Algorithms for piecewise polynomials and splines with free knots

Abstract

We describe an algorithm for computing points a = x 0 > x 1 > ⋯ > x k > x k + 1 = b a = {x_0} > {x_1} > \cdots > {x_k} > {x_{k + 1}} = b which solve certain nonlinear systems d ( x i − 1 , x i ) = d ( x i , x i + 1 ) d({x_{i - 1}},{x_i}) = d({x_i},{x_{i + 1}}) , i = 1 , … , k i = 1, \ldots ,k . 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.