Showing papers on "Convergence (routing) published in 1980"

1 On the Concept of Coordinative Structures as Dissipative Structures: I. Theoretical Lines of Convergence*

Abstract: In this paper we pursue the argument that where a group of muscles functions as a single unit the resulting coordinative structure, to a first approximation, exhibits behavior qualitatively like that of a force-driven mass-spring system. Data are presented illustrating the generative and context-independent characteristics of this system in tasks that require animals and humans to produce accurate limb movements in spite of unpredictable changes in initial conditions, perturbations during the movement and functional deafferentation. Analogous findings come from studies of articulatory compensation in speech production. Finally we provide evidence suggesting that one classically-defined source of information for movement, namely proprioception, may not be dimension-specific in its contribution to coordination and control.

Algorithms for separable nonlinear least squares problems

Abstract: Iterative algorithms of Gauss–Newton type for the solution of nonlinear least squares problems are considered. They separate the variables into two sets in such a way that in each iteration, optimization with respect to the first set is performed first, and corrections to those of the second after that. The linear-nonlinear case, where the first set consists of variables that occur linearly, is given special attention, and a new algorithm is derived which is simpler to apply than the variable projection algorithm as described by Golub and Pereyra, and can be performed with no more arithmetical operations than the unseparated Gauss–Newton algorithm. A detailed analysis of the asymptotical convergence properties of both separated and unseparated algorithms is performed. It is found that they have comparable rates of convergence, and all converge almost quadratically for almost compatible problems. Simpler separation schemes, on the other hand, converge only linearly. An efficient and simple computer impleme...

The Relaxation Method for Solving Systems of Linear Inequalities

Abstract: The relaxation method for solving systems of inequalities is related both to subgradient optimization and to the relaxation methods used in numerical analysis. The convergence theory depends upon two condition numbers. The first one is used mostly for the study of the rate of geometric convergence. The second is used to define a range of values of the relaxation parameter which guarantees finite convergence. In the case of obtuse polyhedra, finite convergence occurs for any value of the relaxation parameter between one and two. Various relationships between the condition numbers and the concept of obtuseness are established.

On global convergence of an algorithm for optimal control

Abstract: This paper presents an algorithm for the solution of optimal control problems with constraints on the control, but without constraints on the trajectory or the terminal state In this algorithm, reduction of a cost at each iteration is guaranteed Global convergence conditions for the algorithm are investigated and an example is worked out

Newton’s Method at Singular Points. I

Abstract: A theorem due to Reddien giving sufficient conditions for convergence of Newton iterates for singular problems is extended.

Convergence of multigrid iterations applied to difference equations

Abstract: Convergence proofs for the multi-grid iteration are known for the case of finite element equations and for the case of some difference schemes discretizing boundary value problems in a rectangular region. In the present paper we give criteria of convergence that apply to general difference schemes for boundary value problems in Lipschitzian regions. Furthermore, convergence is proved for the multi-grid algorithm with Gauss-Seidel's iteration as smoothing procedure.

Recursive square-root ladder estimation algorithms

Abstract: Recursive least-square ladder estimation algorithms have attracted much attention recently because of their excellent convergence behavior and fast parameter tracking capability. We present some recently developed square-root normalized ladder form algorithms that have fewer storage requirements, and lower computational requirements than the unnormalized ones. A Hilbert space approach to the derivations of the normalized recursions is presented. Computer simulation results show that the normalized forms have the same convergence behavior, but even better numerical properties than the unnormalized versions. Other normalized forms, such as joint process estimators and ARMA (pole-zero) models, will also be presented. Applications of these algorithms to fast (or "zero") startup equalizers, adaptive noise- and echo cancellers and inverse models for control problems are also discussed.

Lyapunov techniques for the exponential stability of linear difference equations with random coefficients

Abstract: We consider an approach to studying the exponential stability of linear difference equations with random coefficients through the use of Lyapunov stability techniques. The equations we study are of a form familiar from adaptive estimation algorithms, which motivates the examination. It is necessary to define the almost sure exponential convergence of a random process, and then to derive sufficient conditions on the coefficients of the difference equations to ensure the almost sure exponential convergence of the state. We consider, in particular, two very reasonable types of random coefficients-ergodic and stationary and φ-mixing and nonstationary-which would appear to encompass many engineering situations. An example of the power of the theory is given, where it is applied to a common adaptive filtering algorithm to derive mild conditions for exponential convergence with dependent random inputs.

Analysis of stochastic approximation schemes with discontinuous and dependent forcing terms with applications to data communication algorithms

Abstract: A general convergence result is given for stochastic approximation schemes with (or without) equality constraints. The following features are taken into account. The forcing term is a strongly dependent sequence and may be discontinuous. Many examples are given to illustrate the applicability of the convergence theorem, both classical (recursive least squares scheme) and nonclassical ones (arising in the theory of self-adaptive eqnalizers).

Stability Analysis of an Nth Power Digital Phase-Locked Loop--Part I: First-Order DPLL

Abstract: The behavior of a digital phase-locked loop (DPLL) which tracks the positive-going zero crossings of the incoming signal can be characterized by a nonlinear difference equation in the phaseerror process. This equation was first presented by Gill and Gupta for the CW loop, and modified by Osborne and Lindsey for the N th power loop. Stability results have been previously obtained for first- and second-order loops by linearizing the equation about the steady-state solution. However, in this paper, a mathematically more rigorous and powerful approach is introduced whereby the acquisition behavior is studied by formulating the equation as a fixed-point problem. Stability results can be obtained by studying the nonlinear equation directly, using theorems pertaining to the convergence behavior of the Picard iterates, e.g., Ostrowski's Theorem and the Contraction Mapping Theorem. Using this formulation, we present some new stability results (and rederive some previously obtained results) for the first- and second-order DPLL's. Then, some stability results for the third-order DPLL are derived for the first time. The first-order DPLL results appear in Part I, and the higher order DPLL results appear in Part II.

The limitations of the patch test

Abstract: A simple approximation by nonconforming finite elements is presented that passes the patch test of Irons and Strang but does not yield approximate solutions converging to the solution of the given boundary value problem. It is constructed from continuous piecewise linear functions perturbed by step functions. Further, strange convergence properties of such approximations are explained in all details because they may be typical for the behaviour of nonconforming finite elements violating the basic precondition for convergence.

Feasible direction methods for stochastic programming problems

Abstract: A unified approach to stochastic feasible direction methods is developed. An abstract point-to-set map description of the algorithm is used and a general convergence theorem is proved. The theory is used to develop stochastic analogs of classical feasible direction algorithms.

Analysis of convergence of the $T$-transformation for power series

Abstract: Recently the present author has given some convergence theorems of general nature for Levin's nonlinear sequence transformations. In this work these theorems are extended and sharpened to cover the case of power series, both inside and on their circle of convergence. It is shown that one of the two limiting processes considered in the previous work can be used for analytic continuation and a realistic estimate of its rate of convergence is given. Three illustrative examples are also appended.

Résultats négatifs en accélération de la convergence

Abstract: It is well known that some information is needed for accelerating efficiently the convergence of a sequence. We show in this article that, for several families of sequences, there is no algorithm accelerating the convergence of every sequence of the family.

A class of optimal routing algorithms for communication networks

Abstract: : This report describes an algorithm for minimum delay routing in a communication network. During the algorithm each node maintains a list of paths along which it sends traffic to each destination together with a list of the fractions of total traffic that are sent along these paths. At each iteration a minimum marginal delay path to each destination is computed and added to the current list if not already there. Simultaneously the corresponding fractions are updated in a way that reduces average delay per message. The algorithm is capable of employing second derivatives of link delay functions thereby providing automatic scaling with respect to traffic input level. It can be implemented in both a distributed and a centralized manner, and it can be shown to converge to an optimal routing at a linear rate.

Strong uniqueness and second order convergence in nonlinear discrete approximation

Abstract: Strong uniqueness has proved to be an important condition in demonstrating the second order convergence of the generalised Gauss-Newton method for discrete nonlinear approximation problems [4]. Here we compare strong uniqueness with the multiplier condition which has also been used for this purpose. We describe strong uniqueness in terms of the local geometry of the unit ball and properties of the problem functions at the minimum point. When the norm is polyhedral we are able to give necessary and sufficient conditions for the second order convergence of the generalised Gauss-Newton algorithm.

Algorithm 554: BRENTM, A Fortran Subroutine for the Numerical Solution of Nonlinear Equations [C5]

Abstract: BRENTM is a subroutine designed to solve a system of n nonlinear equations in n variables by using a modification of Brent's method (SIAM J. Numer. Anal., 10, 327-344 (1973)). The subroutine does not use any techniques that attempt to obtain global convergence; therefore, convergence is guaranteed only if the initial estimate for the solution is close enough. On the other hand, the code does seem to have a large region of convergence; convergence occurs only at a zero of the function, and if the iteration is not making satisfactory progress, then BRENTM will attempt to diagnose this situation and stop the iteration with an appropriate message. The use is only required to provide a subroutine that calculates components of the function. (RWR)

Adaptive state estimation using MRAS techniques--Convergence analysis and evaluation

Abstract: Three adaptive state observers for discrete-time systems derived from MRAS techniques are presented. While in a deterministic environment all of these schemes converge toward the linear asymptotic observer, when used in a stochastic environment for adaptive state estimation their performances present noticeable differences. The schemes considered in the paper are analyzed both in a deterministic and stochastic environment using the "equivalent feedback representation" (EFR) method and "ordinary differential equation" (ODE) method, respectively. Conditions for the convergence of the estimated parameters to the desired ones in a stochastic environment are given. The connections with adaptive Kalman filters are discussed. A comparative evaluation of these schemes in a deterministic and stochastic environment based on simulations concludes the paper.

A modified quasilinearization algorithm for the computation of optimal singular control

Abstract: A quasilinearization algorithm is modified to handle a wider class of dynamic optimization problems with singular arcs. That is, the performance index and the system equations may be non-linear in the control variable. Four examples were solved to demonstrate the effectiveness of the modified algorithm. It is shown that the proposed algorithm could offer a larger region of convergence and faster convergence in addition to its ability to handle a wider class of problem.