On the numerical stability of Huang's and related methods

This paper presents an analysis of Huang's and similar methods for solving systems of linear simultaneous equations, which not only derives their termination properties but which also permits bounds on propagated errors to be determined. The accuracy of Huang's method is shown to be proportional to the condition number of the matrix of coefficients of the equations. Finally, a class of methods having optimal stability characteristics is identified.The author is indebted to CNR for financial support while a Visiting Professor at the University of Bergamo, Bergamo, Italy.     

On some questions arising in numerical realization of amplitude-phase methods

For computing rapidly oscillating solutions of certain second order differential equations a new version of amplitude-phase methods has recently been proposed in [11]. Error estimates were given to approximate solutions for large arguments in [15]. One of the most important points in these methods is the introduction of Prüfer transformation modified by auxiliary functions. Their appropriate choice makes the methods applicable and efficient. When implementing and applying the methods to practical problems, we face some further questions. In this paper we describe and try to answer them. Efficiency of the methods is confirmed by numerical experiments on concrete problems. This revised version was published online in June 2006 with corrections to the Cover Date.     

On Newton-type methods with cubic convergence

Recently, there has been some progress on Newton-type methods with cubic convergence that do not require the computation of second derivatives. Weerakoon and Fernando (Appl. Math. Lett. 13 (2000) 87) derived the Newton method and a cubically convergent variant by rectangular and trapezoidal approximations to Newton's theorem, while Frontini and Sormani (J. Comput. Appl. Math. 156 (2003) 345; 140 (2003) 419 derived further cubically convergent variants by using different approximations to Newton's theorem. Homeier (J. Comput. Appl. Math. 157 (2003) 227; 169 (2004) 161) independently derived one of the latter variants and extended it to the multivariate case. Here, we show that one can modify the Werrakoon–Fernando approach by using Newton's theorem for the inverse function and derive a new class of cubically convergent Newton-type methods.     

On the stability of a class of convergence acceleration methods for power series

In this paper we study the problem of evaluating the sum of a power series whose terms are given numerically with a moderate accuracy. For a large class of divergent series a sum may be defined using analytic continuation. This sum may be estimated using the values of a finite number of terms. However, it is established here that the accuracy of this estimate will generally deteriorate if we use an ever-growing number of terms. A result on the stability of product quadrature is also obtained as a corollary of our main stability theorem.Dedicated to professor Germund Dahlquist, on the occasion of his 60th birthday     

On the convergence of multi-grid methods with transforming smoothers

Summary In the present paper we give a convergence theory for multi-grid methods with transforming smoothers as introduced in [31] applied to a general system of partial differential equations. The theory follows Hackbusch's approach for scalar pde and allows a convergence proof for some well-known multi-grid methods for Stokes- and Navier-Stokes equations as DGS by Brandt-Dinar, [5], TILU from [31] and the SIMPLE-methods by Patankar-Spalding, [23].This work was supported in part by Deutsche Forschungsgemeinschaft     

On the convergence of Krylov methods with low-rank truncations

Numerical Algorithms - Low-rank Krylov methods are one of the few options available in the literature to address the numerical solution of large-scale general linear matrix equations. These...     

On the convergence of numerical methods for solving a Volterra bilinear equations of the first kind

A justification is given for the convergence of quadratures (namely, the right rectangle and midpoint rules) for the numerical solution of a Volterra bilinear equation of the first kind. Numerical results for some benchmark problems are presented.     

On the convergence of some alternating series

We establish a necessary and sufficient condition for the convergence of the series $\sum_{n=1}^{\infty} (-1)^{n}|\sin(\pi nx)|n^{-\theta}$ in terms of the rational approximations to x. In particular, it follows from our results that the series $\sum_{n=1}^{\infty} (-1)^{n}|\sin n|/n$ converges.     

On stability concepts in nonlinear programming

Kojima's strong stability of stationary solutions can be characterized by means of first and second order terms. We treat the problem whether there is a characterization of the stability concept allowing perturbations of the objective function only, keeping the feasible set unchanged. If the feasible set is a convex polyhedron, then there exists a characterization which is in fact weaker than that one of strong stability. However, in general it appears that data of first and second order do not characterize that kind of stability. As an interpretation we have that the strong stability is the only concept of stability which both admits a characterization and works for large problem classes.Supported by the Deutsche Forschungsgemeinschaft, Graduiertenkolleg Analyse und Konstruktion in der Mathematik.Partial support under Support Center for Advanced Telecommunications Technology Research.     

On the convergence of primal-dual interior-point methods with wide neighborhoods

We study primal-dual interior-point methods for linear programs. After proposing a new primaldual potential function we describe a new potential reduction algorithm. We make connections between the new potential function and primal-dual interior-point algorithms with wide neighborhoods. Then we describe an algorithm that is a slightly modified version of existing primal-dual algorithms using wide neighborhoods. Assuming the optimal solution is non-degenerate, the algorithm is 1-step Q-quadratically convergent. We also study the degenerate case and show that the neighborhoods of the central path stay large as the iterates approach the optimal solutions.Research performed while the author was a Ph.D. student at Cornell University and was supported in part by the United States Army Research Office through the Army Center of Excellence for Symbolic Methods in Algorithmic Mathematics (ACSyAM), Mathematical Sciences Institute of Cornell University, Contract DAAL03-91-C-0027 and also by NSF, AFOSR and ONR through NSF Grant DMS-8920550.     

On the convergence of multiattribute weighting methods

The convergent validity of five multiattribute weighting methods is studied in an Internet experiment. This is the first experiment where the subjects created the alternatives and attributes themselves. Each subject used five methods to assess attribute weights – one version of the analytic hierarchy process (AHP), direct point allocation, simple multiattribute rating technique (SMART), swing weighting, and tradeoff weighting. They can all be used following the principles of multiattribute value theory. Furthermore, SMART, swing, and AHP ask the decision makers to give directly the numerical estimates of weight ratios although the elicitation questions are different. In earlier studies these methods have yielded different weights. Our results suggest that the resulting weights are different because the methods explicitly or implicitly lead the decision makers to choose their responses from a limited set of numbers. The other consequences from this are that the spread of weights and the inconsistency between the preference statements depend on the number of attributes that a decision maker considers simultaneously.     

On the stability and convergence of nonlocal difference schemes

We study nonlocal boundary value problems of the first and second kind for the heat equation with variable coefficients in the differential and finite-difference settings. By using the method of energy inequalities, we obtain a priori estimates for the corresponding differential and finite-difference problems.     

On some computational orders of convergence

Two variants of the Computational Order of Convergence (COC) of an iterative method for solving nonlinear equations are presented. Furthermore, the way to approximate the COC and the new variants to the local order of convergence is analyzed. The new definitions given here does not involve the unknown root. Numerical experiments using adaptive arithmetic with multiple precision and a stopping criteria are implemented without using any known root.     

On some algebraic problems in connection with general elgenvalue algorithms

Two real matrices A,B are S-congruent if there is a nonsingular upper triangular matrix R such that A = R^TBR. This congruence relation is studied in the set of all nonsingular symmetric and that of all skew-symmetric matrices. Invariants and systems of representation are give. The results are applied to the question of decomposability of a matrix in a product of an isometry and an upper triangular matrix, a problem crucial in eigenvalue algorithms.     

On the convergence of some possibilistic clustering algorithms

In this paper, an analysis of the convergence performance is conducted for a class of possibilistic clustering algorithms (PCAs) utilizing the Zangwill convergence theorem. It is shown that under certain conditions the iterative sequence generated by a PCA converges, at least along a subsequence, to either a local minimizer or a saddle point of the objective function of the algorithm. The convergence performance of more general PCAs is also discussed.     

On the convergence of some Procrustean averaging algorithms

Euclidean “(size-and-)shape spaces” are spaces of configurations of points in R ^N modulo certain equivalences. In many applications one seeks to average a sample of shapes, or sizes-and-shapes, thought of as points in one of these spaces. This averaging is often done using algorithms based on generalized Procrustes analysis (GPA). These algorithms have been observed in practice to converge rapidly to the Procrustean mean (size-and-)shape, but proofs of convergence have been lacking. We use a general Riemannian averaging (RA) algorithm developed in [Groisser, D. (2004) “Newton's method, zeroes of vector fields, and the Riemannian center of mass”, Adv. Appl. Math. 33, pp. 95–135] to prove convergence of the GPA algorithms for a fairly large open set of initial conditions, and estimate the convergence rate. On size-and-shape spaces the Procrustean mean coincides with the Riemannian average, but not on shape spaces; in the latter context we compare the GPA and RA algorithms and bound the distance between the averages to which they converge.     

On the convergence of global methods in multiextremal optimization

A general class of derivative-free optimization procedures is presented including the corresponding convergence theory. This theory turns out to be very constructive, in the sense that the convergence conditions not only can be verified easily for many existing algorithms, but also allow one to construct new procedures. It is shown that popular methods such as branch-and-bound concepts, Pintér's general class of procedures, the algorithms of Pijavskii, Shubert, and Mladineo, and the approach of Zheng and Galperin can not only be subsumed under this class of methods, but also partly be improved by regarding them within the framework presented.     

Mean-square convergence of numerical methods for random ordinary differential equations

Numerical Algorithms - The aim of this work is to analyze the mean-square convergence rates of numerical schemes for random ordinary differential equations (RODEs). First, a relation between the...