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Interpolation and Extrapolation part 3

Rational Function Interpolation and Extrapolation Some functions are not well approximated by polynomials, but are well approximated by rational functions, that is quotients of polynomials. We denote by Ri(i+1).(i+m) a rational function passing through the m + 1 points (xi , yi ) . . . (xi+m , yi+m ). More explicitly, suppose Ri(i+1).(i+m) = p 0 + p1 x + · · · + pµ x µ Pµ (x) = Qν (x) q 0 + q 1 x + · · · + q ν xν (3.2.1) | 3.2 Rational Function Interpolation and Extrapolation 111 3.2 Rational Function Interpolation and Extrapolation Some functions are not well approximated by polynomials but are well approximated by rational functions that is quotients of polynomials. We denote by Ri i i . i m a rational function passing through the m 1 points xi yi xi m yi m . More explicitly suppose 1 Xx po pxx ------- p Ri i i . i m --------- v 3.2.1 Qv x qo qix ---- qv xv Since there are p u 1 unknown p s and q s q0 being arbitrary we must have m 1 p u 1 3.2.2 In specifying a rational function interpolating function you must give the desired order of both the numerator and the denominator. Rational functions are sometimes superior to polynomials roughly speaking because of their ability to model functions with poles that is zeros of the denominator of equation 3.2.1 . These poles might occur for real values of x if the function to be interpolated itself has poles. More often the function f x is finite for all finite real x but has an analytic continuation with poles in the complex x-plane. Such poles can themselves ruin a polynomial approximation even one restricted to real values of x just as they can ruin the convergence of an infinite power series in x. If you draw a circle in the complex plane around your m tabulated points then you should not expect polynomial interpolation to be good unless the nearest pole is rather far outside the circle. A rational function approximation by contrast will stay good as long as it has enough powers of x in its denominator to account for cancel any nearby poles. For the interpolation problem a rational function is constructed so as to go through a chosen set of tabulated functional values. However we should also mention in passing that rational function approximations can be used in analytic work. One sometimes constructs a rational function approximation by the criterion that the rational function of equation 3.2.1 itself have a power series expansion that