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 [14], 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 [4], 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. [9], [8]. 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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