Methods of Approximation
Ideally, every function would be easy to deal with in all mathematical situations, abstract and applied. Unfortunately, this is not always the case. It sometimes becomes convenient to have an approximation to the function, usually a polynomial or some other "well-behaved" series of functional terms. There are several ways to formulate an approximation when this is desirable.
The simplest method of approximation is the least accurate one: estimation. With a certain amount of mathematical intuition, sometimes it is possible to guess a curve (or some related quantity of interest) that is fairly close to the actual desired curve. However, this method is only recommended as a starting point for more exact approximations, since it is an educated guess in the most literal sense of that phrase.
A Taylor series approximation relies upon the derivatives of the function being approximated, and it requires a central point about which to base the approximation. With an infinite number of k values starting at zero, each term in the approximation is (x - x0)k f(k)(x0)/k!. The point x0 is often chosen to be zero itself, making for a simplified series form. Another desirable choice of x0 is such that many of the derivatives will disappear at that point-all of the odd or even derivatives, for example, or all derivatives greater than a certain number. This polynomial series is exact as long as it is infinite; its error rate depends largely on how many terms can be used.
A Legendre polynomial makes it possible to interpolate data between known points into a function. The polynomial in its simplest form is written as y = y0 (x - x1)/(x0 - x1) + y1 (x - x0)/(x1 - x0). This will provide a result between the two selected points x0 and x1, with their functional values y0 and y1. This method may be used for data that do not have a known function associated with them, or for functions that are used to calculate the y-values, and will provide a polynomial interpolation either way. This method has the advantage over the Taylor series approximation in that it uses more than one point in its approximation. However, it is by no means exact. The method may be improved by changing to quadratic or cubic terms, in which case the polynomials each gain a term in both numerator and denominator and the number of points used goes up.
The problem with Legendre polynomials of successively higher accuracy is that each polynomial must be calculated separately. With Newton polynomials, once one polynomial of degree N has been obtained, the next, N + 1, is simply the Nth polynomial plus the term aN (x - x0)(x - x1)...(x - xN - 1). Each a-term is a fit coefficient obtained from the parameters known about the data or function. The Chebyshev polynomials use the same type of techniques as Legendre and Newton polynomials, but they use a limit definition to determine which coefficients of each group are most appropriate for the function over some specified interval, to maximize the accuracy of each method.
The above approximations are the ones most commonly used by human beings directly. There are several other approximations which are sometimes used in computer programs for various reasons. Pade approximations use a "rational approach," meaning that they feature two calculated polynomials of similar types to the above series, dividing one by the other for the net approximation. The largest restriction on these approximations is that the function and all its derivatives must vanish at zero in order for the polynomial to be manageable. Spline functions are not very accurate except on the order of many repetitions; an approximation by spline functions interpolates a chosen degree of polynomial (usually linear or quadratic) between each set of two adjacent points. This choppy method is sometimes useful for obtaining numerical results (especially when computer fitting can make the distance between any two adjacent selected points fairly small), but it does not allow for very much mathematical manipulation of the approximation and is of limited use on that basis.
Of course, there is nothing magical about polynomials as far as approximation goes; they are merely used because of their simplicity and their ease in mathematical use. They are also fairly easy to envision for even the most novice mathematician. However, any Hilbert space - that is, any set of orthogonal polynomials - may be used to obtain an exact series representation of a function over an infinite sum. The second most common set of functions used for approximation is the trigonometric functions, for Fourier approximation. This approximation, using sines and cosines, is favored because it gives approximations over a well-defined and physically useful periodicity.
Approximation is not a skill which is emphasized in some formal portions of mathematics, but it is essential to any branch of applied mathematics. Computer science uses approximate calculations for almost any command, and physics and engineering are almost as dependent upon approximation and interpolation in their experimental phases.
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