Abstract
Nonlinear problems mostly emanate from the work of engineers, physicists, mathematicians and many other scientists. A variety of iterative methods have been developed for solving large scale nonlinear systems of equations. A prominent method for solving such equations is the classical Newton’s method, but it has many shortcomings that include computing Jacobian inverse that sometimes fails. To overcome such drawbacks, an approximation with derivative free line is used on an existing method. The method uses PSB (Powell-Symmetric Broyden) update. The efficiency of the proposed method has been improved in terms of number of iteration and CPU time, hence the aim of this research. The preliminary numerical results show that the proposed method is practically efficient when applied on some benchmark problems.
Highlights
Consider the system of nonlinear equationsThe above system can be denoted by (1).where the function is a nonlinear mapping assumed to satisfy the following conditions, (i) There exists an such that (ii) is a continuously differentiable mapping in a neighborhood of of the system and (iii) The Jacobian matrix of at given by is symmetric
Newton methods is the need to compute and store an matrix at each iteration; this is computationally costly for large scale problems
(9) In general, the PSB method is an iterative method that generates a sequence of via the following from a given initial guess where is a step length determined by any suitable line search
Summary
Consider the system of nonlinear equationsThe above system can be denoted by (1).where the function is a nonlinear mapping assumed to satisfy the following conditions, (i) There exists an such that (ii) is a continuously differentiable mapping in a neighborhood of of the system and (iii) The Jacobian matrix of at given by is symmetric. The Newton method has some shortcomings which includes computation of the Jacobian matrix which may be challenging to compute and solving the Newton system in every iteration. Newton methods is the need to compute and store an matrix at each iteration; this is computationally costly for large scale problems. They are not suitable for solving large scale nonlinear systems of equations. To overcome such deficiencies, a published article [8] have been reviewed and improve it by establishing its global convergence using suitable conditions.
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