A practical two‐term acceleration algorithm for linear systems

In this paper, a practical two‐term acceleration algorithm is proposed, the interval of the parameter which guarantees the convergence of the acceleration algorithm is analyzed in detail. Further, the acceleration ratio of the new acceleration algorithm is obtained in advance. The new acceleration algorithm is less sensitive to the parameter than the Chebyshev semi‐iterative method. Finally, some numerical examples show that the accelerated algorithm is effective. Copyright © 2011 John Wiley & Sons, Ltd.     

Gaussian quadrature and acceleration of convergence

The aim of this paper is to take up again the study done in previous papers, to the case where the integrand possesses an algebraic singularity within the interval of integration. The singularities or poles close to the interval of integration considered in this paper are only real or purely imaginary. This revised version was published online in August 2006 with corrections to the Cover Date.     

An extrapolation theorem for nonlinear approximation and its applications

We prove an extrapolation theorem for the nonlinear m-term approximation with respect to a system of functions satisfying very mild conditions. This theorem allows us to prove endpoint L^p-L^q estimates in nonlinear approximation. As a consequence, some known endpoint estimates can be deduced directly and some new estimates are also obtained. Finally, applications of these new estimates are given to spherical m-widths and m-term approximation of the weighted Besov classes.     

An application of convergence acceleration techniques to a class of two-point boundary value problems on a semi-infinite domain

For boundary value problems posed on unbounded domains it is often appropriate to impose a boundary condition at infinity. For certain classes of boundary value problem obvious numerical difficulties can be avoided by truncating the unbounded domain and solving a sequence of finite domain problems instead. We introduce a novel technique which is straightforward to implement and which exploits information contained in this sequence in order to extrapolate to the unbounded case. The technique introduces a new and interesting application of a variety of convergence acceleration algorithms.     

Some greedyt-intersecting families of finite sequences

Letn, s ₁,s ₂, ... ands _n be positive integers. Assume is an integer for eachi}. For , , and , denotes _p (a)={j|1jn,a _j p}, , and . is called anI _t ^p -intersecting family if, for any a,b ,a _i b _i =min(a _i ,b _i )p for at leastt i's. is called a greedyI _t ^P -intersecting family if is anI _t ^p -intersecting family andW _p (A)W _p (B+A ^c ) for anyAS _p ( ) and any with |B|=t–1.In this paper, we obtain a sharp upper bound of | | for greedyI _t ^p -intersecting families in for the case 2ps _i (1in) ands ₁>s ₂>...>s _n .This project is partially supported by the National Natural Science Foundation of China (No.19401008) and by Postdoctoral Science Foundation of China.     

Asymptotic behavior of the Eckhoff method for convergence acceleration of trigonometric interpolation

Convergence acceleration of the classical trigonometric interpolation by the Eckhoff method is considered, where the exact values of the jumps are approximated by solution of a system of linear equations. The accuracy of the jump approximation is explored and the corresponding asymptotic error of interpolation is derived. Numerical results validate theoretical estimates.     

Acceleration results for the vector E-algorithm

In this paper we are going to study the convergence and acceleration properties of the vector E-algorithm when applied to some families of vector sequences of the form

A method of convergence acceleration of some continued fractions

A new method of convergence acceleration is proposed for continued fractions , where and are polynomials in (, ) for sufficiently large. It uses the fact that the modified approximant approaches the continued fraction value, if is sufficiently close to the th tail . Presented method is of iterative character; in each step, by means of an approximation , it produces a new better approximation of the th tail . Formula for is very simple and contains only arithmetical operations. Hence described algorithm is fully rational.     

A Method of Deducing L-Polyhedra for n-Lattices

We suggest a method for selecting an L-simplex in an L-polyhedron of an n-lattice in Euclidean space. By taking into account the specific form of the condition that a simplex in the lattice is an L-simplex and by considering a simplex selected from an L-polyhedron, we present a new method for describing all types of L-polyhedra in lattices of given dimension n. We apply the method to deduce all types of L-polyhedra in n-dimensional lattices for n=2,3,4, which are already known from previous results.     

An infeasible-interior-point algorithm for linear complementarity problems

We modify the algorithm of Zhang to obtain anO(n²L) infeasible-interior-point algorithm for monotone linear complementarity problems that has an asymptoticQ-subquadratic convergence rate. The algorithm requires the solution of at most two linear systems with the same coefficient matrix at each iteration.This research was supported by the Office of Scientific Computing, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

An approach to the Gummel map by vector extrapolation methods

The numerical approximation of nonlinear partial differential equations requires the computation of large nonlinear systems, that are typically solved by iterative schemes. At each step of the iterative process, a large and sparse linear system has to be solved, and the amount of time elapsed per step grows with the dimensions of the problem. As a consequence, the convergence rate may become very slow, requiring massive cpu-time to compute the solution. In all such cases, it is important to improve the rate of convergence of the iterative scheme. This can be achieved, for instance, by vector extrapolation methods. In this work, we apply some vector extrapolation methods to the electronic device simulation to improve the rate of convergence of the family of Gummel decoupling algorithms. Furthermore, a different approach to the topological ε-algorithm is proposed and preliminary results are presented.     

An acceleration of gradient descent algorithm with backtracking for unconstrained optimization

In this paper we introduce an acceleration of gradient descent algorithm with backtracking. The idea is to modify the steplength t _k by means of a positive parameter θ _k , in a multiplicative manner, in such a way to improve the behaviour of the classical gradient algorithm. It is shown that the resulting algorithm remains linear convergent, but the reduction in function value is significantly improved.     

An SQP algorithm for extended linear-quadratic problems in stochastic programming

Extended Linear-Quadratic Programming (ELQP) problems were introduced by Rockafellar and Wets for various models in stochastic programming and multistage optimization. Several numerical methods with linear convergence rates have been developed for solving fully quadratic ELQP problems, where the primal and dual coefficient matrices are positive definite. We present a two-stage sequential quadratic programming (SQP) method for solving ELQP problems arising in stochastic programming. The first stage algorithm realizes global convergence and the second stage algorithm realizes superlinear local convergence under a condition calledB-regularity.B-regularity is milder than the fully quadratic condition; the primal coefficient matrix need not be positive definite. Numerical tests are given to demonstrate the efficiency of the algorithm. Solution properties of the ELQP problem underB-regularity are also discussed.Supported by the Australian Research Council.     

K-spaces property of product spaces

LetK be a class of spaces which are eigher a pseudo-opens-image of a metric space or ak-space having a compact-countable closedk-network. LetK′ be a class of spaces which are either a Fréchet space with a point-countablek-network or a point-G _δ k-space having a compact-countablek-network. In this paper, we obtain some sufficient and necessary conditions that the products of finitely or countably many spaces in the classK orK′ are ak-space. The main results are that

Spectral invariance,ellipticity, and the Fredholm property for pseudodifferential operators on weighted Sobolev spaces

The pseudodifferential operators with symbols in the Grushin classes \~S _inf0 ^sup, , 0 < 1, of slowly varying symbols are shown to form spectrally invariant unital Frécher-*-algebras (*-algebras) in L(L ²(R ⁿ )) and in L(H ^st ) for weighted Sobolev spaces H _inf ^supst defined via a weight d function . In all cases, the Fredholm property of an operator can be characterized by uniform ellipticity of the symbol. This gives a converse to theorems of Grushin and Kumano-Ta-Taniguchi. Both, the spectrum and the Fredholm spectrum of an operator turn out to be independent of the choices of s, t and .The characterization of the Fredholm property by uniform ellipticity leads to an index theorem for the Fredholm operators in these classes, extending results of Fedosov and Hörmander.     

The fixed point property via dual space properties

A Banach space has the weak fixed point property if its dual space has a weak^∗ sequentially compact unit ball and the dual space satisfies the weak^∗ uniform Kadec-Klee property; and it has the fixed point property if there exists ε>0 such that, for every infinite subset A of the unit sphere of the dual space, A∪(−A) fails to be (2−ε)-separated. In particular, E-convex Banach spaces, a class of spaces that includes the uniformly nonsquare spaces, have the fixed point property.     

An asymptotically hierarchy-consistent, iterative sequence transformation for convergence acceleration of Fourier series

We derive the I transformation, an iterative sequence transformation that is useful for the convergence acceleration of certain Fourier series. The derivation is based on the concept of hierarchical consistency in the asymptotic regime. We show that this sequence transformation is a special case of the J transformation. Thus, many properties of the I transformation can be deduced from the known properties of the J transformation (like the kernel, determinantal representations, and theorems on convergence behavior and stability). Besides explicit formulas for the kernel, some basic convergence theorems for the I transformation are given here. Further, numerical results are presented that show that suitable variants of the I transformation are powerful nonlinear convergence accelerators for Fourier series with coefficients of monotonic behavior. This revised version was published online in August 2006 with corrections to the Cover Date.     

On a conjecture concerningk-Hamilton-nice sequences

In this paper, we prove that a non-negative rational number sequence (a ₁,a ₂, ...,a _k+1) isk-Hamilton-nice, if (1)a _k+12, and (2) _j ^=1/h (i _j –1)k–1 implies for arbitraryi ₁,i ₂,...i _h {1,2,... ,k}. This result was conjectured by Guantao Chen and R.H. Schelp, and it generalizes several well-known sufficient conditions for graphs to be Hamiltonian.This project is supported by the National Natural Science Foundation of China.     

Normally hyperbolic operators, the Huygens property and conformal geometry

In this paper we give a review on normally hyperbolic operators of Huygens type. The methods to determine Huygens operators we explain here were essentially influenced and developed by Paul Günther.