Showing papers on "Steffensen's method published in 2010"
TL;DR: A variant of Steffensen's method of fourth-order convergence for solving nonlinear equations is suggested, and its error equation and asymptotic convergence constant are proven theoretically and demonstrated numerically.
66 citations
TL;DR: In this paper, the convergence of Newton's method and Halley's method for finding the principal pth root of a matrix with no negative real eigenvalues and all zero eigen values of A are semisimple is analyzed.
41 citations
TL;DR: The modification of Newton's method with higher-order convergence is presented, based on King's fourth-order method, and some numerical examples illustrate that the algorithm is more efficient and performs better than classical Newton'smethod and other methods.
40 citations
TL;DR: A novel split Newton iterative algorithm for the numerical solution of nonlinear equations that improves computational efficiency by reducing the computational cost of the Jacobian matrix.
30 citations
TL;DR: An extension of the classical Newton's method for solving non-linear equations that will only require function and first derivative evaluations and can be of practical interest.
Abstract: In this work, we introduce an extension of the classical Newton's method for solving non-linear equations. This method is free from second derivative. Similar to Newton's method, the proposed method will only require function and first derivative evaluations. The order of convergence of the introduced method for a simple root is four. Numerical results show that the new method can be of practical interest.
30 citations
TL;DR: Three fast variants of Steffensen-secant method for solving nonlinear equations achieve cubic convergence or super cubic convergence for finding simple roots by only using three evaluations of the function per step.
23 citations
TL;DR: The convergence results show that this modified Newton method converges cubically in the nonsingular case, and linearly with the rate 3/8 under some sufficient conditions when the Jacobian is singular at the root.
Abstract: The paper presents a convergence analysis of a modified Newton method for solving nonlinear systems of equations. The convergence results show that this method converges cubically in the nonsingular case, and linearly with the rate 3/8 under some sufficient conditions when the Jacobian is singular at the root. The convergence theory is used to analyze the convergence behavior when the modified Newton method is applied to a nonsymmetric algebraic Riccati equation arising in transport theory. Numerical experiment confirms the theoretical results.
23 citations
01 Jan 2010
TL;DR: The Jacobian Computation-free Newton's Method (JCFN) as discussed by the authors is a modification to Newton's method for solving nonlinear equations, which is suitable for small, medium or large scale nonlinear problems with a dense or sparse Jacobian.
Abstract: We propose a modification to Newton's method for solving nonlinear equations, namely a Jacobian Computation-free Newton's Method . Unlike the classical Newton's method, the proposed modification neither requires to compute and store the Jacobian matrix, nor to solve a system of linear equations in each iteration. This is made possible by approximating the Jacobian inverse to a diagonal matrix without computing the Jacobian. The proposed method turns out to be significantly cheaper than Newton's method, much faster than fixed Newton and is suitable for small, medium or large scale nonlinear equations with a dense or sparse Jacobian. After proving the convergence of the reported algorithm, numerical experiments are reported to illustrate the promise of this method.
19 citations
TL;DR: A new iterative method of order of convergence 5 is presented, by composing the Midpoint method with Newton's method and using an approximation for the Jacobian matrix in order to reduce the required number of functional evaluations per iteration.
12 citations
TL;DR: A modified Newton's method for the best rank-one approximation problem to tensor is proposed, using the iterative matrix of Jacobi-Gauss-Newton (JGN) algorithm or Alternating Least Squares (ALS) algorithm and a modified version of GRQ- newton algorithm.
TL;DR: In this paper, a semilocal convergence analysis for Newton's method on a Banach space setting was provided, by splitting the given operator, improving the error bounds, order of convergence, and simplify the sufficient convergence conditions.
Abstract: We provided in [14] and [15] a semilocal convergence analysis for Newton’s method on a Banach space setting, by splitting the given operator. In this study, we improve the error bounds, order of convergence, and simplify the sufficient convergence conditions. Our results compare favorably with the Newton-Kantorovich theorem for solving equations. AMS subject classifications: 65H99, 65H10, 65G99, 49M15, 47J20, 47H04, 90C30, 90C33
TL;DR: In this article, a new one-step iterative method for solving nonlinear equations, which inherits the advantages of both Newton's and Steffensen's methods, was proposed by combining it with the regula falsi method.
Abstract: In this paper, we present a new one-step iterative method for solving nonlinear equations, which inherits the advantages of both Newton’s and Steffensen’s methods. Moreover, two two-step methods of second-order are proposed by combining it with the regula falsi method. These new two-step methods present attractive features such as being independent of the initial values in the iterative interval, or being adaptive for the iterative formulas. The convergence of the iterative sequences is deduced. Finally, numerical experiments verify their merits.
Journal Article•
TL;DR: By analyzing the Newton iterative method for nonlinear equation, a new acceleration technique of Newton method is proposed in this article, and numerical results indicate that the acceleration method is effective. But this method is not suitable for the acceleration of nonlinear equations.
Abstract: By analyzing the Newton iterative method for nonlinear equation,a new acceleration technique of Newton method is proposed.Numerical results indicate that the acceleration method is effective.
TL;DR: In this article, a local convergence analysis for the Newton-Steffensen-type algorithm for solving nonsmooth perturbed variational inclusions in Banach spaces is provided.
Abstract: We provide a local convergence analysis for Newton–Steffensen-type algorithm for solving nonsmooth perturbed variational inclusions in Banach spaces. Under new center–conditions and the Aubin continuity property, we obtain the linear local convergence of Newton–Steffensen method. Our results compare favorably with related obtained in (Argyros and Hilout, 2007 submitted; Hilout in J. Math. Anal. Appl. 339:753–761, 2008).
22 Nov 2010
TL;DR: In this paper, the authors presented a 7th-order convergent Newton-type method for solving nonlinear equations which is free from second derivative, at each iteration it requires three evaluations of the given function and two evaluation of its first derivative.
Abstract: In this paper, we present a seventh-order convergent Newton-type method for solving nonlinear equations which is free from second derivative. At each iteration it requires three evaluations of the given function and two evaluation of its first derivative. Therefore its efficiency index is equal to equation which is better than that of Newton's method √2. Several examples demonstrate that the algorithm is more efficient than classical Newton's method and other existing methods.
TL;DR: In this article, the authors proposed a method to accelerate the convergence from quadratic to cubic by replacing the parameter p in the iteration of Chen and Li (Appl. Numer. Math. 57:80-88, 2007) by a function p(x) defined suitably.
Abstract: A class of quadratically convergent regula falsi iterative methods for solving nonlinear equations f(x)=0 is proposed in Chen and Li (Appl. Numer. Math. 57:80–88, 2007). It is also shown there that both the sequences of diameters and iterative points sequence converge to zero simultaneously. The aim of this paper is to accelerate further the convergence of these methods from quadratic to cubic. This is done by replacing the parameter p in the iteration of Chen and Li (Appl. Numer. Math. 57:80–88, 2007) by a function p(x) defined suitably. A convergence theorem for establishing the cubic convergence of both the sequence of iterates and the sequence of diameters to the root is given. The numerical examples are worked out to demonstrate that our modified methods are more effective and comparable to those given in Chen and Li (Appl. Numer. Math. 57:80–88, 2007) as well as Newton’s method, Steffensen’s method and regula falsi method.
Journal Article•
TL;DR: The Secant method is more efficient than Newton method and the Newton method P.C.iteration format in this article, and the numerical experiments show that the twofold Secant methods are more efficient.
Abstract: Atwofold Secant method iteration formula is given and its order of convergence is proved to be 2.618.Also,some wrong conclusions about the Newton method P.C.iteration format in three references are pointed and analyzed.The efficiency analysis and the numerical experiments show that the twofold Secant method(or Secant method) is more efficient than Newton method and the Newton method P.C.iteration format.
Journal Article•
TL;DR: In this paper, two variants of the Newton's Iteration method are given and it is shown that the new variant methods have some more advantages than the other known Newton's iteration methods.
Abstract: Two Newton's variants Iteration method are given.It is at least sixth order convergence near simple root.In the end,numerical tests show that the new variant methods have some more advantages than the other known Newton's Iteration methods.
TL;DR: In this article, the Steffensen method in R n space is presented, which avoids the calculus of derivative but has the same convergence order 2 with Newton's method and the convergence theorem is established by using the technique of majorizing function.
Abstract: We present the Steffensen method in R n space. The convergence theorem is established by using the technique of majorizing function. Meanwhile, an error estimate is given. It avoids the calculus of derivative but has the same convergence order 2 with Newton's method. Finally, illustrative examples are included to demonstrate the validity and applicability of the technique.
TL;DR: New variants of Newton’s method based on quadrature formula and power mean for solving nonlinear unconstrained optimization problems are proposed and it is proved that the order of convergence of the proposed family is three.
Abstract: In this paper, we propose new variants of Newton’s method based on quadrature formula and power mean for solving nonlinear unconstrained optimization problems. It is proved that the order of convergence of the proposed family is three. Numerical comparisons are made to show the performance of the presented methods. Furthermore, numerical experiments demonstrate that the logarithmic mean Newton’s method outperform the classical Newton’s and other variants of Newton’s method. MSC: 65H05.
Journal Article•
TL;DR: A trust region-based modified Newton method for solving symmetric nonlinear equations is proposed in this article, where the global convergence of the presented method is proved under suitable conditions. But this method is not suitable for the case of the problem in this paper.
Abstract: A trust-region-based modified Newton method for solving symmetric nonlinear equations is proposed.The global convergence of the presented method is prove under suitable conditions.Some numerical results show that this method is effective for the given problem.
TL;DR: In this paper, the authors studied the application of real numbers and conjugate pairs to Newton spectra to understand circumstances in which the Newton inequalities are preserved, and showed that any number of positive real numbers can be added to a Newton spectrum, to retain the Newton property, when the Newton coefficients are positive.
17 Sep 2010
TL;DR: In this article, the authors modify Newton method to solve nonlinear equations by another iteration method, say x=g(x), and using decay method, they presented some conditions of g(x) which their method will be performance.
Abstract: In this paper we will modify Newton method to solve nonlinear equations by another iteration method, say x=g(x), and using decay method. We will present some conditions of g(x) which our method will be performance.
09 Jul 2010
TL;DR: The results show that the method developed in this paper has some more advantages than other well known methods including Newton's method.
Abstract: In this paper we present a new fourth order iterative method for solving nonlinear equations based on linear combination well known third-order variant of Newton methods. Convergence order is proved. Several numerical examples are given and compared with other known Newton type methods. The results show that the method developed in this paper has some more advantages than other well known methods including Newton's method.