Algorithms2016, 9(1), 15; doi:10.3390/a9010015 - published 6 February 2016 Show/Hide Abstract
Abstract: A new orthogonal projection method for computing the minimum distance between a point and a spatial parametric curve is presented. It consists of a geometric iteration which converges faster than the existing Newton’s method, and it is insensitive to the choice of initial values. We prove that projecting a point onto a spatial parametric curve under the method is globally second-order convergence.
Algorithms2016, 9(1), 13; doi:10.3390/a9010013 - published 1 February 2016 Show/Hide Abstract
Abstract: Big data (e.g., [1–3]) has become one of the most challenging research topics in current years. Big data is everywhere, from social networks to web advertisements, from sensor and stream systems to bio-informatics, from graph management tools to smart cities, and so forth. [...]
Algorithms2016, 9(1), 14; doi:10.3390/a9010014 - published 1 February 2016 Show/Hide Abstract
Abstract: In this work, two multi-step derivative-free iterative methods are presented for solving system of nonlinear equations. The new methods have high computational efficiency and low computational cost. The order of convergence of the new methods is proved by a development of an inverse first-order divided difference operator. The computational efficiency is compared with the existing methods. Numerical experiments support the theoretical results. Experimental results show that the new methods remarkably reduce the computing time in the process of high-precision computing.
Algorithms2016, 9(1), 12; doi:10.3390/a9010012 - published 27 January 2016 Show/Hide Abstract
Abstract: Pareto optimization combines independent objectives by computing the Pareto front of the search space, yielding a set of optima where none scores better on all objectives than any other. Recently, it was shown that Pareto optimization seamlessly integrates with algebraic dynamic programming: when scoring schemes A and B can correctly evaluate the search space via dynamic programming, then so can Pareto optimization with respect to A and B. However, the integration of Pareto optimization into dynamic programming opens a wide range of algorithmic alternatives, which we study in substantial detail in this article, using real-world applications in biosequence analysis, a field where dynamic programming is ubiquitous. Our results are two-fold: (1) We introduce the operation of a “Pareto algebra product” in the dynamic programming framework of Bellman’s GAP. Users of this framework can now ask for Pareto optimization with a single keystroke. Careful evaluation of the implementation alternatives by means of an extended Bellman’s GAP compiler demonstrates the dependence of the best implementation choice on the application at hand. (2) We extract from our experiments several pieces of advice to programmers who do not use a system such as Bellman’s GAP, but who choose to hand-craft their dynamic programming recurrences, incorporating Pareto optimization from scratch.
Algorithms2016, 9(1), 10; doi:10.3390/a9010010 - published 22 January 2016 Show/Hide Abstract
Abstract: In the literature, recently, some three-step schemes involving four function evaluations for the solution of multiple roots of nonlinear equations, whose multiplicity is not known in advance, are considered, but they do not agree with Kung–Traub’s conjecture. The present article is devoted to the study of an iterative scheme for approximating multiple roots with a convergence rate of eight, when the multiplicity is hidden, which agrees with Kung–Traub’s conjecture. The theoretical study of the convergence rate is investigated and demonstrated. A few nonlinear problems are presented to justify the theoretical study.