Fast Evaluation and Interpolation at the Chebyshev Sets of Points

Stable polynomial evaluation and interpolation at n Chebyshev or adjusted (expanded) Chebyshev points is performed using O(nlog’ n) arithmetic operations, to be compared with customary algorithms either using on the order of n* operations or being unstable. We also evaluate a polynomial of degree d at the sets of n Chebyshev or adjusted (expanded) Chebyshev points using O(dlog d log n) if n 5 d or O((d log d + n) log d) arithmetic operations ifn>d. 1. INTR~OUCTI~N Consider the set of Fourier points on the unit circle in the complex plane, {W 2a+1 , k=O,l,..., n-l}, (I) w = exp ( > e being a primitive (4n)-th root of 1, and also the Chebyshev set of the real coordinates of these points, {Xk =,,,(yT), k=O,l,... ,n--1) ([41, I519 [71). Fast F ourier transform (FFT) is a very effective means of interpolation to a function at the Fourier set (1) by a polynomial and of the evaluation of a polynomial at such a set of points. We will present fast stable algorithms for the evaluation and interpolation at the Chebyshev set (2), which are almost as efficient as fast Fourier transform (FFT) at the set (1), provided that n + 1 = 2h is an integer power of 2; those computations at the Chebyshev set (2) are highly important for the approximation to functions by polynomials and for stable polynomial evaluation (see Section 4 for our comments on the computational cost and for a discussion). The results apply also to the sets of adjusted (expanded) Chebyshev points {Yk = (‘xk + b, k=O,l,... ,n1) for two real constants a and b. Hereafter all logarithms are to the base 2. 2. EVALUATION Next we will extend the known scheme for FFT to polynomial evaluation at the set (2). Given a polynomial P(x) = Cid,c Pixi, we write P(x) = Pc(x2) + xPr(x2), Pc(x*) = Cj P2j X2j , pI(X2>= Cj pZj+lX 2j, Qh(y) = Ph (9) for j ranging from 0 to [$I, h=O,l, y= 1 2x2, so that P(x) = Qdy) + TQI(Y), Qh(y) = h(z2), Y = 1 2x2, h = O,l. (3) Those equations reduce the evaluation of P(x) at the n point x set (2) to the evaluation of two smaller degree (at most half-degree) polynomials Qc(y), Qi(y) at the Chebyshev 4 points y-set This research has been supported by NSF Grant CCR-8805782