Published in Numerical Algorithms
In this paper, we develop a fourth order method for solving the systems of nonlinear equations. The algorithm is composed of two weighted-Newton steps and requires the information of one function and two first Fréchet derivatives. Therefore, for a system of n equations, per iteration it uses n + 2n2 evaluations. Computational efficiency is compared...
Published in Numerical Algorithms
The classical Falkner methods (Falkner, Phil Mag S 7:621, 1936) are well-known for solving second-order initial-value problems u′′(t) = f(t, u(t), u′(t)). In this paper, we propose the adapted Falkner-type methods for the systems of oscillatory second-order differential equations u′′(t) + Mu(t) = g(t, u(t)) and make a rigorous error analysis. The e...
Published in Numerical Algorithms
A scheme, stemming from the use of pseudospectral approximations to spatial derivatives followed by a time integrator based on trigonometric polynomials, is proposed for the numerical solutions of the N-coupled nonlinear Klein–Gordon equations. Numerical tests on one- and three-coupled Klein–Gordon equations are presented, which are geared towards ...
Published in Numerical Algorithms
In this paper, we present primal-dual interior-point methods for convex quadratic optimization based on a finite barrier, which has been investigated earlier for the case of linear optimization by Bai et al. (SIAM J Optim 13(3):766–782, 2003). By means of the feature of the finite kernel function, we study the complexity analysis of primal-dual int...
Published in Numerical Algorithms
We study the solving of nonlinear equations by an iterative method of Aitken type, which has the interpolation nodes controlled by the Newton method. We obtain a local convergence result which shows that the q-convergence order of this method is 6 and its efficiency index is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \u...
Published in Numerical Algorithms
Discrete ill-posed problems are difficult to solve, because their solution is very sensitive to errors in the data and to round-off errors introduced during the solution process. Tikhonov regularization replaces the given discrete ill-posed problem by a nearby penalized least-squares problem whose solution is less sensitive to perturbations. The pe...
Published in Numerical Algorithms
In this paper we propose a method for computing the roots of a monic matrix polynomial. To this end we compute the eigenvalues of the corresponding block companion matrix C. This is done by implementing the QR algorithm in such a way that it exploits the rank structure of the matrix. Because of this structure, we can represent the matrix in Givens-...
Published in Numerical Algorithms
Published in Numerical Algorithms
We focus on the solution of discrete ill-posed problems to recover the original information from blurred signals in the presence of Gaussian white noise more accurately. We derive seminorms for the Tikhonov–Phillips regularization based on the underlying blur operator H. In this way it is possible to improve the reconstruction using spectral inform...
Published in Numerical Algorithms
We prove that any fat, subanalytic compact subset of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb R^N$\end{document} possesses a nearly optimal (polynomial) a...
