Abstract
A new parametric class of iterative schemes for solving nonlinear systems is designed. The third- or fourth-order convergence, depending on the values of the parameter being proven. The analysis of the dynamical behavior of this class in the context of scalar nonlinear equations is presented. This study gives us important information about the stability and reliability of the members of the family. The numerical results obtained by applying different elements of the family for solving the Hammerstein integral equation and the Fisher’s equation confirm the theoretical results.
Highlights
It is not possible to find the exact solution to this type of equations, so iterative methods are required in order to approximate the desired solution
We can apply the Möbius transformation on the operator associated with the parametric family (2) in order to obtain an operator that does not depend on the constants a and b and, be able to study the dynamical behavior of this family for any quadratic polynomial
The dynamical analysis of the class on quadratic polynomials is done in order to select the members of the family with better stability properties
Summary
It is not possible to find the exact solution to this type of equations, so iterative methods are required in order to approximate the desired solution The essence of these methods is to find, through an iterative process and, from an initial approximation x(0) close to a solution x, a sequence {x(k)} of approximations such that, under different requirements, limk→∞ x(k) = x. Where F (x(k)) denotes the derivative or the Jacobian matrix of function F evaluated in the kth iteration x(k) This method has great importance in the study of iterative methods because it presents quadratic convergence under certain conditions and has great accessibility, that is, the region of initial estimates x(0) for which the method converges is wide, at least for polynomials or polynomial systems. The parametric family of iterative methods for solving nonlinear systems that we propose has the following iterative expression: y(k) = x(k) − F (x(k))−1F(x(k)), H(x(k), y(k), γ). We finish the work with some conclusions and the references used in it
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