A Survey of Methods of Numerical Approximation to Functions

Abstract

This work undertakes a survey of methods of numerical approximation to functions. The functions considered are taken to be continuous within the range of approximation. Some consideration is given to the types of approximating functions in common use and the measurement of goodness of fit. It is seen that these two criteria together decide by what method the unknown coefficients are to be determined. Some properties of orthogonal functions and continued fractions are presented. Methods of deriving interpolating functions are described. Approximations may often be based on series expansions and this is considered, with reference to Chebyshev series, asymptotic series and Pade’ approximants. The next section deals with approximations derived when the measure of fit is chosen as one of the three Holder norms L1, L2, or L.∞. The L, problem is shown to be solved in some cases by treatment as an interpolation problem. The least-squares (L2) problem is best treated using orthogonal polynomials. The minimax (L∞,) approximation.is seen to be found only by means of an iterative process and is the best approach when finding rational function approximations. The method of spline approximations is described. This is basically an interpolative approach, the practical method involves representing the function between the points of agreement, or knots, by cubic polynomials. Finally a general summary covers the types of approximation considered. Some techniques, e.g. range reduction, are mentioned which help in certain cases with finding efficient approximations. An attempt is made to give a general strategy which can be adopted for finding a suitable approximation to a given function and which would be workable in all but exceptional cases.