Abstract:

ESDU 85046 considers two classes of quadrature formulae that may be used for the approximation of definite integrals. They are Newton-Cotes formulae, of both the open and closed type, in which the abscissae are equally spaced across the integration interval, and Gauss-Legendre formulae, in which the abscissae are unequally spaced and determined from the roots of the Legendre polynomials. General expressions are given for these two classes and tables present the abscissae and weights for formulae up to seventh-order. The integration error and choice of formulae are discussed for each method.

This Item further considers an adaptive integration scheme which can distinguish between regions of high and low functional variation and can adapt the abscissae step lengths to meet a specified tolerance.

It is shown that all the methods can be modified in a straightforward manner for the approximation of multiple integrals. It is also shown, through the example of a double integral, that the integration limits can all be constants (i.e. giving a rectangular region of integration in the case of a double integral), or the inner integral limits may be functions of the variable of the outer integral (i.e. the region of integration is non-rectangular). General guidance is given for coping with integrals where it is desired to integrate up to or across a singularity of the integrand.

The Item introduces a computer program together with a full specification for the approximation of both a single definite integral and a double definite integral where the region of integration can be rectangular or non-rectangular. The program uses an adaptive quadrature method based on the standard Simpson's rule. Examples illustrate the use of the computer program.