Algorithms for Piecewise Polynomials and Splines with Free Knots

We describe an algorithm for computing points o = io < ii < • • • < xjt < Zfc+i = b which solve certain nonlinear systems d(z,_i,xt) = d(i,,x,+i), i = 1,... ,fc. In contrast to Newton-type methods, the algorithm converges when starting with arbitrary points. The method is applied to compute best piecewise polynomial approximations with free knots. The advantage is that in the starting phase only simple expressions have to be evaluated instead of computing best polynomial approximations. We finally discuss the relation to the computation of good spline approximations with free knots. 0. Introduction. Let [a, b] be an interval of the real line and k be a natural number. Moreover, let D = {(x,y) e R2: a < x < y < b} and d: D —> R be a function with the following properties: (0.1) d is continuous, (0.2) d(x, x) = 0 for all (x, x) e D, (0.3) d(x,y) < d(x,y) if [x,y] C [x,y] c [a,b]. A partition a = xo < xi < ■ ■ ■ < Xk < Xk+i = b is called a leveled set if (0.4) d(xi-i,Xi) = d(xi,xi+i), i=l,...,k, and it is called an optimal set if (0.5) max d(zt,xt+1) < max d(yi,yi+i) 0<i<k 0<i<fc for all knots a = yo < yi < ■ ■ ■ < yk < 2/fc+i = b. It is easy to verify that every leveled set is optimal. Using the idea of an algorithm for segment approximation in [10], in Section 1 we give an algorithm to compute a sequence of knot sets converging to a leveled (and therefore to an optimal) set. Simultaneously, a sequence converging to the optimal value m*; = min max d(yi,yi+i) {yi,-,yk)o<i<k is determined. In contrast to Newton-type methods, the algorithm converges for arbitrarily chosen (e.g., equidistant) knots. In Section 2 we apply the above algorithm to best uniform approximation of a given function / e C[a, b] by piecewise polynomials with free knots. This approximation problem will be solved in two phases. Received November 20, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 41A15, 41A10, 65D07; Secondary 41A50, 65D15, 65H10. ©1989 American Mathematical Society 0025-5718/89 $1.00 + $.25 per page