Newton's method for the Navier-Stokes equations with finite-element initial guess of stokes equations

It is shown that finite element solutions of Stokes equations may be chosen as the initial guess for the quadratic convergence of Newton's algorithm applied to Navier-Stokes equations provided there are sufficiently small mesh size h and the moderate Reynold's number. We provide also a mixed convergence analysis in terms of iterations and finite-error estimates of the initial guess with a regularity estimate and error analysis for each Newton's step.     

A convergence theorem for the fuzzy subspace clustering (FSC) algorithm

We establish the convergence of the fuzzy subspace clustering (FSC) algorithm by applying Zangwill's convergence theorem. We show that the iteration sequence produced by the FSC algorithm terminates at a point in the solution set S or there is a subsequence converging to a point in S. In addition, we present experimental results that illustrate the convergence properties of the FSC algorithm in various scenarios.     

On the convergence of Uzawa's method for the solution of biharmonic problem

The convergence of Uzawa's method for the solution of biharmonic problem in the mixed finite element method had been proved in [2] and [3]. In this paper, we obtained an asymptotic rate of convergence. Furthermore, in order to keep the same order of error between the solutions of Uzawa's iteration and biharmonic problem, as the order of error between the solutions of the mixed finite element method and biharmonic problem, we estimated the timesn of Uzawa's iteration as follows:n≥≥?(75/vσ ²)h ^?1 lnh for 2-degree Lagrange elements, andn≥?(18/vσ ²)h ^?1 lnh?(12/vσ ²)h ^?1 ln(?lnh) for linear elements.     

Zur Konvergenz des Verfahrens der Tangierenden Hyperbeln und des Tschebyscheff-Verfahrens bei konvexen Abbildungen

For Chebyshev's method and the method of tangential hyperbolas we prove convergence if applied to equationsF(x)=0, for whichF andF′ are both orderconvex.     

Higher-order simultaneous methods for the determination of polynomial multiple zeros

Starting from Laguerre's method and using Newton's and Halley's corrections for a multiple zero, new simultaneous methods of Laguerre's type for finding multiple (real or complex) zeros of polynomials are constructed. The convergence order of the proposed methods is five and six, respectively. By applying the Gauss–Seidel approach, these methods are further accelerated. The lower bounds of the R-order of convergence of the improved (single-step) methods are derived. Faster convergence of all proposed methods is attained with negligible number of additional operations, which provides a high computational efficiency of these methods. A detailed convergence analysis and numerical results are given.     

On the global and superlinear convergence of a discretized version of Wilson's method

This paper discusses an algorithm for the minimization of a nonlinear objective function subject to nonlinear inequality constraints. The considerations are influenced by a paper of Best/Bräuninger/Ritter/Robinson (published in this journal). Their idea of combining a penalty-method with Robinson's method can be generalized by extending the principle of coupling to a whole class of locally convergent algorithms. An example is given by using a discretized version of Wilson's method, advantageously in the following sense: During the second phase, only linear equations occur in the subproblems. After a sufficiently large number of iterations, these systems are uniquely solvable. The minimization of penalty functions, necessary in the first phase, is asymptotically exact. Altogether, the implementability of the method can be guaranteed. The given convergence results are verified by using Banach's fixed-point theorem mainly. On the whole, they correspond with the paper mentioned above. The assumptions for proving global convergence are permitted to be weaken. By using different consistent approximations of the Hessian of the Lagrange function several methods arise, which have estimates of theR-order well-known from the treatment of nonlinear equations.     

An updated version of the Kantorovich theorem for Newton's method

An affine invariant version of the Kantorovich theorem for Newton's method is presented. The result includes the Gragg-Tapia error bounds, as well as recent optimal and sharper upper bounds, new optimal and sharper lower bounds, and new inequalities showingq-quadratic convergence all in terms of the usual majorizing sequence. Closed form expressions for these bounds are given.     

An iterative method for algebraic equation by Padé approximation

In this paper, we consider iterative formulae with high order of convergence to solve a polynomial equation,f(z)=0. First, we derive the numerator of the Padé approximant forf(z)/f′(z) by combining Viscovatov's and Euclidean algorithms, and then calculate the zeros of the numerator so as to apply one of the zeros for the next approximation. Regardless of whether the root is simple or multiple, the convergence order of this iterative formula is always attained for arbitrary positive integerm with the Taylor polynomial of degreem for a given polynomialf(z). Since it is easy to systematically obtain formulae of different order, we can choose formulae of suitable order according to the required accuracy.     

On the improved Newton-like methods for the inclusion of polynomial zeros

The aim of this paper is to present some modifications of Newton's type method for the simultaneous inclusion of all simple complex zeros of a polynomial. Using the concept of the R-order of convergence of mutually dependent sequences, the convergence analysis shows that the convergence rate of the basic method is increased from 3 to 6 using Jarratt's corrections. The proposed method possesses a great computational efficiency since the acceleration of convergence is attained with only few additional calculations. Numerical results are given to demonstrate convergence properties of the considered methods.     

On the convergence order of a modified method for simultaneous finding polynomial zeros

Using Newton's corrections and Gauss-Seidel approach, a modification of single-step method [1] for the simultaneous finding all zeros of ann-th degree polynomial is formulated in this paper. It is shown thatR-order of convergence of the presented method is at least 2(1+τ _n ) where τ _n ∈(1,2) is the unique positive zero of the polynomial \(\tilde f_n (\tau ) = \tau ^n - \tau - 1\) . Faster convergence of the modified method in reference to the similar methods is attained without additional calculations. Comparison is performed in the example of an algebraic equation.     

Global convergence of Oja's PCA learning algorithm with a non-zero-approaching adaptive learning rate

A non-zero-approaching adaptive learning rate is proposed to guarantee the global convergence of Oja's principal component analysis (PCA) learning algorithm. Most of the existing adaptive learning rates for Oja's PCA learning algorithm are required to approach zero as the learning step increases. However, this is not practical in many applications due to the computational round-off limitations and tracking requirements. The proposed adaptive learning rate overcomes this shortcoming. The learning rate converges to a positive constant, thus it increases the evolution rate as the learning step increases. This is different from learning rates which approach zero which slow the convergence considerably and increasingly with time. Rigorous mathematical proofs for global convergence of Oja's algorithm with the proposed learning rate are given in detail via studying the convergence of an equivalent deterministic discrete time (DDT) system. Extensive simulations are carried out to illustrate and verify the theory derived. Simulation results show that this adaptive learning rate is more suitable for Oja's PCA algorithm to be used in an online learning situation.     

一类求解非线性奇异方程组的牛顿改进算法

受一个求解非线性奇异方程组迭代格式的启示,将两种牛顿改进算法推广成一般形式,并将其发展为一类求解具有奇异雅可比矩阵的非线性方程组的牛顿改进算法.首先,描述这类新算法的迭代格式,并导出其收敛阶,该新格式每步迭代仅需计算一次函数值和一次导函数值;然后,对测试函数进行检验,并与牛顿算法及其他奇异牛顿算法进行比较,从而验证该算法的快速收敛性;最后,通过两个实际问题验证所提出算法的有效性.     

A family of root-finding methods with accelerated convergence

A parametric family of iterative methods for the simultaneous determination of simple complex zeros of a polynomial is considered. The convergence of the basic method of the fourth order is accelerated using Newton's and Halley's corrections thus generating total-step methods of orders five and six. Further improvements are obtained by applying the Gauss-Seidel approach. Accelerated convergence of all proposed methods is attained at the cost of a negligible number of additional operations. Detailed convergence analysis and two numerical examples are given.     

On the equivalence of the k-step iterative euler methods and successive overrelaxation (SOR) methods for k-cyclic matrices

For the solution of the linear system x = Tx + c (1), where T is weakly cyclic of index k ≥ 2, the block SOR method together with two classes of monoparametric k-step iterative Euler methods, whose (optimum) convergence properties were studied in earlier papers, are considered. By establishing the existence of the matrix analog of the Varga's relation, connecting the eigenvalues of the SOR and the Jacobi matrices associated with (1), it is proved that the aforementioned SOR method is equivalent to a certain monoparametric k-step iterative Euler method derived from (1). By suitably modifying the existing theory, one can then determine (optimum) relaxation factors for which the SOR method in question converges, (optimum) regions of convergence etc., so that one can obtain, what is known, several new results. Finally, a number of theoretical applications of practical importance is also presented.     

On a modified Numerov's method for inverse Sturm–Liouville problems

In this paper, we discuss the convergence of modified Numerov's method in Q. Gao, X.L. Cheng and Z.D. Huang [Modified Numerov's method for inverse Sturm–Liouville problems, J. Comput. Appl. Math. 253 (2013), pp. 181–199] for computing symmetric potentials from finite Dirichlet eigenvalues. A sufficient condition for convergence of the estimate to the true potential is given and the rate of convergence is investigated. The proof relies on the asymptotics of eigenvalues of the Sturm–Liouville operator and the errors in the finite difference eigenvalues obtained by Numerov's approach. Some numerical experiments are presented to confirm the theoretically predicted convergence properties.     

A note on parallel multisplitting TOR method for H-matrices

In 2001, Chang studied the convergence of parallel multisplitting TOR method for H-matrices [D.W. Chang, The parallel multisplitting TOR(MTOR) method for linear systems, Comput. Math. Appl. 41 (2001), pp. 215–227]. In this paper, we point out some gaps in the proof of Chang's main results solving them. Moreover, we improve some of Chang's convergence results. A numerical example is presented in order to illustrate the improvement of Chang's convergence region.     

Performance analysis of a distributed power control algorithm for shared and split spectrum femtocell networks

In this paper, we study the performance of a distributed power adjustment algorithm for shared and split spectrum allocation setups. The theoretical analysis reveals that the convergence of the power control algorithm is guaranteed under different spectrum allocation schemes and the convergence rate is exponential. The performance analysis is also carried out via simulations which demonstrate the algorithm fairness under Jain''s and Atkinkons'' fairness indices.     

Efficient iterative algorithms for bounding the inverse of a matrix

In this paper we are considering iterative methods for bounding the inverse of a matrix, which make use of interval arithmetic. We present a class of methods as a combination of ordinary Schulz's methods for only approximating the inverse matrix (see [3]) and of well-known interval Schulz's methods (see [1]). Two convergence theorems are proved. Our methods are shown to be asymptotically of the same order of convergence as the ordinary Schulz's methods being part of them. Therefore we are getting considerably more efficient interval methods by our approach than by the classical interval Schulz's methods in [1] or [5]. A numerical example is given.     

On optimal parameter of Laguerre's family of zero-finding methods

ABSTRACT

A one parameter Laguerre's family of iterative methods for solving nonlinear equations is considered. This family includes the Halley, Ostrowski and Euler methods, most frequently used one-point third-order methods for finding zeros. Investigation of convergence quality of these methods and their ranking is reduced to searching optimal parameter of Laguerre's family, which is the main goal of this paper. Although methods from Laguerre's family have been extensively studied in the literature for more decades, their proper ranking was primarily discussed according to numerical experiments. Regarding that such ranking is not trustworthy even for algebraic polynomials, more reliable comparison study is presented by combining the comparison by numerical examples and the comparison using dynamic study of methods by basins of attraction that enable their graphic visualization. This combined approach has shown that Ostrowski's method possesses the best convergence behaviour for most polynomial equations.     

Approximation for bounded functions and absolutely continuous functions by beta operators

In this paper, the pointwise approximation properties of Beta operators _n are studied to the bounded functions and the absolutely continuous functions, respectively. First, we use the asymptotic form of the central limit theorem in probability theory to derive an asymptotic estimate on the rate of convergence of Beta operators _n for the bounded functions. Next, we give the optimal estimate on the first-order absolute moment of the Beta operators B_n(|t−χ|,χ) by direct computations. Then, we use this estimate and Bojanic-Cheng-Khan's method combining with some analysis techniques to derive an asymptotically optimal estimate on the rate of convergence of Beta operators _n, for the absolutely continuous functions.