Abstract
We present a third-order method for solving the systems of nonlinear equations. This method is a Newton-type scheme with the vector extrapolation. We establish the local and semilocal convergence of this method. Numerical results show that the composite method is more robust and efficient than a number of Newton-type methods with the other vector extrapolations.
Highlights
Finding the solution of nonlinear equations is important in scientific and engineering computing areas
We will establish the semilocal convergence of method (4). This convergence may be derived by using recurrence relations, which have been used in establishing the convergence of Newton’s method and some third-order methods [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]
An attempt is made to use recurrence relations to establish the semilocal convergence for the method (4)
Summary
Finding the solution of nonlinear equations is important in scientific and engineering computing areas. Though the PPM can reduce the computational cost of Jacobian matrix, in some cases, the sequences produced by PPM converge slowly and even cannot converge because of the accumulation of the computational error Many vector extrapolation methods have been developed, such as the minimal polynomial extrapolation (MPE) method [3], the reduced rank extrapolation (RRE) method [4, 5], the modified minimal polynomial extrapolation (MMPE) method [6,7,8], the topological ε-algorithm (TEA) [6], and vector ε-algorithms (VEA) [9, 10]; see [11, 12] and the references therein These methods could be applied to the solvers of linear and nonlinear systems and accelerate their convergence. The local and semilocal convergence are established for the method
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