On the complexity of extending the convergence region for Traub’s method

The convergence region of Traub’s method for solving equations is small in general. This fact limits its applicability. We locate a more precise region containing the Traub iterations leading to at least as tight Lipschitz constants as before. Our convergence analysis is finer, and obtained without additional conditions. The new theoretical results are tested on numerical examples that illustrate their superiority over earlier results.     

On the semilocal convergence behavior for Halley’s method

The present paper is concerned with the semilocal convergence problems of Halley’s method for solving nonlinear operator equation in Banach space. Under some so-called majorant conditions, a new semilocal convergence analysis for Halley’s method is presented. This analysis enables us to drop out the assumption of existence of a second root for the majorizing function, but still guarantee Q-cubic convergence rate. Moreover, a new error estimate based on a directional derivative of the twice derivative of the majorizing function is also obtained. This analysis also allows us to obtain two important special cases about the convergence results based on the premises of Kantorovich and Smale types.     

Semilocal convergence for the Super-Halley’s method

The semilocal convergence of Super-Halley’s method for solving nonlinear equations in Banach spaces is established under the assumption that the second Frëchet derivative satisfies the ω-continuity condition. This condition is milder than the well-known Lipschitz and Hölder continuity conditions. The importance of our work lies in the fact that numerical examples can be given to show that our approach is successful even in cases where the Lipschitz and the Hölder continuity conditions fail. The difficult computation of second Frëchet derivative is also avoided by replacing it with the divided difference containing only the first Frëchet derivatives. A number of recurrence relations based on two parameters are derived. A convergence theorem is established to estimate a priori error bounds along with the domains of existence and uniqueness of the solutions. The R-order convergence of the method is shown to be at least three. Two numerical examples are worked out to demonstrate the efficacy of our method. It is observed that in both examples the existence and uniqueness regions of solution are improved when compared with those obtained in [7].     

On convergence of the method of fundamental solutions for solving the Dirichlet problem of Poisson’s equation

In this paper the convergence of using the method of fundamental solutions for solving the boundary value problem of Laplaces equation in R² is established, where the boundaries of the domain and fictitious domain are assumed to be concentric circles. Fourier series is then used to find the particular solutions of Poissons equation, which the derivatives of particular solutions are estimated under the L² norm. The convergent order of solving the Dirichlet problem of Poissons equation by the method of particular solution and method of fundamental solution is derived. Dedicated to Charles A. Micchelli with esteem on the occasion of his 60th birthdayAMS subject classification 35J05, 31A99     

On the local convergence of Newton’s method under generalized conditions of Kantorovich

Following an idea similar to that given by Dennis and Schnabel (1996) in [2], we prove a local convergence result for Newton’s method under generalized conditions of Kantorovich type.     

Strong convergence of a regularization method for Rockafellar’s proximal point algorithm

In this paper, for a monotone operator T, we shall show strong convergence of the regularization method for Rockafellar’s proximal point algorithm under more relaxed conditions on the sequences {r _k } and {t _k }, $$\lim\limits_{k\to\infty}t_k = 0;\quad \sum\limits_{k=0}^{+\infty}t_k = \infty;\quad\ \liminf\limits_{k\to\infty}r_k > 0.$$ Our results unify and improve some existing results.     

On the convergence rate of a parallel nonoverlapping domain decomposition method

In recent years,a nonoverlapping domain decomposition iterative procedure,which is based on using Robin-type boundary conditions as information transmission conditions on the subdomain interfaces,has been developed and analyzed.It is known that the convergence rate of this method is 1-O(h),where h is mesh size.In this paper,the convergence rate is improved to be 1-O(h1/2 H-1/2)sometime by choosing suitable parameter,where H is the subdomain size.Counter examples are constructed to show that our convergence estimates are sharp,which means that the convergence rate cannot be better than 1-O(h1/2H-1/2)in a certain case no matter how parameter is chosen.     

A modified Brent’s method for finding zeros of functions

Brent’s method, also known as zeroin, has been the most popular method for finding zeros of functions since it was developed in 1972. This method usually converges very quickly to a zero; for the occasional difficult functions encountered in practice, it typically takes $O(n)$ iterations to converge, where $n$ is the number of steps required for the bisection method to find the zero to approximately the same accuracy. While it has long been known that in theory Brent’s method could require as many as $O(n^2)$ iterations to find a zero, such behavior had never been observed in practice. In this paper, we first show that Brent’s method can indeed take $O(n^2)$ iterations to converge, by explicitly constructing such worst case functions. In particular, for double precision accuracy, Brent’s method takes $2{,}914$ iterations to find the zero of our function, compared to the $77$ iterations required by bisection. Secondly, we present a modification of Brent’s method that places a stricter complexity bound of $O(n)$ on the search for a zero. In our extensive testing, this modification appears to behave very similarly to Brent’s method for all the common functions, yet it remains at worst five times slower than the bisection method for all difficult functions, in sharp contrast to Brent’s method.     

Local convergence of Newton’s method for subanalytic variational inclusions

This paper concerns variational inclusions of the form where f is a single locally Lipschitz subanalytic function and F is a set-valued map acting in Banach spaces. We prove the existence and the convergence of a sequence (x _k ) satisfying where lies to which is the Clarke Jacobian of f at the point x _k .      

On the complexity of the closed fragment of Japaridze’s provability logic

We consider the well-known provability logic GLP. We prove that the GLP-provability problem for polymodal formulas without variables is PSPACE-complete. For a number n, let \({L^{n}_0}\) denote the class of all polymodal variable-free formulas without modalities \({\langle n \rangle,\langle n+1\rangle,...}\) . We show that, for every number n, the GLP-provability problem for formulas from \({L^{n}_0}\) is in PTIME.     

On the rate of convergence for infinite server Erlang–Sevastyanov’s problem

Polynomial convergence rates in total variation are established in Erlang–Sevastyanov type problems with an infinite number of servers and a general distribution of service under assumptions on the intensity of service.     

On the convergence of the Lavrent’ev method for an integral equation of the first kind with involution

The convergence in the mean-square metric of the Lavrent’ev regularizationmethod for an integral equation with involution is established. The proof of the convergence is based on studying the behavior of the resolvent of a certain integro-differential equation related to the original equation.     

Concerning the radius of convergence of Newton’s method and applications

We present local and semilocal convergence results for Newton’s method in a Banach space setting. In particular, using Lipschitz-type assumptions on the second Fréchet-derivative we find results concerning the radius of convergence of Newton’s method. Such results are useful in the context of predictor-corrector continuation procedures. Finally, we provide numerical examples to show that our results can apply where earlier ones using Lipschitz assumption on the first Fréchet-derivative fail.     

On the convergence of continued fractions at Runckel’s points

We consider the limit periodic continued fractions of Stieltjes type

On the quadratic convergence of Newton’s method under center-Lipschitz but not necessarily Lipschitz hypotheses

We provide new local and semilocal convergence results for Newton’s method in a Banach space. The sufficient convergence conditions do not include the Lipschitz constant usually associated with Newton’s method. Numerical examples demonstrating the expansion of Newton’s method are also provided in this study.