Rational Function Approximation
Rational function approximation Rational function of degree N = n + m is written as p(x) p + p x + + p xn r(x) = = 0 1 ··· n q(x) q + q x + + q xm 0 1 ··· m Now we try to approximate a function f on an interval containing 0 using r(x). WLOG, we set q0 = 1, and will need to determine the N + 1 unknowns p0,..., pn, q1,..., qm. Numerical Analysis I – Xiaojing Ye, Math & Stat, Georgia State University 240 Pad´eapproximation The idea of Pad´eapproximation is to find r(x) such that f (k)(0) = r (k)(0), k = 0, 1,..., N This is an extension of Taylor series but in the rational form. i Denote the Maclaurin series expansion f (x) = i∞=0 ai x . Then i m i n i ∞ a x q x P p x f (x) r(x) = i=0 i i=0 i − i=0 i − q(x) P P P If we want f (k)(0) r (k)(0) = 0 for k = 0,..., N, we need the − numerator to have 0 as a root of multiplicity N + 1. Numerical Analysis I – Xiaojing Ye, Math & Stat, Georgia State University 241 Pad´eapproximation This turns out to be equivalent to k ai qk i = pk , k = 0, 1,..., N − Xi=0 for convenience we used convention p = = p = 0 and n+1 ··· N q = = q = 0. m+1 ··· N From these N + 1 equations, we can determine the N + 1 unknowns: p0, p1,..., pn, q1,..., qm Numerical Analysis I – Xiaojing Ye, Math & Stat, Georgia State University 242 Pad´eapproximation Example x Find the Pad´eapproximation to e− of degree 5 with n = 3 and m = 2.
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