Journal of Engineering and Applied Sciences 14 (Special Issue 7): 10080-10086, 2019 ISSN: 1816-949X © Medwell Journals, 2019 A New Line Search Method to Solve the Nonlinear Systems of Monotone Equations Karrar Habeeb Hashim, Nabiha Kahtan Dreeb, Hasan Hadi Dwail, Mohammed Maad Mahdi, H.A. Wasi, Mushtak A.K. Shiker and Hussein Ali Hussein Department of Mathematics, College for Pure Science, University of Babylon, Hillah, Iraq Abstract: In this study, we suggest a new line search algorithm for solving nonlinear systems of equations such that we combine a monotone technique into a modified line search rule. The new proposed algorithm can decrease the CPU time, the number of iterations and the function evaluations and can increase the efficiency of the approach. Under some standard conditions, the global convergence of the algorithm is proved. Preliminary numerical results shows that the new algorithm is promised for solving nonlinear systems of equations monotone equations. Key words: Nonlinear system of equations, line search method, Monotone strategy, global convergence, numerical results, iterative method INTRODUCTION The nonlinear systems are one of the problems that arise in different fields of science and computational geometry, especially in the interpretation of nonlinear partial differential equations, the problem of specified value, etc. There are situations in which thousands of nonlinear equations can be solved in some independent variables effectively. Thus, finding the roots of nonlinear systems of equations has many applications in numerical and applied mathematics. Therefore, the focus of many researchers is to find and provide appropriate ways and means to solve these non-linear systems and thus some common algorithms are suggested to solve these problem. Nonlinear equations are one of the most important problems of multiple scientific uses such as computer science tremolo systems (Ortega and Rheinboldt, 1970; Zeidler, 2013), the first-order necessary condition for the problem of unconstrained convex optimization and also some sub-problems in generalization (Iusem and Solodov, 1997; Shiker and Sahib, 2018). Since, the fixed points that can be found from the problem of improvement are equal to find the answer of a non-linear system of equations and the systems of nonlinear equations can be converted into problem of the lower squares this indicates a close relationship between the problems of unconstrained optimization and systems of nonlinear equations, so, it is appropriate to use unconstrained optimization algorithms to solve this problem. One of the two important iterative methods that is used to solve nonlinear system of equations is the line search strategy, the other method is trust region. Here, we focus on the line search method and its framework. This method is fairly simple, so, its understanding and application is easy. However, they are ineffective and have some disadvantage, for example, if the array being searched for contains 30.000 items, to find the value of the last element, the algorithm will have to look at all those 30.000 elements. Typically, if we have a matrix of M elements, the linear search will identify an element in M/2 attempts. For example, if we have a matrix of 40.000 items, the linear search will compare with 20.000 items in a typical case. This is through the possibility to find the search element constantly in the array, so, the number M is always maximum in comparisons. An another disadvantage, on the large scale, the research and convergence of the line search method are slow. So, most of researchers used the monotone strategy to address that problem. Consider the nonlinear system of equations: (1) F(x) = 0 where, F: R n 6R n is continuous and monotone, i.e: n F(x)-F(y), x-y 0, x,y R    By fixed point map or a natural map, some monotone variational inequality can be converted into nonlinear monotonous equations but before that there are some coercive conditions that the basic function has to achieve. Corresponding Author: Karrar Habeeb Hashim, Department of Mathematics, College for Pure Science, University of Babylon, Hillah, Iraq 10080