Search

From the eigenvalue problem theory, we see that the convergence rate of the biharmonic eigenvalues obtained by the mixed method in I. Bab u ˇ ska and J. Osborn, [Eigenvalue Problems, Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam, 1991.] is h 2 k − 2 for k ≥ 2. In this paper, we give a presentation of the lowest-order virtual element method for the approximation of Kirchhoff plate vibration problem. This discrete scheme is based on a conforming H 1 (Ω) × H 1 (Ω) formulation, following the variational formulation of Ciarlet–Raviart method, which allows us to make use of simpler and lower-regularity virtual element space. By using the classical spectral approximation theory in functional analysis, we prove the spectral approximation and optimal convergence order h 2 for the eigenvalues. Finally, some numerical experiments are presented, which show that the proposed numerical scheme can achieve the optimal convergence order. [ABSTRACT FROM AUTHOR]

In this paper we consider rational approximation of functions in the Hardy spaces on the strip with . Observing that can be regarded as the direct sum of and , we give three kinds of adaptive rational approximation in enhancing the corresponding approximations in and where and We also obtain a type of approximation for subspaces in by making use of Riesz bases in these subspaces. [ABSTRACT FROM AUTHOR]

In this paper, we provide sufficient conditions for the existence of periodic solutions emerging of the equilibrium points of the spatial Hill lunar problem having the following equations of motion:ϵ is a small parameter andFi,i∈ {1, 2, 3}, are smooth periodic functions intwhich define a perturbation in resonancep:qwith some of the periodic solutions of the spatial Hill lunar problem beingpandqpositive relatively prime integers. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generalization of the Kantorovich-type Bernstein operators based on q-integers and get a Bohman-Korovkin-type approximation theorem of these operators. We also compute the rate of convergence using the first modulus of smoothness. [ABSTRACT FROM AUTHOR]

Ant colony optimization (ACO) is a kind of powerful and popular randomized search heuristic for solving combinatorial optimization problems. This paper investigates the performance of two ACO algorithms, calledand, on the travelling salesman problem with distance one and two (TSP(1,2)) which is an NP-complete problem. It is shown that two ACO algorithms obtain an approximation ratio ofwith regard to the optimal solutions in expected polynomial runtimes. We also study the influence of pheromone information and heuristic information on the approximation performance. Finally, we construct an instance and demonstrate that ACO outperforms the local search algorithms on this instance. [ABSTRACT FROM PUBLISHER]

A dynamic capacity allocation problem is considered in this paper. A fixed amount of daily processing capacity is allowed. Jobs of different priorities arrive randomly over time and a decision is required on which jobs should be scheduled on which days. The jobs that are waiting to be processed incur a holding cost depending on their priority levels. The objective is to minimize the total expected cost over a planning horizon. In this paper the focus is on a class of policies that are characterized by a set of protection levels. The role of the protection levels is to “protect” a portion of the capacity from the lower priority jobs so as to make it available for the future higher priority jobs. A stochastic approximation method to find a good set of protection levels is developed and its convergence is proved. Computational experiments indicate that protection level policies perform especially well when the coefficient of variation for the job arrivals is high. [Supplementary materials are available for this article. Go to the publisher's online edition of IIE Transactions for the following free supplemental resource: Technical appendix detailing the proofs of propositions.] [ABSTRACT FROM AUTHOR]

Renshaw and Verrall [ 11 ] specified the generalized linear model (GLM) underlying the chain-ladder technique and suggested some other GLMs which might be useful in claims reserving. The purpose of this paper is to construct bounds for the discounted loss reserve within the framework of GLMs. Exact calculation of the distribution of the total reserve is not feasible, and hence the determination of lower and upper bounds with a simpler structure is a possible way out. The paper ends with numerical examples illustrating the usefulness of the presented approximations. [ABSTRACT FROM AUTHOR]

A new Pareto front approximation method is proposed for multiobjective optimization problems (MOPs) with bound constraints. The method employs a hybrid optimization approach using two derivative-free direct search techniques, and intends to solve black box simulation-based MOPs where the analytical form of the objectives is not known and/or the evaluation of the objective function(s) is very expensive. A new adaptive weighting scheme is proposed to convert a multiobjective optimization problem to a single objective optimization problem. Another contribution of this paper is the generalization of the star discrepancy-based performance measure for problems with more than two objectives. The method is evaluated using five test problems from the literature, and a realistic engineering problem. Results show that the method achieves an arbitrarily close approximation to the Pareto front with a good collection of well-distributed nondominated points for all six test problems. [ABSTRACT FROM AUTHOR]

Sung [On inverse moments for a class of nonnegative random variables. J Inequal Appl. 2010;2010:1–13. Article ID 823767, doi:10.1155/2010/823767] obtained the asymptotic approximation of inverse moments for a class of nonnegative random variables with finite second moments and satisfying a Rosenthal-type inequality. In the paper, we further study the asymptotic approximation of inverse moments for a class of nonnegative random variables with finite first moments, which generalizes and improves the corresponding ones of Wu et al. [Asymptotic approximation of inverse moments of nonnegative random variables. Statist Probab Lett. 2009;79:1366–1371], Wang et al. [Exponential inequalities and inverse moment for NOD sequence. Statist Probab Lett. 2010;80:452–461; On complete convergence for weighted sums of ϕ mixing random variables. J Inequal Appl. 2010;2010:1–13, Article ID 372390, doi:10.1155/2010/372390], Sung (2010) and Hu et al. [A note on the inverse moment for the nonnegative random variables. Commun Statist Theory Methods. 2012. Article ID 673677, doi:10.1080/03610926.2012.673677]. [ABSTRACT FROM PUBLISHER]

Given a collection ofncurves that are independent realizations of a functional variable, we are interested in finding patterns in the curve data by exploring low-dimensional approximations to the curves. It is assumed that the data curves are noisy samples from the vector space span f1, …,fm, wheref1, …,fmare unknown functions on the real interval (0,T) with square-integrable derivatives of all ordersmor less, andm

Subdivision schemes are special multi-resolution analysis (MRA) methods that have become prevalent in computer-aided geometric design. This paper draws useful analogies between the mathematics of subdivision schemes and the hierarchical structures of music compositions. Based on these analogies, we propose new methods for music synthesis and analysis through MRA, which provide a different perspective on music composition, representation and analysis. We demonstrate that the structure and recursive nature of the recently proposed subdivision models [S. Hed and D. Levin, Subdivision models for varying-resolution and generalized perturbations, Int. J. Comput. Math. 88(17) (2011), pp. 3709–3749; S. Hed and D. Levin, A ‘subdivision regression’ model for data analysis, 2012, in preparation] are well suited to the synthesis and analysis of monophonic and polyphonic musical patterns, doubtless due in large part to the strongly hierarchical nature of traditional musical structures. The analysis methods demonstrated enable the compression and decompression (reconstruction) of selected musical pieces and derive useful features of the pieces, laying groundwork for music classification. [ABSTRACT FROM PUBLISHER]

In this paper, we consider geometric aspects of a rational, planar system of difference equations defined on the open first quadrant and whose behaviour is governed by four independent, non-negative parameters. This system, indexed as (23, 23) in the notation of Ladas (Open problems and conjectures, J. Differential Equ. Appl. 15(3) 2009, pp. 303–323), is one of the 200 systems from Ladas about which little is known. Using geometric techniques, we answer several questions concerning the behaviour of this system. [ABSTRACT FROM AUTHOR]

Let X be a normed space, V be an open convex subset of X and let be a given function. A function is called -midconvex if By the result of Tabor and Tabor, we know that under respective conditions on , if f is -midconvex and locally bounded above at a point then there exists a continuous function such that In this paper we determine the smallest function satisfying the above inequality. The required conditions on are such that the functions , satisfy them. As the main tool we use de Rham theorem. [ABSTRACT FROM AUTHOR]

In this paper, it is shown that the necessary conditions for stability of Kharitonov polynomials can be checked by simply inspecting few upper and lower bound coefficients. [ABSTRACT FROM AUTHOR]

We consider projections from the space of continuous functions on polyhedron with n + 2 vertices in n onto subspace of affine functions. We give necessary and sufficient condition for a projection to be minimal. Then, for each n we find the polyhedron, for which the norm of a minimal projection is maximal. This is a generalization of the result from Shekhtman's and Skrzypek's paper [9], where only the case n = 2 was considered. It is worth noticing that our proof in the even case is significantly different from our proof in the odd case. [ABSTRACT FROM AUTHOR]

We show that spherical truncations of the 1/r interactions in models for water and acetonitrile yield very accurate results in bulk simulations for all site-site pair correlation functions as well as dipole-dipole correlation functions. This good performance in bulk simulations contrasts with the generally poor results found with the use of such truncations in nonuniform molecular systems. We argue that Local Molecular Field (LMF) theory provides a general theoretical framework that gives the necessary corrections to simple truncations in most nonuniform environments and explains the accuracy of spherical truncations in uniform environments by showing that these corrections are very small. LMF theory is derived from the exact Yvon-Born-Green (YBG) hierarchy by making physically-motivated and well-founded approximations. New and technically interesting derivations of both the YBG hierarchy and LMF theory for a variety of site-site molecular models are presented in appendices. The main paper focuses on understanding the accuracy of these spherical truncations in uniform systems both phenomenologically and quantitatively using LMF theory. [ABSTRACT FROM AUTHOR]

This paper focuses on the issue of description of the set of parametric perturbation which preserves the spectrum of a discrete linear time varying system. We prove that all perturbations with the Lyapunov exponent lower then a regularity coefficient are characterized with this feature. Moreover, in case of a diagonal system, the perturbations with Lyapunov exponent equal to the regularity coefficient also reveal this feature. [ABSTRACT FROM AUTHOR]

Gupta and Shabbir 2 have suggested an alternative form of ratio-type estimators for estimating the population mean. In this paper, we obtained a corrected version for the mean square error (MSE) of the Gupta-Shabbir estimator, up to first order of approximation, and the optimum case is discussed. We expand this estimator to the stratified random sampling and propose general classes for combined and separate estimators. Also an empirical study is carried out to show the properties of the proposed estimators. [ABSTRACT FROM AUTHOR]

In this paper, the condition number of singular value is discussed. The expressions of the zero-structured and patterned conditioning are also studied. The numerical tests are reported to provide interesting information about the singular value sensitivity when the perturbations of the matrix have arbitrarily assigned zero-structured and patterned case. [ABSTRACT FROM AUTHOR]

In this paper, several equivalent systems without constraints of the indefinite least squares problem with equality constraints (ILSE) are established. We also derive the perturbation results for the ILSE problem and illustrate our results with numerical tests. [ABSTRACT FROM AUTHOR]

Chemists often come across triangle domains - usually with the basic simplex in 2. A smooth surface is needed for approximating the chemical properties between the measured data for solving some model (differential) equations numerically. Our research group has been working on approximating ternary chemical surfaces of two special fields by smooth functions (vapour - Liquid equilibrium data and explosion-limit surfaces of ternary gas systems). A mathematical solution was given in both fields by special spline surfaces, and for visualization, our own software (TRIGON) was used. In this paper the summarized chemical background of each problem is provided, the mathematical solutions, the newest theoretical developments and their results are discussed. [ABSTRACT FROM AUTHOR]

Saddle point problems arise in a wide variety of applications in computational and engineering. The aim of this paper is to present a SSOR-like iterative method for solving the saddle point problems. Here the convergence of this method is studied and specifically, the spectral radius and the optimal relaxation parameter of the iteration matrix are also investigated. Numerical experiments show that the SSOR-like method with a proper preconditioning matrix is better than SOR-like method presented by Golub et al. [G.H. Golub, X. Wu, and J.-Y. Yuan, SOR-like methods for augmented systems, BIT 41 (2001), pp. 71-85]. [ABSTRACT FROM AUTHOR]

This paper describes an approximate analytical method for evaluating the average values of throughput and buffer levels of closed three-machine production systems with finite buffers. The method includes a new set of decomposition equations and a new building block model. The machines have deterministic processing times and geometrically distributed probabilities of failure and repair. The numerical results of the method are close to those from simulation. The method performs well because it takes into account the correlation among the numbers of parts in the buffers. Extensions to larger systems are discussed. [Supplementary materials are available for this article. Go to the publisher's online edition of IIE Transactions for the following free supplemental resource: Appendix of additional numerical results] [ABSTRACT FROM AUTHOR]

In this paper, the study of best simultaneous Lp-approximation by algebraic polynomials on a union of intervals, I, is considered. We obtain convergence of the best simultaneous approximations when the measure of I tends to zero. We also get an interpolation result. [ABSTRACT FROM AUTHOR]

A new parallel symmetric successive over-relaxation (PSSOR) preconditioner is proposed in this paper by the multi-type partition techniques introduced in SIAM J. Scientific Computing 20, 2006, pp. 1513-1533. In a general matrix expression, it is proved to be symmetric and positive-definite (SPD) if the coefficient matrix of a linear system is SPD. It is also proved to be equivalent to the SSOR preconditioner using the multi-type ordering. Thus, it works for the preconditioned conjugate gradient method (PCG) and can be analysed by the classic SOR theory. Numerical tests on an anisotropic model problem show that the PSSOR preconditioner can make the PCG have a faster rate of convergence and better parallel performance than the red-black SSOR preconditioner. They also confirm that the PSSOR preconditioner can have a rate of convergence that is nearly the same as the classic sequential SSOR preconditioner when the problem has large anisotropy. [ABSTRACT FROM AUTHOR]

Spearman's rank correlation test is a widely used statistical method for testing the correlation between two samples. Unfortunately, the evaluation of the exact distribution of this statistic is difficult and approximation obtained by using the normal distribution is not accurate for moderate sample sizes. In this paper, the critical values for samples 11≤n≤25 are provided. On the basis of the calculated exact values, the accuracy of different approximation methods is discussed. The use of approximation obtained by using the Edgeworth series is recommended in cases when the exact distribution is not available. [ABSTRACT FROM AUTHOR]

This paper presents a framework for the study of generalizing the standard notion of rough set approximation space. We propose new definitions of lower and upper approximations, which are basic concepts of the rough set theory. These definitions follow naturally from a particular property on the universe and are called general lower and upper approximations. Properties of general approximations are investigated, and their connections are examined. [ABSTRACT FROM AUTHOR]

This paper provides closed form expressions for the sample size for two-level factorial experiments when the response is the number of defectives. The sample sizes are obtained by approximating the two-sided test for no effect through tests for the mean of a normal distribution, and borrowing the classical sample size solution for that problem. The proposals are appraised relative to the exact sample sizes computed numerically, without appealing to any approximation to the binomial distribution, and the use of the sample size tables provided is illustrated through an example. [ABSTRACT FROM AUTHOR]

This paper presents a numerical study of a micropolar fluid flowing past a stretching surface involving both heat and mass transfer with Ohmic heating and viscous dissipation. A similarity transformation is employed to change the governing equations into nonlinear coupled higher-order ordinary differential equations. These equations are solved numerically using a finite element method with linear shape functions and a Chebyshev finite difference method that involves higher-order polynomials. Numerical results of both methods have been compared with the closed-form solution of a particular case when the material parameter is taken to be zero. Good agreement between the numerical results of both methods, together with excellent agreement with the closed-form solution, ensures the reliability of using linear shape functions in the finite element method. The effect of the parameters governing the problem on the velocity, microrotation, temperature and concentration functions has been studied for different boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper we investigate linear systems with multiple right-handed sides in the form of AX=B and (A ⊗ B)X=C. We derive normwise, mixed and componentwise condition numbers for these linear systems. Examples are given to evaluate the tightness of the first-order perturbation bounds. [ABSTRACT FROM AUTHOR]

The little-known Belgian engineer Bernard Chait contributed to business cycle theory by means of a mathematical model, of which the 'law of divergence' was an important building block. The law of divergence has been interpreted as a generalization of the acceleration principle. This paper draws upon published and archival sources to examine the sources of Chait's thinking, explain the basics of his model and assess the impact of his work. His relations with François Divisia and Jan Tinbergen are explored and his claim that Leontief's dynamic input - model was an unacknowledged reformulation of his own work is analysed. [ABSTRACT FROM AUTHOR]

In this paper, we propose a three-term conjugate gradient method which can produce sufficient descent condition, that is, [image omitted] . This property is independent of any line search used. When an exact line search is used, this method reduces to the standard Hestenes-Stiefel conjugate gradient method. We also introduce two variants of the proposed method which still preserve the sufficient descent property, and prove that these two methods converge globally with standard Wolfe line search even if the minimization function is nonconvex. We also report some numerical experiment to show the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we prove an a posteriori and an a priori convergence theorem for Newton-Kantorovich approximations starting from an initial point x0. We apply these results to operators that are analytic at interior points of a closed ball centered at x0 and of radius R. We obtain some theorems on approximate zeros and on approximate zeros of second kind for these operators, which improve previous results. [ABSTRACT FROM AUTHOR]

This paper is concerned with asymptotic behaviour of solutions of perturbed dynamic systems on time scales. A time scale version of the Hartman-Wintner theorem is established for a class of time scales. [ABSTRACT FROM AUTHOR]

The class of nonlinear reproductive dispersion mixed models (NRDMMs) is an extension of nonlinear reproductive dispersion models and generalized linear mixed models. This paper discusses the influence analysis of the model based on Laplace approximation. The equivalence of case-deletion models and mean-shift outlier models in NRDMMs is investigated, and some diagnostic measures are proposed via the case-deletion method. We also investigate the assessment of local influence of various perturbation schemes. The proposed method is illustrated with an example. [ABSTRACT FROM AUTHOR]

This paper considers the problem of approximating an integrable function by piecewise linear functions that keep the integral and positivity of the original function. [ABSTRACT FROM AUTHOR]

In this paper, we study the distributed and boundary optimal control of the Stokes equations in the presence of pointwise control constraints. The analysis of the problems involve existence results, as well as necessary conditions and optimality systems. A primal–dual active set strategy is studied for the numerical solution of the problems. For the distributed case, global and local convergence results for the infinite dimensional method are presented. For the boundary control case, due to the lack of regularity of the pointwise constraint multiplier, the method is applied to a discretized version of the original problem. Finally, some numerical experiments, which illustrate the performance of the method for both cases, are commented. [ABSTRACT FROM AUTHOR]

In the growing literature of factor analysis, little is done to understand the finite sample properties of an approximate factor model solution. In empirical applications with relatively small samples, the asymptotic theory might be a poor approximation and the resulting distortions might affect the estimation (bias in the point estimate and the standard errors) and the statistical inference. The present paper uses the estimation method of Bai and Ng [Bai, J. and Ng, S., 2002, Determining the number of factors in approximate factor models. Econometrica , 70, 191–221.] and assesses the sampling behavior of the estimated common components, common factors and factor loadings. The study compares the empirical distributions to the asymptotic theory of Bai [Bai, J., 2003, Inference on factor models of large dimension. Econometrica , 71, 135–171.]. Simulation results suggest that the point estimates have a Gaussian distribution for panels with relatively small dimensions. However, these estimates have a significant finite sample bias and the dispersion of their sampling distribution is severely underestimated by the asymptotic theory. [ABSTRACT FROM AUTHOR]

A method of designing a state feedback gain achieving a specified insensitivity of the closed-loop trajectory by the singularly perturbed unified system using the d operators is proposed. The order of system is reduced by the singular perturbation technique by ignoring the fast mode in it. The proposed method takes care of the actual trajectory variations over the range of the singular perturbation parameter. Necessary conditions for optimality are also derived. The previous study was done in the continuous time system. The present paper extends the previous study to the discrete system and the d -operating system that unifies the continuous and discrete systems. Advantages of the proposed method are shown in the numerical example. [ABSTRACT FROM AUTHOR]

Recent work has formulated an MPC strategy for a sustainable development problem, focused on the control of a stochastic, non-linear, economic model called PROMETHEUS. This work also takes PROMETHEUS to be the system. To use MPC an analytic approximation of PROMETHEUS is required, with previous work focusing on linear and non-linear time invariant models. Global optimality of the solution could not be guaranteed, since to achieve reasonable accuracy in iterative fitting was required which destroyed convexity of the optimization. In this paper an LTV model is considered and is shown to reduce the approximation error. It is demonstrated that the model can be finitely parameterized and that the predominant time variation in PROMETHEUS is from a decaying exponential weighting on the predicted input sequence. It is suggested that this variation is a representation of inflation. The sustainable development MPC methodology is recast for such a model. Convexity of the MPC optimization with an LTV model guarantees global optimality; results are presented which demonstrate the improvement in performance which this allows. [ABSTRACT FROM AUTHOR]

Synchronous generators have a natural different time scale dynamics. That is why, for modelling and control design in such systems, the methods of singular perturbations are widely used. In this paper the possibilities of sliding mode control design for synchronous generators are analysed. With this aim the concept of singular perturbation is revised in order to use it for relay control systems. The obtained results are used for sliding mode control of synchronous generator. [ABSTRACT FROM AUTHOR]

Testing conditional symmetry against various alternative diagonals-parameter symmetry models often provides a point of departure in studies of square contingency tables with ordered categories. Typically, chi-square or likelihood-ratio tests are used for such purposes. Since these tests depend on the validity of asymptotic approximation, they may be inappropriate in small-sample situations where exact tests are required. In this paper, we apply the theory of UMP unbiased tests to develop a class of exact tests for conditional symmetry in small samples. Oesophageal cancer and longitudinal income data are used to illustrate the approach. [ABSTRACT FROM AUTHOR]

A number of approximate analysis techniques are based on matching moments of continuous time phase type (PH) distributions. This paper presents an explicit method to compose minimal order continuous time acyclic phase type (APH) distributions with a given first three moments. To this end we also evaluate the bounds for the first three moments of order n APH distributions (APH( n )). The investigations of these properties are based on a basic transformation, which extends the APH( n - 1) class with an additional phase in order to describe the APH( n ) class. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of whether the best approximation to any x in a Hilbert space X from the set where C is a closed convex subset of X , S is a closed convex cone that does not necessarily have nonempty interior, Y is a Banach space, and g : X ? Y is a continuous S -convex function, can be characterized by the best approximation to a perturbation x - l of x from the set C for some l ? X . We provide a global approach to this problem by presenting a dual global constraint qualification, which is less restrictive than the Slater type (or interior-point) conditions, guaranteeing the strong conical hull intersection property (CHIP). We then show that the strong CHIP characterizes the perturbation property under a mild closure condition. The closure condition, for instance, holds whenever the explicit constraint set is described by finitely many linear inequality constraints. We also establish easily verifiable dual conditions that are equivalent to the best approximation from the set K . We finally demonstrate that our results recapture the corresponding known results in the particular case where Y is a finite dimensional space. [ABSTRACT FROM AUTHOR]

This paper investigates the problem of Hankel norm model reduction for linear systems with time-varying delay in the state. For a given stable system, our attention is focused on the construction of reduced-order model, which guarantees the corresponding error system to be asymptotically stable and has a specified Hankel norm error performance. Two different approaches are proposed to solve this problem. One casts the model reduction into a convex optimization problem by using a linearization procedure, and the other is based on the cone complementarity linearization idea, which casts the model reduction into a sequential minimization problem subject to linear matrix inequality constraints. Both continuous and discrete time eases are considered. A numerical example is provided to show the effectiveness of the proposed theory. [ABSTRACT FROM AUTHOR]

Stable indirect and direct adaptive controllers are presented for a class of input-output feedback linearizable timevarying non-linear systems. The radial basis function neural networks are used as on-line approximators to learn the time-varying characteristics of system parameters. Stability results are given in the paper, and the performance of the indirect and direct adaptive schemes is demonstrated through a fault-tolerant engine control problem where the faults are naturally time-varying. [ABSTRACT FROM AUTHOR]

An important subtask of the pattern discovery process is pattern matching, where the pattern sought is already known and we want to determine how often and where it occurs in a sequence. In this paper we review the most practical techniques to find patterns of different kinds. We show how regular expressions can be searched for with general techniques, and how simpler patterns can be dealt with more simply and efficiently. We consider exact as well as approximate pattern matching. Also we cover both sequential searching, where the sequence cannot be preprocessed, and indexed searching, where we have a data structure built over the sequence to speed up the search. [ABSTRACT FROM AUTHOR]

A novel technique for the construction of positive weight high-precision rational approximations to a class of transcendental curves is presented. The approximations are induced from the rational parametrisations of the circle. The previously published rational parametrisations of the circle are not suited to the induction process and the new parametrisations are constructed in the paper for the purpose. Explicit rational approximations of a number of transcendental curves are then given. The work is a development of the authors' previous work on induced rational parametrisations of special algebraic curves. [ABSTRACT FROM AUTHOR]

One of the most popular algorithms for solving minimax approximation problems is the Barrodale and Phillips simplex-based method [I. Barrodale and C. Phillips (1974). An improved algorithm for discrete Chebyshev linear approximation. In: Proceedings of the Fourth Manitoba Conference on Numerical Mathematics , pp. 177-190, University of Manitoba, Winnipeg, Canada.]. Our new research has found that the modified pivoting strategy in the first stage of the Barrodale and Phillips algorithm could result in an invalid starting point and system failure in later stages [D. Lei (2002). Robust and efficient algorithms for ℓ 1 and ℓ ∞ approximations. PhD Thesis, School of Computing and Engineering, University of Huddersfield, Huddersfield, UK.]. In contrast, a primal method that we propose in this paper is numerically more stable, and the approximating function does not suffer the effects of wild oscillation. However, on average, the primal method requires more searching steps to find an optimal solution. To improve the efficiency of the primal method, we have considered using the least squares solution to provide a better starting point. Experimental results show that if the least squares estimator is available, then it can speed up the convergence rate significantly in most cases. The number of steps required by using the least squares solution for the primal method also compares favourably with that of the dual method. [ABSTRACT FROM AUTHOR]

This paper considers the robust stability of descriptor systems with structured perturbations. It presents a maximum perturbation bound for keeping a system regular, impulse-free and stable, by using the technique of Kronecker product and the structured singular value μ. [ABSTRACT FROM AUTHOR]