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An efficient fourth order weighted-Newton method for systems of nonlinear equations. (English) Zbl 1283.65051
Given a system of $$n$$ nonlinear equations in $$n$$ unknowns, the quadratically convergent Newton’s method is a basic procedure for finding its zero set. This method requires the inverse of the first Fréchet derivative, and hence $$n+n^2$$ evaluations per iteration.
Many efforts have been made to improve the convergence without increasing the number of evaluations by too much, thereby speeding up the overall calculation. In this paper, using Taylor’s expansion on vector functions, the authors show that
Theorem 1: Let $$\mathbf{F}:D \subseteq \mathbb R^n \rightarrow \mathbb R^n$$ be four times Fréchet differentiable in a convex set $$D$$ containing the root $$\mathbf{r}$$ of $$\mathbf{F}(\mathbf{x}) = 0$$. Then the sequence $$[\mathbf{x}^{(k)}]_{k \geq 0}$$ ($$\mathbf{x}^{(0)} \in D$$) defined by $\mathbf{y}^{(k)} = \mathbf{x}^{(k)}-\theta\mathbf{F}'(\mathbf{x}^{(k})^{-1} \mathbf{F}(\mathbf{x}^{(k)})$ and $\mathbf{x}^{(k+1)} = \mathbf{x}^{(k)}-[a_1\mathbf{I}+a_2\mathbf{F}' (\mathbf{y}^{(k)})^{-1}\mathbf{F}'(\mathbf{x}^{(k)})+a_3\mathbf{F}' (\mathbf{x}^{(k)})^{-1}\mathbf{F}'(\mathbf{y}^{(k)})]\mathbf{F}' (\mathbf{x}^{(k)})^{-1}\mathbf{F}(\mathbf{x}^{(k)})$ converges to $$\mathbf{r}$$ with convergence order four, provided $$a_1 = -1/2, a_2 = 9/8, a_3 = 3/8$$, and $$\theta = 2/3$$. They note that their method requires only $$n+2n^2$$ evaluations per iteration, and they call their method the “weighted- Newton method”.
To compare their method with other, higher order convergence methods, the authors use the efficiency index defined in [W. Gautschi, Numerical Analysis. An Introduction. Boston: Birkhäuser (1997; Zbl 0877.65001)]. Additionally they carry out computational experiments in Mathematica to show that theory and practice generally agree. Besides Newton’s method, the authors compare their method to the third order method by Homeier, the fourth order method by Cordero et. al., and the fourth order method by Darvishi et. al. They find that their method requires almost the same number of iterations as the other, fourth order mathods, but fewer total function evaluations than all of the other methods. Explicit computations of the efficiency index for each method, as well as the systems of equations for the computational tests are given.

##### MSC:
 65H10 Numerical computation of solutions to systems of equations
Mathematica
Full Text:
##### References:
 [1] Amat, S., Busquier, S., Gutiérrez, J.M.: Geometrical constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157, 197–205 (2003) · Zbl 1024.65040 · doi:10.1016/S0377-0427(03)00420-5 [2] Cordero, A., Torregrosa, J.R.: Variants of Newton’s method for functions of several variables. Appl. Math. Comput. 183, 199–208 (2006) · Zbl 1123.65042 · doi:10.1016/j.amc.2006.05.062 [3] Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007) · Zbl 1122.65350 · doi:10.1016/j.amc.2007.01.062 [4] Cordero, A., Martínez, E., Torregrosa, J.R.: Iterative methods of order four and five for systems of nonlinear equations. Appl. Math. Comput. 231, 541–551 (2009) · Zbl 1173.65034 · doi:10.1016/j.cam.2009.04.015 [5] Darvishi, M.T., Barati, A.: A fourth-order method from quadrature formulae to solve systems of nonlinear equations. Appl. Math. Comput. 188, 257–261 (2007) · Zbl 1118.65045 · doi:10.1016/j.amc.2006.09.115 [6] Frontini, M., Sormani, E.: Some variant of Newton’s method with third-order convergence. Appl. Math. Comput. 140, 419–426 (2003) · Zbl 1037.65051 · doi:10.1016/S0096-3003(02)00238-2 [7] Frontini, M., Sormani, E.: Third-order methods from quadrature formulae for solving systems of nonlinear equations. Appl. Math. Comput. 149, 771–782 (2004) · Zbl 1050.65055 · doi:10.1016/S0096-3003(03)00178-4 [8] Gautschi, W.: Numerical Analysis: An Introduction. Birkhäuser, Boston (1997) · Zbl 0877.65001 [9] Grau-Sánchez, M., Grau, À., Noguera, M.: On the computational efficiency index and some iterative methods for solving systems of nonlinear equations. J. Comput. Appl. Math. 236, 1259–1266 (2011) · Zbl 1231.65090 · doi:10.1016/j.cam.2011.08.008 [10] Gutiérrez, J.M., Hernández, M.A.: A family of Chebyshev–Halley type methods in Banach spaces. Bull. Aust. Math. Soc. 55, 113–130 (1997) · Zbl 0893.47043 · doi:10.1017/S0004972700030586 [11] Homeier, H.H.H.: A modified Newton method for rootfinding with cubic convergence. J. Comput. Appl. Math. 157, 227–230 (2003) · Zbl 1070.65541 · doi:10.1016/S0377-0427(03)00391-1 [12] Homeier, H.H.H.: A modified Newton method with cubic convergence: the multivariable case. J. Comput. Appl. Math. 169, 161–169 (2004) · Zbl 1059.65044 · doi:10.1016/j.cam.2003.12.041 [13] Kelley, C.T.: Solving Nonlinear Equations with Newton’s Method. SIAM, Philadelphia (2003) · Zbl 1031.65069 [14] Noor, M.A., Wassem, M.: Some iterative methods for solving a system of nonlinear equations. Appl. Math. Comput. 57, 101–106 (2009) · Zbl 1165.65349 · doi:10.1016/j.camwa.2008.10.067 [15] Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970) · Zbl 0241.65046 [16] Ostrowski, A.M.: Solution of Equations and Systems of Equations. Academic Press, New York (1966) · Zbl 0222.65070 [17] Özban, A.Y.: Some new variants of Newton’s method. Appl. Math. Lett. 17, 677–682 (2004) · Zbl 1065.65067 · doi:10.1016/S0893-9659(04)90104-8 [18] Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs (1964) · Zbl 0121.11204 [19] Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13, 87–93 (2000) · Zbl 0973.65037 · doi:10.1016/S0893-9659(00)00100-2 [20] Wolfram, S.: The Mathematica Book, 5th edn. Wolfram Media (2003)
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