Further remarks on line segment curve-fitting using dynamic programming

Abstract

In a recent paper, Bellman showed how dynamic programming could be used to determine the solution to a problem previously considered by Stone. The problem comprises the determination, given N , of the N points of subdivision of a given interval ( α , β and the corresponding line segments, that give the best least squares fit to a function g ( x ) in the interval. Bellman confined himself primarily to the analytical derivation, suggesting briefly, however, how the solution of the equation derived for each particular point of subdivision u _i could be reduced to a discrete search. In this paper, the computational procedure is considered more fully, and the similarities to some of Stone's equations are indicated. It is further shown that an equation for u ₂ involving no minimization may be found. In addition, it is shown how Bellman's method may be applied to the curve-fitting problem when the additional constraints are added that the ends of the line segments must be on the curve.