An unconstrained dual program for computing convex \(C^ 1\)-spline approximants
type: Item
instance of: scholarly article
title: An unconstrained dual program for computing convex \(C^ 1\)-spline approximants
zbMATH Open document ID: 0633.65008
DOI: 10.1007/BF02310102
author: Jochen W. Schmidt
published in: Computing
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 65D07
Mathematics Subject Classification ID: 65D10
Mathematics Subject Classification ID: 41A15
zbMATH DE Number: 4030805
zbMATH Keywords: smoothing
zbMATH Keywords: curve fitting
zbMATH Keywords: primary and dual optimization problems
zbMATH Keywords: cubic convex splines
zbMATH Keywords: Newton method
reviewed by: Ryszard Smarzewski
MaRDI profile type: Publication
cites work: Monotone and Convex Approximation by Splines: Error Estimates and a Curve Fitting Algorithm
cites work: Convex spline interpolants with minimal curvature
cites work: A practical guide to splines
cites work: Algorithms for computing shape-preserving spline approximations to data
cites work: Constrained interpolation and smoothing
cites work: A decomposition-dualization approach for solving constrained convex minimization problems with applications to discretized obstacle problems
cites work: Uniform approximation by some Hermite interpolating splines
cites work: Smoothing by spline functions.
cites work: Convex Analysis
cites work: Convex interval interpolation with cubic splines
cites work: On tridiagonal linear complementarity problems
cites work: Quadratic and related exponential splines in shape preserving interpolation
cites work: SPLINE FUNCTIONS AND THE PROBLEM OF GRADUATION
cites work: Q5603731
full work available at URL: https://doi.org/10.1007/bf02310102
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description: scientific article; zbMATH DE number 4030805