In § and § we learned how to find good polynomial approximations to a given function f (x) in a given interval a ≤ x ≤ b. Here, we want to generalize the task to find good approximations that are rational functions (see §). | 204 Chapter5. Evaluation ofFunctions Rational ChebyshevApproximation In and we learned how to find good polynomial approximations to a given function f x in a given interval a x b. Here we want to generalize the task to find good approximations that are rational functions see . The reason for doing so is that for some functions and some intervals the optimal rational function approximation is able to achieve substantially higher accuracy than the optimal polynomial approximation with the same number of coefficients. This must be weighed against the fact that finding a rational function approximation is not as straightforward as finding a polynomial approximation which as we saw could be done elegantly via Chebyshev polynomials. Let the desired rational function R x have numerator of degree m and denominator of degree k. Then we have D x _ p P1X -------- pmxm R x -----------------------t f x tor a x b 1 qi x ---- qk xk The unknown quantities that we need to find are p0 . . pm and q1 . qk that is m k 1 quantities in all. Let r x denote the deviation of R x from f x and let r denote its maximum absolute value r x R x f x r max r x a x 6 The ideal minimax solution would be that choice of p s and q s that minimizes r. Obviously there is some minimax solution since r is bounded below by zero. How can we find it or a reasonable approximation to it A first hint is furnished by the following fundamental theorem If R x is nondegenerate has no common polynomial factors in numerator and denominator then there is a unique choice of p s and q s that minimizes r for this choice r x has m k 2 extrema in a x b all of magnitude r and with alternating sign. We have omitted some technical assumptions in this theorem. See Ralston 1 for a precise statement. We thus learn that the situation with rational functions is quite analogous to that for minimax polynomials In we saw that the error term of an nth order approximation with n 1 Chebyshev .
