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Interpolation and Extrapolation part 1

We sometimes know the value of a function f(x) at a set of points x1, x2, . . . , xN (say, with x1 | Chapter 3. Interpolation and Extrapolation 3.0 Introduction We sometimes know the value of a function f x at a set of points x1 x2 . . xN say with x1 . xN but we don t have an analytic expression for f x thatlets us calculate its value at an arbitrary point. For example the f x s might result from some physical measurement or from long numerical calculation that cannot be cast into a simple functional form. Often the xi s are equally spaced but not necessarily. The task now is to estimate f x for arbitrary x by in some sense drawing a smooth curve through and perhaps beyond the xi. If the desired x is in between the largest and smallest of the xi s the problem is called interpolation if x is outside that range it is called extrapolation which is considerably more hazardous as many former stock-market analysts can attest . Interpolation and extrapolation schemes must model the function between or beyond the known points by some plausible functional form. The form should be sufficiently general so as to be able to approximate large classes of functions which might arise in practice. By far most common among the functional forms used are polynomials 3.1 . Rational functions quotients of polynomials also turn out to be extremely useful 3.2 . Trigonometric functions sines and cosines give rise to trigonometric interpolation and related Fourier methods which we defer to Chapters 12 and 13. There is an extensive mathematical literature devoted to theorems about what sort of functions can be well approximated by which interpolating functions. These theorems are alas almost completely useless in day-to-day work If we know enough about our function to apply a theorem of any power we are usually not in the pitiful state of having to interpolate on a table of its values Interpolation is related to but distinct from function approximation. That task consists of finding an approximate but easily computable function to use in place of a more complicated one. In the case of .