e^{-u^{2} }du\) and \(\int _{0}^{1}(\sin u)(u + 1)^{-1}du\). We then have to settle for numerical approximations of \(\int _{a}^{b}f(u)du\). The process of approximately computing definite integrals with a sufficient degree of precision is called numerical integration.
$$\displaystyle{ \int _{a}^{b}f(u)du = F(b) - F(a), }$$
(4.1)