1. Interpolation and Approximation To the question, "Why approximate?", we can only answer, "Because we must!" Mathematical models of physical or natural processes inevitably contain some inherent errors. -result from incomplete understanding of natural phenomena - the stochastic or random nature of many processes -uncertainties in experimental measurements. Often, a model includes only the most pertinent features of the physical process and is deliberately stripped of superfluous detail related to second-level effects.

2. he most common approximating functions g(x) are those involving linear combinations of simple functions drawn from a class of funqtions {g,(x)} of the form The classes of functions most often encountered are the monomials {Xi}, i = 0, I,, n, the Fourier functions {sin kx, cos kx}, k = 0, I,, n, Linear combination of the Fourier functions leads to approximations of the form

3. -Approximations employing exponentials are usually of the form the form' Rational approximations, The theory of polynomial approximation is well developed and fairly simple. Polynomials are easy to evaluate and their sums, products, and differences are also polynomials. Polynomials can be differentiated and integrated with little difficulty, yielding other polynomials in both cases. In addition, if the origin of the coordinate system is shifted or if the scale of the independent variable is changed, the transformed polynomials remain polynomials, that is, if pn(x) is a polynomial, so are pn(x + a) and pn(ax). Some, but not all, of these favorable properties are possessed by the Fourier approximations as well.

4. Despite the fact

7. One reason for not using the simultaneous equations approach is that solving a linear system of any size is not an easy task, particularly if hand methods are being used. More important, perhaps, the development of an interpolating formula often produces an error term as a by-product.

8. Most numerical analysis textbooks cover in great detail the subject of interpolation by formulas derived from the criterion of (1.2). All can be classified into one of two groups: those applicable for arbitrarily spaced base points and those for evenly spaced base points, that is, for base points,

14. A similar symmetric form for the nth divided difference in terms of the tabulated arguments and functional values:

15. This symmetric form can be written more compactly as From geometric considerations alone. it is apparent that, for this case,

16. Equation (1.19) is then, of course, also only an approximation. This situation is shown schematically in Fig. 1.7.

17. To account for any discrepancy and to restore the desired equality, an error or remainder term R1(x) can be appended to (1.19).Then Solving (1.21) for R1(x) and collecting factors in terms of the finite divided differences yields Then (1.2l) has the form It is, of course, impossible to compute f[x,x I,xo] exactly since f(x), required for its evaluation, is unknown, if an additional value of f(x) is known, say f(x2) at x=x2, then, on the assumption that f[x,xI,xo] is not a rapidly changing function on the interval containing x2,x1 and xo, that is,

18. inadequate. To introduce some curvature into the approximating function, assume for the moment that the second divided difference f[x,x1,xo) is constant and given by f[x,x1,xo] = f[x2,xI,x0] (1.25) Then the remainder term given by (1.24) can be incorporated into (1.23) to yield a second-degree polynomial

20. The procedure already employed to generate the first, en the second, then the third-degree interpolating polyomial leads by induction to the general form for nth degree interpolation, Newlon's fundamental formula : Here the nth-degree divided-difference interpolating polynomial p,,(x) has the form Following the procedure used to develop (1.22) and (1.28), the error term for the third-degree polynomial (1.29) can be shown to have the form

27. Polynomial Interpolation with Equally Spaced Base points When the base-point values are equally spaced so that

34. Equation (1.54) is known as Newton's forward formula (NFF). Notice that NFF uses only the differences along the upper diagonal of the difference table. Consequently, NFF is most useful for interpolation near the beginning of an equal interval table. Of course, the same formula can be applied in other parts of a table by a suitable translation of the zero subscript.

35. If Newton's fundamental formula is written in terms of the divided differences along the lower diagonal path of a divided-difference table, (1.30) has the form

36. Equation (1.61) is known as Newton's backward formula (NBF). Since NBF uses differences along the lower diagonal of the difference table, it is most useful for interpolation near the end of a set of tabulated values.

37. Central Differences In Section 1.6 it was shown that the interpolation error tends to be smallest when we fit the interpolating polynomial with base points on both sides of the interpolation argument. This can be effected by choosing a path through the divided-difference table which zigzags about some base point near the interpolation argument. Stated another way, this means that if a central path is taken across the difference table, more of the functional value is represented by leading terms in the difference formulation (the sequence converges faster). Consequently, a low-order central interpolation may pro duce answers with remainder terms no larger than a higher-order fit using the forward or backward paths. Of course, if the same base points are used to produce polynomials of equivalent degree, then all paths are equivalent.