INTRODUCTION In cases where the indefinite integral@??f(x) dx can be expressed in terms of well-known simple algebraic or trigonometrical functions@ evaluation of the definite integral is straightforward. In other cases it may be difficult or indeed impossible to evaluate an integral by analytical means and for such cases numerical methods are used. The extension of these methods to multiple integrals is also considered. Frequently the function to be integrated is only specified by a number of discrete values of the function at stations within the range of integration. Such cases usually arise from experimental data and can be treated@ for example@ by fitting a least-squares curve through the data and integrating the resulting curve. This general topic@ and others associated with curve fitting@ is not considered in this Item. Numerous numerical integration formulae have been devised. Those finding most common application are formulae in which the definite integral is expressed solely in terms of the integrand@ f(x)@ at selected values of the variable@ x. A common feature of these methods is the essential approximation of the function@ f(x)@ by a polynomial of a certain degree over the range of integration or a part thereof. Two main classes of formulae worthy of more detailed description are described in the text. The first comprises the Newton-Cotes formulae wherein the selected values of the independent variable are usually chosen to be equally spaced across the integration range. The second class of formulae are termed Gaussian@ in which the intervals in x are determined by the condition that the integration formulae have the highest possible degree of accuracy within the limits set by the total number of stations. Selected sources of information supplementary to that in this Item are References 1 to 4 and 6 to 11. The compromise between effort expended and accuracy of results is of considerable importance and so is discussed in this Item. A problem that may occasionally arise is that it is desired to integrate up to or across a singularity of the integrand. No general method of coping with this is forthcoming@ but some guidance is given in the Item. Organisations having access to reasonably powerful computation facilities may have ready access to some suitable library programs for evaluating definite integrals. However@ to cater for the needs of people not so placed@ a computer program is provided and is described in Section 7 of this Item. Examples illustrating the use of the program are given in Section 8.
ESDU 85046 D-2010 history
2010ESDU 85046 D-2010 Quadrature methods for the evaluation of definite integrals