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The purpose of this paper is to define and study conditional pressure for subadditive potentials which is an extension of conditional entropy. This study reveals a variational principle of topological conditional pressure for subadditive potentials. Besides, we discuss the topological conditional pressure for asymptotically subadditive potentials. The main idea of the proof is quite similar to that of Cao, Feng and Huang's approximations. [ABSTRACT FROM AUTHOR]

Modified direct methods for the computation of Zernike and pseudo-Zernike moments are presented in this paper. The presence of many factorial terms in direct methods for computing Zernike-type moments makes their computation process a very time consuming task. Although the computational power of the modern computers is impressively increasing, the calculation of the factorial of a big number is still an inaccurate numerical procedure. The main concept of this paper is that, using Stirling's approximation for the factorial and applying some suitable mathematical properties, novel factorial-free direct methods can be developed. The resulting moments are not equal to those computed using the original direct methods, but they are a sufficiently accurate approximation of them. In addition, their variability does not affect their ability to uniquely describe and distinguish the objects that they represent. This is verified by appropriate pattern recognition experiments. [ABSTRACT FROM AUTHOR]

This paper is the fourth in a series whose goal is to develop a fundamentally new way of building theories of physics. The motivation comes from a desire to address certain deep issues that arise in the quantum theory of gravity. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics arises when the topos is the category of sets. Other types of theory employ a different topos. The previous papers in this series are concerned with implementing this program for a single system. In the present paper, we turn to considering a collection of systems; in particular, we are interested in the relation between the topos representation for a composite system and the representations for its constituents. We also study this problem for the disjoint sum of two systems. Our approach to these matters is to construct a category of systems and to find a topos representation of the entire category. [ABSTRACT FROM AUTHOR]

The present paper investigates approximation spaces in the context of mathematical structures which axiomatise the notion of nearness. Starting with the framework of information quanta which distinguishes two levels of information structures, namely property systems (the first level) and information quantum relational systems (the second level), we shall introduce the notion of Pawlak property system. These systems correspond bijectively to finite approximation spaces, i.e. their respective information quantum relational systems. Then we characterise Pawlak property systems in terms of symmetric topological spaces. In the second part of the paper, these systems are defined by means of topological structures based on the concept of nearness. We prove that the category of Pawlak property systems is isomorphic to the category of finite topological nearness spaces and provide its additional topological characterisation. [ABSTRACT FROM AUTHOR]

Abstract The present paper does not introduce a new approximation but it modifies a certain known method. This method for obtaining a periodic approximation of a periodic solution of a linear nonhomogeneous differential equation with periodic coefficients and periodic right-hand side is used in technical practice. However, the conditions ensuring the existence of a periodic solution may be violated and therefore the purpose of this paper is to modify the method in order that these conditions remain valid. [ABSTRACT FROM AUTHOR]

Motivated by a recent paper by Cichon and Macyna [1], who introduced m counters (instead of just one) in the approximate counting scheme first analysed by Flajolet [2], we analyse the moments of the sum of the m counters, using techniques that proved to be successful already in several other contexts [11]. [ABSTRACT FROM AUTHOR]

In this paper, we continue studying the so-called non-linear best m-term one-sided approximation problems and obtain the asymptotic estimations of non-linear best m-term one-sided trigonometric approximation under the norm L p (1 ≤ p ≤ ∞) of multiplier function classes and the corresponding m-term Greedy-liked one-sided trigonometric approximation results. [ABSTRACT FROM AUTHOR]

The aim of this paper is threefold: 1) to propose a simple accurate closed-form approximation to the probability density function of the sum of arbitrarily distributed Nakagami-m random variables; 2) to propose a simple accurate closed-form approximation to the level crossing rate for the sum of Nakagami-m random processes; and 3) to show some possible applications for the proposed formulations. With such an aim, we choose the α-μ distribution for which the parameters are estimated from the sum of the Nakagami-m envelopes. As shall be shown from sample representative examples, the proposed approximations are simple, versatile, and highly accurate. The approach used here can easily be extended to other applications such as bit error rate and channel capacity calculations among others. [ABSTRACT FROM AUTHOR]

In this paper, we show that approximation spaces are basic structures for knowledge discovery from multi-relational data. The utility of approximation spaces as fundamental objects constructed for concept approximation is emphasized. Examples of basic concepts are given throughout this paper to illustrate how approximation spaces can be beneficially used in many settings. [ABSTRACT FROM AUTHOR]

In this paper, the so-called η-approximation approach is used to obtain the sufficient conditions for a nonlinear multiobjective programming problem with univex functions with respect to the same function η. In this method, an equivalent η-approximated vector optimization problem is constructed by a modification of both the objective and the constraint functions in the original multiobjective programming problem at the given feasible point. Moreover, to find the optimal solutions of the original multiobjective problem, it sufficies to solve its associated η-approximated vector optimization problem. Finally, the description of the η-approximation algorithm for solving a nonlinear multiobjective programming problem involving univex functions is presented. [ABSTRACT FROM AUTHOR]

In this paper we analyze the behavior of the solution of the dissipative Boussinesq systems where α, β, c > 0 are parameters. Those systems model two-dimensional small amplitude long wavelength water waves. For α ≤ 1, this equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for almost every β, c and almost every α ≤ 1, it admits solutions that are quasiperiodic in time. The proof uses the fact that the equation leaves invariant a smooth center manifold and for the restriction of the Boussinesq system to the center manifold, uses arguments of classical perturbation theory by considering the Hamiltonian formulation of the problem and studying the Birkhoff normal form. [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]

Letbe a unit sphere of thed-dimensional Euclidean space Rd and let(0 < p= 1) denote the real Hardy space onFor 0 < p= 1 andletEj(f,Hp) (j= 0, 1, ...) be the best approximation offby spherical polynomials of degree less than or equal toj, in the spaceGiven a distributionfonits Cesàro mean of order d>-1 is denoted byFor 0p. In this paper, the following result is proved:TheoremLet0N(f)˜BN(f)means that there’s a positive constant C, independent of N and f, such thatIn the cased= 2,this result was proved by Belinskii in 1996. [ABSTRACT FROM AUTHOR]

In this paper, we consider the finite-dimensional approximations of Tikhonov regularization for nonlinear ill-posed problems with approximately given right-hand sides. We propose an a posteriori parameter choice strategy, which is a modified form of Morozov's discrepancy principle, to choose the regularization parameter. Under certain assumptions on the nonlinear operator, we obtain the convergence and rates of convergence for Tikhonov regularized solutions. This paper extends the results, which were developed by Plato and Vainikko in 1990 for solving linear ill-posed equations, to nonlinear problems. [ABSTRACT FROM AUTHOR]

Mean-field Boltzmann machine learning is recognized as a practical method to circumvent the difficulty that Boltzmann machine learning is very time-consuming. However, its theoretical meaning is still not clear. In this paper, based on information geometry, we give an information-theoretic interpretation of mean-field Boltzmann machine learning and a clear geometrical explanation of the approximation used there. Based on this interpretation, computer simulations for evaluating the effectiveness of mean-field Boltzmann machine learning are carried out for two-unit Boltzmann machines. The necessity of geometrical analysis in demonstrating the effectiveness of mean-field Boltzmann machine learning is discussed. © 1999 Scripta Technica, Electron Comm Jpn Pt 3, 82(8): 30–39, 1999 [ABSTRACT FROM AUTHOR]

The time-series analysis (called indirect method) based on the Perron-Frobenius operator is well known as a theoretical method which can determine the statistics for various chaos in one-dimensional distance dynamical systems without determining the trajectory. However, it is rare that a strict solution for the statistics can be determined explicitly and consequently, the indirect method is not practical at present. To remedy this situation, previously we proposed an indirect method based on Galerkin's approximation to the operator, and verified its usefulness. However, the method was not effective for the intermittent chaos with 1/f power spectrum. This paper extends the indirect method using Galerkin's method by introducing the singular function in the approximation to the unbounded invariant density. As a result, the power spectrum is density. As a result, the power spectrum is obtained which agrees fairly well with the data calculated directly from the trajectory over a wide frequency range. An interesting result was obtained by the recent study wherein the spectrum of the intermittent chaos has the form of 1/fdelta; in the limit of zero frequency, where the exponent δ is related explicitly to the nonlinearity of the one-dimensional discrete dynamical system. However, the forementioned result refers only to the spectrum of the chaos in the limit. By contrast, the proposed method can serve as a practical method to estimate the power spectrum of 1/f noise over a wide range. [ABSTRACT FROM AUTHOR]

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 the paper, the authors established two best approximation formulas for the Bateman's G-function. Also, they studied the completely monotonicity of some functions involving G(x). Some new inequalities are deduced for the function and its derivative such as 1/2 ln [1 + 2x + a/x2 + 2x + 4/3 ] < G(x + 2) < 1/2 ln [ 1 + 2x + b/x2 + 2x + 4/3 ], x > 0 where a = 3 and b = e4-16/12 are the best possible constants. Our results improve some recent inequalities about the function G(x). [ABSTRACT FROM AUTHOR]

In this paper, using the concept of A-statistical convergence for double sequences, we investigate a Korovkin-type approximation theorem for sequences of positive linear operator on the space of all continuous real valued functions defined on any compact subset of the real two-dimensional space. Then we display an application which shows that our new result is stronger than its classical version. We also obtain a Voronovskaya-type theorem and some differential properties for sequences of positive linear operators constructed by means of the Bernstein polynomials of two variables. [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]

This paper brings together two fundamental topics: polyhedral projection and parametric linear programming. First, it is shown that, given a parametric linear program (PLP), a polyhedron exists whose projection provides the solution to the PLP. Second, the converse is tackled and it is shown how to formulate a PLP whose solution is the projection of an appropriately defined polyhedron described as the intersection of a finite number of halfspaces. The input to one operation can be converted to an input of the other operation and the resulting output can be converted back to the desired form in polynomial time-this implies that algorithms for computing projections or methods for solving parametric linear programs can be applied to either problem class. [ABSTRACT FROM AUTHOR]

A robust reliable control with integral quadratic constraint (IQC) performance for a class of uncertain systems with state and input delays is considered in this paper. Two classes of failure situations for sensor or actuator are studied. In the first class, a delay-dependent criterion for time-delay systems without perturbations is proposed to design the reliable control with IQC performance. Next, a criterion for uncertain time-delay systems with parameter uncertainties is obtained via simple derivations. The linear matrix inequality (LMI) approach is used to design a robust reliable state feedback control with IQC performance. In the second class, a reliable control with IQC performance is also provided from he previous method. A numerical example is given to illustrate the effectiveness of the procedure. [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]

In this paper, we study Levitin–Polyak type well-posedness for generalized vector variational inequality problems with abstract and functional constraints. Various criteria and characterizations for these types of well-posednesses are given. [ABSTRACT FROM AUTHOR]

The paper considers the problem of finding a bounded solution of a one-parametric family of systems of ordinary differential equations. Using the parametrization method, the author proves necessary and sufficient conditions for the existence of a unique solution of the problem considered that is bounded on the whole axis in terms of a two-sided, infinite block-band matrix composed with respect to integrals over intervals of length h > 0 of the matrix of the system of differential equations. Also, the author constructs a family of two-point boundary-value problems on a finite interval that approximates the problem of finding the bounded solution and finds an interconnection between the correct solvability of the initial and approximating problems. [ABSTRACT FROM AUTHOR]

The problem considered in this paper is the extension of an approximation space to include a nearness relation. Approximation spaces were introduced by Zdzis?aw Pawlak during the early 1980s as frameworks for classifying objects by means of attributes. Pawlak introduced approximations as a means of approximating one set of objects with another set of objects using an indiscernibility relation that is based on a comparison between the feature values of objects. Until now, the focus has been on the overlap between sets. It is possible to introduce a nearness relation that can be used to determine the "nearness" of sets of objects that are possibly disjoint and, yet, qualitatively near to each other. Several members of a family of nearness relations are introduced in this article. The contribution of this article is the introduction of a nearness relation that makes it possible to extend Pawlak's model for an approximation space and to consider the extension of generalized approximations spaces. [ABSTRACT FROM AUTHOR]

In the classical risk model with initial capital u, let τ(u) be the time of ruin, X+(u) be the risk reserve just before ruin, and Y+(u) be the deficit at ruin. Gerber and Shiu (1998) defined the function mδ(u) = E[e-δτ(u)w(X+(u), Y+(u)) 1(τ(u) < ∞)], where δ ⩾ 0 can be interpreted as a force of interest and w(r, s) as a penalty function, meaning that mδ(u) is the expected discounted penalty payable at ruin. This function is known to satisfy a defective renewal equation, but easy explicit formulae for mδ(u) are only available for certain special cases for the claim size distribution. Approximations thus arise by approximating the desired mδ(u) by that associated with one of these special cases. In this paper a functional approach is taken, giving rise to first-order correction terms for the above approximations. [ABSTRACT FROM AUTHOR]

In this paper, we prove direct and inverse theorems of approximation theory in the space of p-absolutely continuous functions which generalize Terekhin’s results in the same way as Timan’s results in L p generalize the classical theorems of approximation theory. The main theorems are refined for functions with quasimonotone Fourier coefficients and, in a number of cases, the resulats are shown to be sharp. [ABSTRACT FROM AUTHOR]

The article analyzes the development of the variational Tikhonov regularization method for ill-posed problems. A. N. Tikhonov published papers on the development of efficient solution methods for a broad class of ill-posed problems in 1963. He was also credited for coming up with the generalization of the notion of approximate solution. Definitions of the regularizing algorithm and regularized solution are provided.

This paper is focused on approximate ( $$\varepsilon$$ -efficient) solutions of multiobjective mathematical programs. We introduce a new $$\varepsilon$$ -efficiency concept which extends and unifies different notions of approximate solution defined in the literature. We characterize these $$\varepsilon$$ -efficient solutions in convex multiobjective programs through approximate solutions of linear scalarizations, which allow us to obtain parametric representations of different $$\varepsilon$$ -efficiency sets. Several classical $$\varepsilon$$ -efficiency notions are considered in order to show the concepts introduced and the results obtained. [ABSTRACT FROM AUTHOR]

Numerical solutions of the differential equations system describing the transitions between energy levels can help in the understanding of the physical mechanisms governing thermoluminescence (TL) emission but they are not suitable for the analysis of complex experimental TL glow curves. On the other hand, simplified descriptions, as mixed or general order kinetics, require many additional assumptions that may limit the validity of the results or are mostly empirical. In this paper, the accuracy of such approximations has been evaluated for different retrapping-recombination ratios and it has been found that differences between the fitted and the simulated parameters arise from the simplification of the models because quasi- equilibrium condition seems to be valid in all the considered cases. [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]

This paper presents a perturbation approach for performing sensitivity analysis of mathematical programming problems. Contrary to standard methods, the active constraints are not assumed to remain active if the problem data are perturbed, nor the partial derivatives are assumed to exist. In other words, all the elements, variables, parameters, Karush-Kuhn-Tucker multipliers, and objective function values may vary provided that optimality is maintained and the general structure of a feasible perturbation (which is a polyhedral cone) is obtained. This allows determining: (a) the local sensitivities, (b) whether or not partial derivatives exist, and (c) if the directional derivative for a given direction exists. A method for the simultaneous obtention of the sensitivities of the objective function optimal value and the primal and dual variable values with respect to data is given. Three examples illustrate the concepts presented and the proposed methodology. Finally, some relevant conclusions are drawn. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the Blackman-type sampling series. In Signal Analysis the Blackman window has been used over 40 years [1], [4]. Main goal of this paper is to present a mathematical treatment of approximation problems by the Blackman-type sampling series. We considered cases when we have a very good order of approximation. In some cases we are able to compute exact values of those operator norms. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present an application of the local linear approximation technique in tracking power system frequency. Samples of the voltage phase angle are fitted using linear function. This approximation is valid locally on the left-sided window with variable length. The window slides with each new sample. The linear coefficient of the locally fitted function is the point estimate of the frequency deviation from the nominal value. For each new sample this technique automatically selects optimal window length in order to maximize estimation accuracy. A long window is selected if the frequency is slow varying, to increase efficiency in filtering noise and harmonics. For a fast varying frequency the window length automatically reduces in order to make frequency tracking more accurate but sacrificing on filtering efficiency. Automatic selection of the optimal window length that balances between tracking and filtering performance makes this technique very powerful in tracking frequency in a wider range, as required in generator control and protection applications. The paper concludes with the presentation of the representative simulation results. [Copyright &y& Elsevier]

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]

In this paper, we present a fast and high-performance algorithm for color filter array (CFA) demosaicing. CFA demosaicing is formulated as a problem of reconstructing correlated signals from their downsampled versions with an opposite phase. The major contributions of this work include 1) a new iterative demosaicing algorithm in the color difference domain and 2) a spatially adaptive stopping criterion for suppressing color misregistration and zipper artifacts in the demosaiced images. We have compared the proposed demosaicing algorithm with two current state-of-the-art techniques reported in the literature. Ours outperforms both of them on demosaicing performance and computational cost. [ABSTRACT FROM AUTHOR]

In this paper we present two approximation algorithms for the permutation flow shop problem with makespan objective. The first algorithm has an absolute performance guarantee, the second one is an $$O(\sqrt {m{\text{ log }}m} )$$ -approximation algorithm. The last result is almost best possible approximation algorithm which can be obtained using the trivial lower bound (maximum of the maximum machine load and the maximum job length) (Potts, Shmoys, and Williamson, 1991). [ABSTRACT FROM AUTHOR]

Let f be an entire function of finite type with respect to finite order p in Cn and let E be a subset of an open cone in a certain n-dimensional subspace R2n (= Cn) (the smaller p, the sparser E). We assume that this cone contains a ray {z = tz0 ∈ Cn: t>0}. It is shown that the radial indicator hf (z0) of f at any point z0 ∈ Cn \ {0} may be evaluated in terms of function values at points of the discrete subset E. Moreover, if f tends to zero fast enough as z → ∞ over E, then this function vanishes identically. To prove these results, a special approximation technique is developed. In the last part of the paper, it is proved that, under certain conditions on p and E, which are close to exact conditions, the function f bounded on E is bounded on the ray. [ABSTRACT FROM AUTHOR]

In this paper, the following inverse heat conduction problem: ut = uxx, x≥0, t≥0, u(x,0) = 0, x≥0 u(1,t) = g(t), t≥0, u&z.sfnc;x→∞ is bounded, is considered again. This problem is severely ill-posed: its solution (if it exists) does not depend continuously on the data; a small perturbation in the data may cause a dramatically large error in the solution for 0 < x < 1. In this paper, a new wavelet regularization method for this problem is given. Moreover, we can easily find the regularization parameter J such that some sharp stable estimates between the exact solution and the approximate one in Hr(R)-norm meaning is given. [Copyright &y& Elsevier]

Inner and outer approximation of numerical ranges of n × n complex matrices and matrix polynomials is investigated in this paper, which is based on the numerical ranges of matrices of smaller or double dimensions. [ABSTRACT FROM AUTHOR]

This paper investigates the performances of dynamic econometric models in relationship to their size. More precisely, the central issue addressed in this paper is whether there exists a procedure that systematically associates with every large-scale model a small-scale model that constitutes a reasonably good approximation of the large-scale model. Such a procedure is shown to exist for both endogenous and exogenous variables. This result is applied to show that a model with approximately twenty endogenous and four hundred exogenous variables can do almost as well as the current models with thousands of variables. This result also implies that a necessary condition for small models of twenty to thirty variables to perform satisfactorily is that only regular patterns of variations for the exogenous variables be considered. [ABSTRACT FROM AUTHOR]

This paper shows that seemingly different 'adjustment' procedures are equivalent, if viewed in appropriate coordinate systems. It extends previous results concerning sequential adjustments that are constrained to be linear functions of observed impact points to the class of translation-invariant procedures. It also reviews properties of the optimal sequential adjustment procedure, including some related to stochastic approximation. [ABSTRACT FROM AUTHOR]

Reliability functions have the property that if they intersect the diagonal y = x it is done in an S-shaped form (Moore and Shannon). In this paper a wider class of polynomials is presented, the members of which form a dense grid on the unit square. The S-shapedness intersection property holds whenever a reliability function intersects a polynomial in this set. The members of this set are all reliability polynomials. The diagonal y = x is also a member of this set, thus a generalization of Moore-Shannon result is achieved. [ABSTRACT FROM AUTHOR]

The viewshed of a point on an irregular topographic surface is defined as the area visible from the point. The area visible from a set of points is the union of their viewsheds. We consider the problems of locating the minimum number of viewpoints to see the entire surface, and of locating a fixed number of viewpoints to maximize the area visible, and possible extensions. We discuss alternative methods of representing the surface in digital form, and adopt a TIN or triangulated irregular network as the most suitable data structure. The space is tesselated into a network of irregular triangles whose vertices have known elevations and whose edges join vertices which are Thiessen neighbours, and the surface is represented in each one by a plane. Visibility is approximated as a property of each triangle: a triangle is defined as visible from a point if all of its edges are fully visible. We present algorithms for determination of visibility, and thus reduce the problems to variants of the location set covering and maximal set covering problems. We examine the performance of a variety of heuristics. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to describe a new gradient method for maximizing general functions. After a brief discussion of various known gradient methods the mathematical foundation is laid for the new algorithm which rests on maximizing a quadratic approximation to the function on a suitably chosen spherical region. The method requires no assumptions about the concavity of the function to be maximized and automatically modifies the step size in the light of the success of the quadratic approximation to the function. The paper further discusses some practical problems of implementing the algorithm and presents recent computational experience with it. [ABSTRACT FROM AUTHOR]