An unconstrained dual program for computing convex \(C^ 1\)-spline approximants
Given the approximate or exact values \(f_ 0,...,f_ n\) of a function f at data points \(x_ 0<x_ 1<...<x_ n\), and positive parameters \(\sigma\), \(p_ 0,...,p_ n\), \(w_ 1,...,w_ n\), the author considers the cubic convex spline \(s=s^*\in C^ 1[x_ 0,x_ n]\) which minimizes \[ \sum^{n}_{i=1}\{\alpha w_ i\int^{x_ i}_{x_{i- 1}}[s''(x)]^ 2dx+p_{i-1}[s(x_{i-1})-f_{i-1}]^ 2+p_ i[s(x_ i)-f_ i]^ 2\} \] over all cubic convex \(C^ 1\)-splines s with knots \(x_ i\). In this paper, it is shown that the unique solution \(s^*\) of the minimization problem can be determined by applying the Newton method to an unconstrained dual problem.
scientific article; zbMATH DE number 4030805
scientific article