Abstract—In this paper we present a substantiation of a new Laguerre’s type iterative method for solving of a nonlinear polynomial equations systems with real coefficients. The problems of its implementation, including relating to the structural choice of initial approximations, were considered. Test examples demonstrate the effectiveness of the method at the solving of many practical problems solving. Keywords—Iterative method, Laguerre’s method, Newton's method, polynomial equation, system of equations I. INTRODUCTION T is well known that the problem of roots finding of a nonlinear equations and their systems attracted the attention of researchers for several centuries, so a variety of methods of its solution were developed and published in the scientific literature. Despite this, it remains one of the most important tasks of computational mathematics, due to the necessity of solving a large number of applications, whose models are presented by systems of nonlinear equations. A special case of nonlinear systems are nonlinear systems of polynomial equations, which solving algorithms substantiated and investigated in most detail. Nevertheless, a general method for such systems solving, which could be considered universal for most practical problems, has not been developed yet. This is a motivation to search for new algorithms that are adapted, at least for typical applications. For example, mathematical models of many problems in kinematics and dynamics of multilink mechanisms with a finite number of degrees of freedom, in the numerical solution of which the most commonly used algorithms of Newton- Raphson or derivatives [1] - [7] etc., can be reduced to the systems of a nonlinear polynomial equations. But it is known that these algorithms in some cases may not be effective enough. O. Poliakov is with the Sevastopol National Technical University, Laboratory of Biomechanics, Sevastopol, 99053 Ukraine (Corresponding Author Work Phone: +38-0692-435-161; e-mail: alex_polyakov@inbox.ru). Y. Pashkov is with the Sevastopol National Technical University, Sevastopol, 99053 Ukraine. M. Kolesova is with the Sevastopol National Technical University, Sevastopol, 99053, Ukraine. O. Chepenyuk is with the Sevastopol National Technical University, Sevastopol, 99053 Ukraine. M. Kalinin is with the Sevastopol National Technical University, Laboratory of biomechanics, Sevastopol, 99053 Ukraine. V. Kramar is with the Sevastopol National Technical University, Sevastopol, 99053 Ukraine. The main aim of this work is to develop a unified solution algorithm of polynomial equations and systems, effective in solving the problems of analysis and synthesis of multilink mechanisms with an absolute rigid or elastic links. In [8] is presented a process of withdrawal of the iterative formula for the ε -estimates search of the real roots of polynomial equations of finite degree. It turns out that a constructive approach used for it, based on the continuation by the parameter [9], leads to one of the Laguerre’s type Hansen- Patrick family formulas [10], which can be written as follows: ( 29 k n k k n k n k k r p r r p n r p r r ' 1 ) ( ) 1 ( ) ( - ⋅ - + = + or ∑ ∑ = - = - + ⋅ ⋅ - - ⋅ ⋅ - = n i i k i n n i i k i n k k r a i n r a i n r r 0 0 1 ) 1 ( ) ( , (1) where i a , n i , 0 = - the real coefficients of an n - degree polynomial equation ( 29 0 = x p n ; k r - ε - estimation of the real root of the equation by k - iteration. As shown by O. Tikhonov, a formula of the form (1) is most effective compared with others in selecting a relatively large r [11]. However, the results of a set of numerical experiments and solutions of the practical problems demonstrate its effectiveness in other cases [12]. Later it was shown that generalization of (1) for the system of nonlinear polynomial equations is possible. In [13] has been proved a new iterative method for finding the ε - estimates vector of the real roots of a finite system of polynomial equations with real coefficients, and also a comparative analysis with the Newton’s method is presented. This paper presents a generalization of the method and discusses its practical implementation. II. ITERATIVE LAGUERRE’S TYPE METHOD FOR SOLVING OF A POLYNOMIAL EQUATIONS SYSTEMS A. Substantiation of the method In the derivation of the iterative formula (1) were used an LRP-polynomials of the form ( 29 ( 29 (29 (29 x g x p x a i n x a x p n n i i n n i n i i i n n 1 0 0 , - - = = - - = - - = ∑ ∑ ν ν ν , where ν - is a some real parameter [14]. The value of ν changes during the iterative process and is determined from Oleksandr Poliakov, Yevgen Pashkov, Marina Kolesova, Olena Chepenyuk, Mykhaylo Kalinin, Vadym Kramar New Laguerre’s Type Method for Solving of a Polynomial Equations Systems I World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:6, No:9, 2012 1323 International Scholarly and Scientific Research & Innovation 6(9) 2012 scholar.waset.org/1307-6892/15431 International Science Index, Mathematical and Computational Sciences Vol:6, No:9, 2012 waset.org/Publication/15431