Jacobian-free diagonal Newton's method for solving nonlinear systems with singular Jacobian

The basic requirement of Newton’s method in solving systems of nonlinear equations is, the Jacobian must be non-singular. This condition restricts to some extent the application of Newton method. In this paper we present a modification of Newton’s method for systems of nonlinear equations where the Jacobian is singular. This is made possible by approximating the Jacobian inverse into a diagonal matrix by means of variational techniques. The anticipation of our approach is to bypass the point in which the Jacobian is singular. The local convergence of the proposed method has been proven under suitable assumptions. Numerical experiments are carried out which show that, the proposed method is very encouraging

A New Hybrid Approach for Solving Large-scale Monotone Nonlinear Equations

In this paper, a new hybrid conjugate gradient method for solving monotone nonlinear equations is introduced. The scheme is a combination of the Fletcher-Reeves (FR) and Polak-Ribiére-Polyak (PRP) conjugate gradient methods with the Solodov and Svaiter projection strategy. Using suitable assumptions, the global convergence of the scheme with monotone line search is provided. Lastly, a numerical experiment was used to enumerate the suitability of the proposed scheme for large-scale problems

Two-step diagonal Newton method for large-scale systems of nonlinear equations

We propose some improvements on a diagonal Newton's method for solving large-scale systems of nonlinear equations. In this approach, we use data from two preceding steps to improve the current approximate Jacobian in diagonal form. Via this approach, we can achieve a higher order of accuracy for Jacobian approximation when compares to other existing diagonal-type Newton's method. The results of our numerical tests, demonstrate a clear enhancement in numerical performance of our proposed method

Two-step derivative-free diagonally Newton's method for large-scale nonlinear equations

In this study, we extend the technique of Waziri et al. (2010a) via incorporating the two-step scheme in the framework of the diagonal Jacobian updating method to solve large-scale systems of nonlinear equations. In this approach we used points from two previous steps unlike one step approach in most Newton’s-like methods. The anticipation has been to improve the current Jacobian approximation into a diagonal matrix. Under mild assumptions local convergence of the proposed method is proved. The results of numerical tests are provided to demonstrate the distinctive qualities of this new approach in contrast with other available variants of Newton’s method. The method proposed in this paper has out performs some Newton-like methods in terms of computation cost and storage requirements

Barzilai-Borwein gradient method for solving fuzzy nonlinear

In this paper, we employ a two-step gradient method for solving fuzzy nonlinear equations. This method is Jacobian free and only requires a line search for . The fuzzy coefficients are presented in parametric form. Numerical experiments on well-known benchmark problems have been presented to illustrate the efficiency of the proposed method

A dai-liao hybrid conjugate gradient method for unconstrained optimization

One of todays’ best-performing CG methods is Dai-Liao (DL) method which depends on non-negative parameter  and conjugacy conditions for its computation. Although numerous optimal selections for the parameter were suggested, the best choice of  remains a subject of consideration. The pure conjugacy condition adopts an exact line search for numerical experiments and convergence analysis. Though, a practical mathematical experiment implies using an inexact line search to find the step size. To avoid such drawbacks, Dai and Liao substituted the earlier conjugacy condition with an extended conjugacy condition. Therefore, this paper suggests a new hybrid CG that combines the strength of Liu and Storey and Conjugate Descent CG methods by retaining a choice of Dai-Liao parameterthat is optimal. The theoretical analysis indicated that the search direction of the new CG scheme is descent and satisfies sufficient descent condition when the iterates jam under strong Wolfe line search. The algorithm is shown to converge globally using standard assumptions. The numerical experimentation of the scheme demonstrated that the proposed method is robust and promising than some known methods applying the performance profile Dolan and Mor´e on 250 unrestricted problems.  Numerical assessment of the tested CG algorithms with sparse signal reconstruction and image restoration in compressive sensing problems, file restoration, image video coding and other applications. The result shows that these CG schemes are comparable and can be applied in different fields such as temperature, fire, seismic sensors, and humidity detectors in forests, using wireless sensor network techniques

Characterization of GMG-ITC isolated from aerial parts of moringa oleifera tree

Isothicyanate is a major bioactive compound in Moringa oleifera Lam. There are numerous literatures that report the therapeutic effects of isothiocyanates. It is for this reason that the local consumption of the plant is increasing. In the current study, we devised a rapid protocol for the extraction of ITC from the aerial parts of M. oleifera and also determined the plant part with the highest yield. The purity of the ITCs was confirmed by High Performance Liquid Chromatography analysis. Our findings revealed that the seeds contain the highest proportion of isothiocyanate. This implies that the seeds of M. oleifera could serve as invaluable feedstock for large scale extraction of the ITCs for pharmaceutical and food industries

Newton method for nonlinear system with singular Jacobian using diagonal updating

It is well known that when the Jacobian of nonlinear systems is nonsingular in the neighborhood of the solution, the convergence of Newton method to a solution x* of F(x) = 0 is guaranteed and the rate is quadratic. Violating this condition, i.e. the Jacobian to be singular the convergence may be unsatisfactory and may even lost. In this paper we present a modification of Newton method for systems of nonlinear equations with singular Jacobian which is very much faster and significantly cheaper than both Newton and Fixed Newton methods. This is made possible by approximating the Jacobian into a nonsingular diagonal matrix. Numerical experiments are carried out which shows that, the proposed method is very encouraging

A new Newtons method with diagonal Jacobian approximation for systems of nonlinear equations

Problem statement: The major weaknesses of Newton method for nonlinear equations entail computation of Jacobian matrix and solving systems of n linear equations in each of the iterations. Approach: In some extent function derivatives are quit costly and Jacobian is computationally expensive which requires evaluation (storage) of n×n matrix in every iteration. Results: This storage requirement became unrealistic when n becomes large. We proposed a new method that approximates Jacobian into diagonal matrix which aims at reducing the storage requirement, computational cost and CPU time, as well as avoiding solving n linear equations in each iterations. Conclusion/Recommendations: The proposed method is significantly cheaper than Newton’s method and very much faster than fixed Newton’s method also suitable for small, medium and large scale nonlinear systems with dense or spa rse Jacobian. Numerical experiments were carried out which shows that, the proposed method is very encouraging