We present a new third order convergence iterative method for m multiple roots of nonlinear equation. The proposed method requires one evaluation of function and two evaluations of the first derivative of function. In numerical tests exhibit that the present method provides provides high accuracy numerical result as compared to other methods. The stability of the dynamical behaviour of iterative method is investigated by displaying the basin of attraction. Basin of attraction displays less black points which give us wider choices of initial guess in computation. Keywords: Basin of attraction, Multi-point iterative methods, Multiple roots, Nonlinear equations, Order of convergence.
In a Riemannian manifold framework, an analogue of the well-known Kantorovich’s the- orem was given in  for Newton’s method for vector ﬁelds on Riemannian manifolds while the extensions of the famous Smale’s α-theory and γ -theory in  to analytic vec- tor ﬁelds and analytic mappings on Riemannian manifolds were done in . In the re- cent paper , the convergence criteria in  were improved by using the notion of the γ -condition for the vector ﬁelds and mappings on Riemannian manifolds. The radii of uniqueness balls of singular points of vector ﬁelds satisfying the γ -conditions were esti- mated in , while the local behavior of Newton’s method on Riemannian manifolds was
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A modiﬁed conjugate gradient method to solve unconstrained optimization problems is proposed which satisﬁes the suﬃcient descent condition in the case of the strong Wolfe line search, and its global convergence property is established simply. The numerical results show that the proposed method is promising for the given test problems.
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To solve problems of higher computational cost and lower convergence speed in collaborative optimization, a new relaxation cooperative optimization method of accelerating convergence is presented. The optimizations in this method are divided into two stages. In the accelerating convergence stage, the calculation method of relaxation factors is improved, relaxation factors are constructed according to the inconsistent information between each disciplinary optimal solution and their average value; in the optimization solving stage, the optimal solution in the former stage is taken as the initial point, relaxation factors with consistent precision are collaboratively optimized to obtain the final optimal solution. Finally, this optimization method is tested by a typical numerical example. Experimental results show that this method can reduce the computational cost and accelerate the convergence speed greatly. 1
However, while the Hardy Cross method with the simultaneous joint balancing distribution sequence provides remarkable convergence in practice, a proof of its convergence was not published until much more recently [Volokh, 2002]. Volokh characterizes the method with the simultaneous joint balancing distribution sequence as a Jacobi iterative scheme. He starts with the classical displacement method of a structure and then shows an incremental form of the Jacobi iterative scheme that can be used to solve these simultaneous equations. By using a specific starting point, the incremental form of Jacobi iterative scheme can represent the process of applying the Hardy Cross method with the simultaneous joint balancing distribution sequence. Because of the diagonal dominance of the stiffness matrix from the displacement method’s simultaneous equations, the Jacobi iteration—and equivalently the Hardy Cross method—converges for any loading condition. Section 2.4 illustrates these mathematical transformations in detail.
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Abstract. In this paper, based on the method first considered by Bai, Golub and Ng [Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J.Matrix Anal. Appl.}, 24(2003): 603-626], the relaxed Hermitian and skew-Hermitian (RHSS) splitting method is presented and then we prove the convergence of the method for solving positive definite, non-Hermitian linear systems. Moreover, we find that the new scheme can outperform the standard HSS method and can be used as an effective preconditioner for certain linear systems, which is shown through numerical experiments.
ever, there was almost no rigorous proof in the literature until recently , in which the author has proved the first order accuracy of the IB method for the Stokes equations with a periodic boundary condition. The proof is based on some known inequalities between the fundamental solution and the discrete Green function with a periodic boundary condition for Stokes equations. In , the author showed that the pressure obtained from IB method has O(h 1/2 ) order of convergence in the L 2 norm for a 1D model. In [19, 20], the authors designed some level set methods based on discrete
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In this paper, convergence of Adomian decomposition method (ADM) when applied to KdV–Burgers equation is proved. Two approaches for extracting the soliton solutions to the nonlinear dispersive and dissipative KdV– Burgers equation are implemented. The first one is the classical ADM while, the second is the modified ADM which is called the general iteration method. Test examples are given and a comparison between the two approaches is carried out to illustrate the pertinent feature of the general iteration method.
3) Tri-dimensional filter method can make full use of the information we get along the algorithm process. This paper is divided into 4 sections. The next section introduces the concept of a Modified tri-dimensional filter and the NCP function. In Section 3, an algorithm of line search filter is given. The global convergence properties are proved in the last section.
Note that the p-Jarratt-type method (p ∈ [, ]) given in  uses (.)-(.), but the suﬃ- cient convergence conditions are diﬀerent from the ones given in the study and guarantees only third-order convergence (not fourth obtained here) in the case of the Jarratt method (for p = /).
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In this paper, motivated by the research work going on in this direction, see, for instance, 2, 3, 7–21, we introduce an iterative method for finding a common element of the set of fixed points of a strict pseudocontraction and of the set of solutions to the problem 1.14 with multivalued maximal monotone mapping and relaxed δ, r -cocoercive mappings. Strong convergence theorems are established in the framework of Hilbert spaces.
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ABSTRACT: In this work, we show a new method of generating convergence acceleration algorithms, by performing an adequate approximation technique. The Aitken’s 2 algorithm can be derived by this way, as well as the more general E-algorithm. Moreover, a family of algorithms for logarithmic sequences have been derived and tested with success on a set of logarithmic sequences.
Motivated by T. Suzuki, we show strong convergence theorems of the CQ method for nonexpansive semigroups in Hilbert spaces by hybrid method in the mathematical programming. The results presented extend and improve the corresponding results of Kazuhide Nakajo and Wataru Takahashi (2003).
In this paper, we consider an iterative method for ﬁnding a ﬁxed point of continuous mappings on an arbitrary interval. Then, we give the necessary and suﬃcient conditions for the convergence of the proposed iterative methods for continuous mappings on an arbitrary interval. We also compare the rate of convergence between iteration methods. Finally, we provide a numerical example which supports our theoretical results.
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mathematical programming for the sum of two convex functions. In inﬁnite Hilbert space, we establish two strong convergence theorems as regards this problem. As applications of our results, we give strong convergence theorems as regards the split feasibility problem with modiﬁed CQ method, strong convergence theorem as regards the lasso problem, and strong convergence theorems for the mathematical programming with a modiﬁed proximal point algorithm and a modiﬁed
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We get the following proposition by comparing the coefficients of ( x k − α ) 2 of formula (4.5) and (4.12). Proposition 4.6. Let the equation h(x) = 0 be deformed from f(x) = 0. Let f ( ) α = h ( ) α = 0 , and α(≠0) a simple root. Then the necessary and sufficient condition for the convergence to α of q-th power of TH-method of f(x) to be equal to or faster than that r-th power of TH-method of h(x) is that the real numbers q and r satisfy the following condition.
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In this paper, we aim to ﬁnd a common solution of the minimization problem (MP) for a convex and continuously Fréchet diﬀerentiable functional and the common ﬁxed point problem of an inﬁnite family of nonexpansive mappings in the setting of Hilbert spaces. Motivated and inspired by the research going on in this area, we propose two iterative schemes for this purpose. One is called a multi-step implicit iterative method with regu- larization which is based on three well-known methods: extragradient method, approxi- mate proximal method and gradient projection algorithm with regularization. Another is an implicit hybrid method with regularization which is based on the CQ method, extra- gradient method and gradient projection algorithm with regularization. Weak and strong convergence results for these two schemes are established, respectively. Recent results in this direction can be found, e.g., in [–].
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The paper describes the theory of fractional derivative and specific application examples in the field of engineering sciences. On this basis, this paper mainly studies the low reaction-diffusion equations. First using compact operator, the paper constructs a higher-order finite difference scheme. Then the paper proves the existence and uniqueness of the difference solution by matrix method and analyzes the stability and convergence of the scheme by Fourier method.
the root to the M-estimating equations. Techniques justified by uniform convergence are used here. Uniform convergence also lends itself to the use of a graphical method of plotting "expectation curves". It can be used for either identifying the M-estimator from multiple solutions of the defining equations or in large samples (e.g. > 50) as a visual indica tion of whether the fitted model is a good approximation for the under lying mechanism. Theorems based on uniform convergence are given that show a domain of convergence (numerical analysis interpretation) for the Newton-Raphson iteration method applied to M-estimating equations for the location parameter when redescending loss functions are used.
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convergence various diculties. In another context this aspect has been the focus of other research activities, see e.g. , . The estimate in Theorem 3.5 enables us to show a result on the identication of the set of active indices. It estimates the measure of the set on which the active set at the current iterate diers from the active set of the solution by the distance of the iterate from the solution in the X-norm. This result is the key for the convergence analysis of the algorithm.
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