Dynamics of some one-point third-order methods for the solution of nonlinear equations

The article of record as published may be found at https://doi.org/10.5899/2018/cna-00362In this paper we have considered 32 one-point methods of cubic order to obtain simple zeros of a nonlinear function. These schemes are constructed by decomposition of previously known schemes. We have used the idea of basins of attractions to compare the performance of these methods with Halley's method on 4 polynomial functions and one non-polynomial function. Based on 3 quantitative criteria, namely average number of iterations per point, CPU time required and the number of points for which the method diverge, we have found 4 methods that performed close to best. We also show that decomposing good methods does not necessarily lead to a better one or even to a scheme as good as the original. We found only one example that gave reasonable results and it is the only one with repelling extraneous fixed points on the imaginary axis

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Basins of attraction for several third order methods to find multiple roots of nonlinear equations

The article of record as published may be found at http://dx.doi.org/10.1016/j.amc.2015.06.068There are several third order methods for solving a nonlinear algebraic equation having roots of a given multiplicity m. Here we compare a recent family of methods of order three to Euler-Cauchy's method which is found to be the best in the previous work. There are fewer fourth order methods for multiple roots but we will not include them here.Basic Science Research Program through the National Reserach Foundation of Korea (NRF)Ministry of Education (NRF-2013R1A1A2005012

Software for the parallel solution of systems of ordinary differential equations

In this report we supply software for the numerical solution of systems of ordinary differential equations (ODEs) on an INTEL iPSC/2 hypercube. The first program can only be used to solve linear initial or boundary value systems of ODEs and based on an algorithm developed by Katti and Neta (1989) and improved by Lustman et al (1990). The second program is based on polynomial extrapolation and Gragg's scheme and is useful for nonlinear ODEs as well. This algorithm is described in Lustman, Neta and Gragg (1991)http://archive.org/details/softwareforparal00lustApproved for public release; distribution is unlimited

Software for the parallel conservative scheme for the shallow water equations

This report contains the host and node programs for the solution of the shallow water equations with topography on an INTEL iPSC/2 hypercube. Finite difference scheme conserving potential enstrophy and energy is employed in each subdomain. In this report we supply software for the numerical solution of the shallow water equations on an INTEL iPSC/2 hypercube. The method is based on domain decomposition with overlap. Finite difference scheme is used to solve in each subdomain. The scheme conserves potential enstrophy and energy. The efficiency of the algorithm is 81% when using 8 processorshttp://archive.org/details/softwareforparall00lustN

Stability analysis for Eulerian and semi-Lagrangian finite-element formulation of the advection-diffusion equation

The article of record as published may be located at http://dx.doi.org/10.1016/S0898-1221(99)00185-6This paper analyzes the stability of the finite-element approximation to the linearized two-dimensional advection-diffusion equation. Bilinear basis functions on rectangular elements are considered. This is one of the two best schemes as was shown by Neta and Williams [1]. Time is discretized with the theta algorithms that yield the explicit (theta = 0), semi-implicit (theta = 1/2), and implicit (theta = 1) methods. This paper extends the results of Neta and Williams [1] for the advection equation. Giraldo and Neta [2] have numerically compared the Eulerian and semi-Lagrangian finite-element approximation for the advection-diffusion equation. This paper analyzes the finite element schemes used there. The stability analysis shows that the semi-Lagrangian method is unconditionally stable for all values of a while the Eulerian method is only unconditionally stable for 1/2 < theta < 1. This analysis also shows that the best methods are the semi-implicit ones (theta = 1/2). In essence this paper analytically compares a semi-implicit Eulerian method with a semi-implicit semi-Lagrangian method. It is concluded that (for small or no diffusion) the semi-implicit semi-Lagrangian method exhibits better amplitude, dispersion and group velocity errors than the semi-implicit Eulerian method thereby achieving better results. In the case the diffusion coefficient is large, the semi-Lagrangian loses its competitiveness. Published by Elsevier Science Ltd

High order nonlinear solver

J. Computational Methods in Science and Engineering, 8 No. 4-6, (2008), 245–250.An eighth order method for finding simple zeros of nonlinear functions is developed. The method requires two function- and three derivative-evaluation per step. If we define informational efficiency of a method as the order per function evaluation, we find that our method has informational efficiency of 1.6

Audio detection algorithms

Audio information concerning targets generally includes direction, frequencies and energy levels. One use of audio cueing is to use direction information to help determine where more sensitive visual direction and acquisition sensors should be directed. Generally, use of audio cueing will shorten times required for visual detection, although there could be circumstances where the audio information is misleading and degrades visual performance. Audio signatures can also be useful for helping classify the emanating platform, as well as to provide estimates of its velocity. The Janus combat simulation is the premier high resolution model used by the Army and other agencies to conduct research. This model has a visual detection model which essentially incorporates algorithms as described by Hartman(1985). The model in its current form does not have any sound cueing capability. This report is part of a research effort to investigate the utility of developing such a capabilityhttp://archive.org/details/audiodetectional00netaN

Explicit Analytical Expression for a Lanchester Attrition-Rate Coefficient for Bonder and Farrell’s m-Period Target-Engagement Policy

Working Paper #5, DTRA Project, July 9, 2001The purpose of this working paper is to give an explicit analytical expression for a Lanche s- ter-type attrition-rate coefficient for direct-fire combat in a heterogeneous-target environment with serial acquisition of targets for Bonder and Farrell’s m-period target-acquisition policy1. It develops this result (its main result) from Taylor’s [2001d] new important general result (that does not depend on the target-engagement policy of a firer type or even on the particulars of the target-acquisition process) for a Lanchester attrition-rate coefficient for serial acquisition by developing explicit ana- lytical expressions for the two key intermediate quantities on which the coefficient depends: namely, (1) expected time to acquire a target that will be engaged, (2) next-target-type-to-be-engaged probability. An analytical expression for the former quantity (the expect value) was recently developed by one of the authors (Taylor [2001e]), while the paper at hand develops such an expression for the latter probability. These two new important intermediate results have allowed us to develop the explicit analytical expression for a Lanchester attrition-rate coefficient for Bonder and Farrell’s target- acquisition policy via Taylor’s general expression for direct-fire combat in a heterogeneous-target environment with serial acquisition of targets. These analytical results are then verified against simulation results