Examples on an Algorithm for Least Squares Data Fitting by Nonnegative Differences

A smooth function is measured at equally spaced abscissae and the measurements contain random errors. We address the problem of making the least sum of squares change to the data by requiring nonnegative differences of order r for the smoothed values. The problem is a strictly convex quadratic programming calculation, where each of the constraint functions depends on r+1 adjacent components of the smoothed values, which are the binomial coefficients with alternating signs that arise in the expansion of . We take account of this structure and describe a special active set method that is much faster than general quadratic programming algorithms. We present two examples that illustrate our approach and that although have a common development they follow different solution paths. The first of them starts from the point that satisfies all the constraints as equalities and the second starts from the unconstrained minimum of the problem. (1 1) −