Approximate product-integration

Abstract

Methods of obtaining numerical integration formulae of the type ∫ b a f ( x ) ϕ ( x )d x = Ʃ r α r f ( x r ) + correction terms are derived. A main feature of the development is that the calculations of the weights α r and corrections are made to depend on the inversion of certain standard matrices. Tables of the inverse matrices required for certain generalized formulae of the Newton-Cotes type are given, together with tables of the coefficients required in the correction terms. If f ( x ) is a polynomial of degree n , and n + 1 ordinates are used, then, whatever the form of ϕ ( x ), the formulae are exact, that is, they introduce no analytical error. When it is convenient to do so, ordinates outside the range of integration may be used, no change in the method of deriving the formulae being necessary. If the integrand is singular within or near the range of integration, conventional methods of numerical integration break down, but in the formulae derived here, the presence of singularities in ϕ ( x ) does not affect the derivation if certain conditions are satisfied. The methods are not affected by discontinuities in ϕ ( x ). The methods described here have an important application to the numerical solution of all types of integral equations: this application will be discussed in the following paper.