Showing papers on "Minimax approximation algorithm published in 1972"

Synthesis of recursive digital filters using the minimum p-error criterion

Abstract: The problem of designing a stable recursive digital filter to have an arbitrarily prescribed frequency response may be considered as an approximation problem. Using the minimum p - error criterion, a new problem of minimizing a function of n variables results, which is successfully solved using the Fletcher-Powell algorithm. An important theorem guaranteeing the existence of a stable optimum for a large class of synthesis problems is stated, and necessary modifications to the Fletcher-Powell algorithm to assure stability are considered. Finally a number of results of the application of this method are given.

A Course on Optimization and Best Approximation

Abstract: Preliminaries- Theory of optimization- Theory of best approximation- Comments on the problems- Selected special topics

Theory of Semiclassical Transition Probabilities for Inelastic and Reactive Collisions. V. Uniform Approximation in Multidimensional Systems

Abstract: An integral semiclassical expression for the S matrix of inelastic and reactive collisions was formulated earlier in this series. In the present paper a uniform approximation for the expression is derived for the case of multidimensional systems. The method is an extension of that employed in Part II for the case of one internal coordinate. The final result, Eq. (2), is highly symmetrical, thus making some of its properties immediately clear.

Rate of Convergence of Lawson's Algorithm

Abstract: The algorithm of Charles L. Lawson determines uniform approximations of functions as limits of weighted L2 approximations. Lawson noticed from experimental evi- dence that the algorithm seemed to converge linearly and convergence was related to a factor which was the ratio of the largest nonmaximum error of the best uniform approximation to the maximum error. This paper proves the linear convergence and explores the relation of the rate of convergence to this ratio. 1. Introduction. In his Ph.D. dissertation of 1961, Charles L. Lawson discussed an algorithm for solving uniform approximation problems by means of limits of weighted p-norm solutions. Since then, this algorithm has been explored further by several authors. The algorithm is mentioned in Rice (3), and a variation on Lawson's algorithm was shown to produce p-norm approximations (p > 2) as a limit of weighted L2 norm solutions in Rice and Usow (4). In the Ph.D. dissertation of this author (1), Lawson's algorithm (originally defined for approximation on finite sets) was extended to the case of approximation on compact Hausdorff spaces. Presently, attempts are underway extending Lawson's algorithm in a different fashion for solving L, approximation problems. In his dissertation, Lawson gave conditions for convergence of the weighted LP solutions to the uniform solution. In some cases (theoretically possible but com- putationally highly unlikely), the algorithm may have to be restarted a finite number of times before it converges to the proper solution. When it converges to the uniform solution, Lawson noticed experimental results indicating linear convergence with a convergence factor linked to a certain ratio of error at a point to maximum error of the solution. It is the purpose of this paper to show that the Lawson algorithm does have linear convergence and demonstrate the importance of the convergence factor which

Converse theorems and extensions in Chebyshev rational approximation to certain entire functions in [∗∗∗(∗∗0,+∞)∗∗∗∗∗

Abstract: Recently, it has been shown that the problem of rational approximation to e~~ in [O, + <*>) arises naturally in numerical methods for approximating solutions of heat-conduction-type partial differential equations [l] , [2]. This special approximation problem leads one to the general question of approximating functions on the half line [0, + 00 ). In this paper we wish to announce two results of this study which are in the spirit of work done by S. N. Bernstein. A complete description of this work with proofs of these results and additional results will appear elsewhere. In order to state these results, we need the following notation. For any nonnegative integer w, let irm denote the collection of all real polynomials of degree at most m. For given r>0 and s> 1, let S(r, s) denote the unique open ellipse in the complex plane with foci at x = 0 and x = r and semimajor and semiminor axes a and b such that b/a = (s — l)/(s+l). Finally, if f(z) is any entire function, we set

Minimax solution of the multiple-target problem

Abstract: The convexization procedure developed for a class of minimax problems is applied to the determination of the minimax solution of the multiple-target problem. The method is parallel to that used in earlier works, but the results are completely independent. It is shown that the state space may be partitioned into subregions in which the minimax strategy is a pure strategy and into subregions in which it is a mixed strategy in the terminology developed in the theory of games, which aptly characterizes the nature of the minimax solution in this problem. It is also shown that the minimax strategy in open-loop form is a piecewise linear function of the initial state and a linear function of the state along the resulting trajectory. In feedback form, it is a piecewise linear function of the state and the cost incurred in the elapsed interval of play.

Minimax Optimization of Networks by Grazer Search

Abstract: A new optimization method called grazer search has been developed. This method is suitable for nonlinear minimax optimization of network and system responses. A linear programming problem using gradient information of one or more highest ripples in the response error function to produce a downhill direction followed by a linear search to find a minimum in that direction is central to the algorithm. Unlike the razor search method due to Bandler and Macdonald, the present method overcomes the problem of discontinuous derivatives characteristic of minimax objectives without using random moves. It can fully exploit the advantages of the adjoint network method of evaluating partial derivatives of the response function with respect to the variable parameters. Sufficient details are given to enable the grazer search method to be readily programmed and used. Although the method is intended for the computer-aided solution of an extremely wide range of design problems, it is largely compared with other methods on microwave network design problems, for which the solutions are known. Its reliability and efficiency on more arbitrary problems, examples of which are also included, is thereby established.

On a class of best nonlinear approximation problems

Abstract: i = 0 \\ i = 0 \\ k = l 1 = 0 \\ where the knots a, /? and rjkeT are of multiplicity n, m and /ik respectively, and the total multiplicity of the interior knots is stipulated to be YJ=i fa — > The functions (2) display each rjt as a simple knot with the terms involving the knots a and jS omitted. The class of functions of the form (3) are designated as ^ m , r . Our main objective is to characterize the best approximation to h(Ç) in the L(T) norm from among the functions in ^„,m>^. Formally stated, we wish to establish criteria for evaluating {at}, {bj}, {ct} and {rjt} achieving

A minimax filter for systems with large plant uncertainties

Abstract: A minimax filter is derived in order to estimate the state of a system when large uncertainties in the plant dynamics and process noise are present. If the system dynamics and measurements are uncoupled and the noise covariance matrices are diagonal, simple results occur.

On the normal approximation for a certain class of statistics

Abstract: We consider the class C(a) of asymptotic tests proposed by Neyman [6]. The term of order n 1/2 in the normal approximation for the distributions of the test statistics is obtained. Moreover several theorems on conditional distributions are proved. They are used in deriving the main result but they also seem to be of independent interest. Neyman [6] proposed a class of asymptotic tests called C(a) for the following statistical problem. Let a random variable (r.v.) X have a distribution depending on parameters 0 = (O., , O.) and 4 which take their values in open sets O c RS and R1 respectively. (We denote by Rs. s = 1. 2. , the space of real row vectors x = (x1, l , x5) with the Euclidean norm ||x|| = (xx')"I2, a prime denoting the transposition.) The hypothesis H: = 0, where OCE is a specified value, is to be tested on the basis of n independent observations X1, *-, Xn of the r.v. (In the sequel, we put '0 = 0.) The distribution of X is assumed to have a density f(x; 0, 4) (with respect to an appropriate measure) which satisfies certain regularity conditions. The C(a) tests are constructed as follows. Let a function g(x, 0) be such that

A unified approach to uniform real approximation by polynomials with linear restrictions

Abstract: Problems concerning approximation of real-valued continuous functions of a real variable by polynomials of degree smaller than n with various linear restrictions have been studied by several authors. This paper is an attempt to provide a unified approach to these problems. In particular, the notion of restricted derivatives approximation is seen to fit into the theory and includes as special cases the notions of monotone approximation and restricted range approximation. Also bounded coefficients approximation, c-interpolator approximation, and polynomial approximation with interpolation fit into our scheme.

The convergence of rational functions of best approximation to the exponential function. II

Abstract: The object of the paper is to establish convergence throughout the entire complex plane of sequences of rational functions of prescribed types which satisfy a certain degree of approximation to the function aeyz on the disk \z\ S p. It is assumed that the approximating rational functions have a bounded number of free poles. Estimates are given for the degree of best approximation to the exponential function by rational functions of prescribed types. The results obtained in the paper imply that the successive rows of the Walsh array for aeyz on |z|ap converge uniformly to aerz on each bounded subset of the plane.

Uniform approximation of the remainder term in the dirichlet divisor problem

Abstract: In this paper we study the average value of the function τk(n), the number of representations of n as a product of k natural factors, n≤x, with a remainder term which is uniform in x and k.

Inequalities for differentiable periodic functions and best approximation of one class of functions by another

Abstract: New results are obtained in this paper which elucidate properties of differentiable periodic functions connected with rearrangements. These results are applied in order to obtain a sharp estimate of the best uniform approximation of functions of the class by functions of the class .

On decision-directed estimation and stochastic approximation (Corresp.)

Abstract: A decision-directed scheme for estimating the mean is expressed as a stochastic approximation algorithm. The algorithm is shown to converge, but not to the true value, by means of the theory of stochastic approximation. A modification of the algorithm that converges to the true value is presented.

Chebyshev approximation of a constant group delay with constraints at the origin

Abstract: This paper deals with low-pass filter functions approximating a constant delay in an equiripple manner which does not yield a standard delay error curve. This type of Chebyshev approximation is obtained by imposing a constraint on the error curve at \omega = 0 . It is shown that using the constrained approximation, the delay approximation bandwidth for n odd and a prescribed ripple factor \epsilon may be equal to, or even larger than, that obtained by Abele's polynomials; the latter solution is neither unique nor the best approximation. The magnitude characteristics of the constrained approximants are very much improved and the transient responses to a unit step input compare favorably with those for the other known systems including Schussler's functions with equiripple step response. Tables are presented which include the pole locations of some selected constrained approximants of 3, 5, 7, and 9 degrees, the comparative stopband attenuation relative to the Abele case, and the most important quantities associated with a step response.

Approximation by Alternating Families on Subsets

Abstract: A method of determining the best approximation by an alternating family on an interval is by approximating on finite subsets of the interval. In this note we show that this method can fail to converge, particularly in the case of polynomial rational approximation and exponential approximation when the best approximation is degenerate.

Minimax Problems, Saddle-Functions and Duality

Abstract: : Minimax problems are fundamented to nonlinear programming, because of the way constraints can be represented using Lagrange multipliers. Better ways of solving minimax problems would lead thus lead to breakthroughs in solving most other problems of optimization. The dissertation opens a new avenue to the study of minimax problems by developing a theory of dual operations on saddle- functions convex-concave functions parallel to that already known for (purely) convex functions. Results are thereby obtained concerning minimax problems which are dual to each other. It is expected that these results will find computational applications analogous to those already acclaimed in the convex case, for instance in decomposition of large-scale problems.