Showing papers on "Linear approximation published in 1970"

Two Simple Algorithms for Discrete Rational Approximation

Abstract: This paper reports on computational experience with algorithms due to Loeb and Appel for rational approximation on discrete point sets. Following a brief review of the linear discrete approximation problem, the two rational algorithms are stated in a general setting. Finally, several numerical examples of applications to 1l, 12, and 14 approxi- mation are supplied and discussed. 1. Introduction. Rational functions can often provide very satisfactory approxi- mations to discrete data. However, as with most other nonlinear approximating functions, effective algorithms that produce best discrete rational approximations are few in number and are often complicated or time-consuming. The purpose of this paper is to give further exposure to two rational algorithms due to Loeb and Appel, to extend their applicability to each of the three norms 11, 12, and I., and to test their effectiveness on a variety of problems. Both methods are simple in the sense that they employ only a linear approximation algorithm and possibly a straight- forward iteration. For the sake of completeness, the remainder of this introductory section consists of some remarks on the general problem of best approximation on a discrete point set. Given a set X = {xl, x2, * * *, xv} of real numbers and a function f(x) defined on X, we choose an approximating function F(A, x) and select a particular form F(A*, x) which approximates f(x) satisfactorily on X, according to some criterion. Here, A = {al, a2, * . , an} is a set of free parameters, and F(A, x) is a linear ap- proximating function only if it depends linearly upon these parameters. Thus, a rational function F(A, x) = (a, + a2x)/(1 + a3x) is nonlinear, and the most general linear function is F(A, x) = E-, ajoi(x), where the 47(x)'s are given linearly in- dependent functions defined on X. F(A*, x) is called a best approximation in a norm I I 1I1I if, for all choices of A, I If(x) - F(A*, x)I I <| IIf(x) - F(A, x)I 1. The three norms used in practice are: N ( TV ) ~~~~~~~1/2

Equation for the Shock Adiabat

Abstract: The main difficulty in using a linear relationship to fit the experimental data on shock and particle velocities is the fact that the Hugoniot pressure tends to infinity as the relative compression tends to a maximum whose value depends on the material. It is shown in this paper that this difficulty is overcome by the use of a quadratic fit, since this leads to an expression free from infinity points for the Hugoniot as a function of volume. It is demonstrated, by the introduction of complex variable methods, that in the quadratic approximation the shock adiabat is a two branch function, with the branch point located precisely at the infinity point resulting from the linear approximation. Series expansions of the Hugoniot suggest the introduction of a quasilinear approximation as a limiting case. Numerical results obtained using the quadratic approximation for iron and cadmium are in very good agreement with the experimental data. The point of highest pressure reported for cadmium confirms the existence of the two branches for the Hugoniot.

Computation of best one-sided ₁ approximation

Abstract: A computational procedure based on linear programming is presented for finding the best one-sided L1 approximation to a given function. A theorem which ensures that the computational procedure yields approximations which converge to the best approximation is proved. Some numerical examples are discussed.

Critical Phenomena and Zeroes of the Partition Function in Two Complex Variables

Abstract: By an extension of the Yang-Lee picture of phase transitions to the case of two complex variables, we have been able to study the properties of a ferromagnetic system in the vicinity of the critical point. A linear approximation is used, which yields a satisfactory result close to that point. Especially, we see how a critical point and a phase boundary can arise from statistical mechanical laws. It is shown that scaling laws are generally valid, although there may be special cases when they are not.

Letter to the Editor—On Stochastic Linear Approximation Problems

Abstract: This paper considers the extension of the linear approximation problem minimize ‖ϵ‖ subject to AX + ϵ = b to the case where the elements of b are independent random variables with known distributions. This extension is accomplished by the use of chance constraints. An analysis of this stochastic problem shows that the problem can be solved by some of the powerful computational methods of approximation theory.

A simple linear approximation for perturbed motion about moderately elliptic orbits.

Abstract: Moderately elliptic reference orbit perturbed motion, applying linearized perturbation equations for circular orbit

The variance function of the Erlang process

Abstract: A relatively simple exact expression of closed form is obtained for the varianceσ 2(t) of the asynchronous counting distribution for a counting period of lengtht,t>0, in an Erlang process. Useful bounds are placed upon the error of the linear approximation toσ 2(t). Implications of these results are examined. In particular, a new exact expression and related bounds are obtained for the mean function of the synchronous counts (also known as the renewal function of the process). All bounds given are sharp in asymptotic order of magnitude as the length of the counting period is allowed to increase.

Ray Tracing with Horizontal and Vertical Gradients

Abstract: Ray paths are found in a medium with a sound speed varying in two dimensions. The velocity field is approximated in a triangular region by a linear function of range and depth. The assumed velocity is continuous, but the gradient of the velocity is discontinuous across the boundaries of the triangular region. Results are compared with exact solutions for a tilted bilinear profile, and tilted and nontilted hyperbolic cosine profiles. The errors in ray position are due to the linear approximation of the velocity, not to the existence of horizontal gradients.

Optimum line shape analysis by least-square approximation and linear transformation processes

Abstract: The problem of optimum analysis of overlapping signals by means of least-square approximation techniques and by linear transformation processes, like resolution enhancement by inverse convolution, is considered. It is shown that optimum accuracy is obtained by employing, before doing a least-square fit, a transformation process which transforms the random noise into white noise. This particularly implies that in any process of artificial resolution enhancement, a loss of accuracy occurs. Examples from the field of spectroscopy are given to demonstrate the achievable accuracy and the requirements with respect to signal-to-noise ratio.

Generalized Linear Trajectory Approximations and the Constant Acceleration Approximation

Abstract: The evaluation of correlation functions by means of approximation of the time evolution operator is discussed It is shown that different approximations may be obtained depending upon the particular factorization of the equilibrium distribution function in the averages to be computed With the approximation of free particle dynamics, generalized linear trajectory approximations for correlation functions are obtained The circumvention in the generalized approximation of the separation of the intermolecular potential employed in the linear trajectory approximation introduced by Helfand is discussed For low density, it is demonstrated that the constant acceleration approximation introduced by Bloom and Oppenheim is exactly equivalent to a generalized linear trajectory approximation An explicit expression for the deviation of the constant acceleration approximation result from the exact correlation function expression is obtained The differences between the constant acceleration and generalized linear trajectory approximations at higher densities are discussed

An extended criterion for statistical linearization

Abstract: The random-input describing function is the best linear approximation to an instantaneous nonlinearity in the sense of minimum mean-squared error. It is shown here that the describing function also satisfies a more comprehensive criterion which is important to the accuracy of approximate analysis and synthesis techniques for systems with nonlinear elements.

An approximation with poles along the negative real axis of the complex frequency plane

Abstract: A time domain approximation of the type e-etis developed and shown to be a rational function of s in the frequency domain. This approximation is considered as a transfer function which may be realized with RC or RL elements. Two examples illustrate the approximation method.