Showing papers on "Linear approximation published in 1976"
TL;DR: An efficient algorithm for obtaining solutions is given and shown to be closely related to a well-known algorithm of Levinson and the Jury stability test, which suggests that they are fundamental in the numerical analysis of stable discrete-time linear systems.
Abstract: It is common practice to partially characterize a filter with a finite portion of its impulse response, with the objective of generating a recursive approximation. This paper discusses the use of mixed first and second information, in the form of a finite portion of the impulse response and autocorrelation sequences. The discussion encompasses a number of techniques and algorithms for this purpose. Two approximation problems are studied: an interpolation problem and a least squares problem. These are shown to be closely related. The linear systems which form the solutions to these problems are shown to be stable. An efficient algorithm for obtaining solutions is given and shown to be closely related to a well-known algorithm of Levinson and the Jury stability test. The close connection between these algorithms suggests that they are fundamental in the numerical analysis of stable discrete-time linear systems.
196 citations
TL;DR: In this article, a two-variable expansion technique is used to solve for the mean Green's function from the Dyson equation under the nonlinear approximation, and the modified radiative transfer (MRT) equations give rise to simple and useful solutions for both active and passive microwave remote sensing.
Abstract: A two-variable expansion technique is used to solve for the mean Green's function from the Dyson equation under the nonlinear approximation. The Bethe-Salpeter equation then gives rise to a set of modified radiative transfer (MRT) equations which accommodate coherent effects essential to bounded media. It is found that the nonlinear approximation, instead of the more popular bilocal approximation, should be used for the case of bounded media. The two approximations yield identical results for unbounded media. The MRT equations are then solved for a two-layer random medium. The MRT equations give rise to simple and useful solutions which are applicable to both active and passive microwave remote sensing.
38 citations
26 citations
TL;DR: A fast scheme using a sphtand-merge procedure is described for functions of both one and two variables for piecewise constant and piecewise linear approximations without continuity constraints.
Abstract: Algorithms for piecewise approximations with variable breakpomts can be used for picture segmentation. The high computational requirements of many of the algorithms whmh search for optimal solutions make them unsuitable for such apphcations A fast scheme using a sphtand-merge procedure is described for functions of both one and two variables. Examples of its application on pictures are given. The present implementation is only for piecewise constant and piecewise linear approximations without continuity constraints. I t can be readily extended to higher order approximations without constraints. Theoretically the spht-and-merge procedure can also be used with continuity constraints.
21 citations
TL;DR: A variational approach in conjunction with the finite element method is used to solve the groundwater flow equation in this article, and Galerkin's approach coupled with the FEM method was used to calculate the hydraulic head distribution and concentration of solute in the flow region.
Abstract: . Transient, two-dimensional solutions are developed which describe the movement and distribution of a conservative substance in a stream-aquifer system. The solutions are obtained by solving sequentially the groundwater flow and mass transport equations. A variational approach in conjunction with the finite element method is used to solve the groundwater flow equation. Galerkin's approach coupled with the finite element method is used to solve the mass transport equation. Linear approximated triangular elements and a centered scheme of numerical integration are employed to calculate the hydraulic head distribution and the concentration of solute in the flow region. The linear approximation used to define the concentration function within each element is not appropriate for cases involving steep concentration gradients. For such cases, higher order approximations are necessary to assure the continuity of gradients across interelemental boundaries. Numerical examples that illustrate the applicability of the model are presented.
9 citations
01 Jan 1976
TL;DR: In this article, a short proof is given for the pointwise estimate of | f ( t ) − φ n ( f ; t )| ( t ∈ [ a, b ] in terms of the second modulus of continuity, i.e. with increment depending on t.
Abstract: This note is concerned with positive linear approximation processes φ n ( f ; t ) ( n = 1, 2, …) on C [ a , b ] which are exact for the functions l, t . A short proof will be given for the pointwise estimate of | f ( t ) − φ n ( f ; t )| ( t ∈ [ a , b ]) in terms of the second modulus of continuity, i.e. with increment depending on t .
8 citations
TL;DR: In this article, linearized equations of magnetohydrodynamics are used to investigate the tearing mode, for arbitrary values of the conductivity, through a consideration of the additional effect of the electron-inertia contribution to Ohm's law.
Abstract: Linearized equations of magnetohydrodynamics are used to investigate the tearing mode, for arbitrary values of the conductivity, through a consideration of the additional effect of the electron-inertia contribution to Ohm's law. A description is provided of the equilibrium and subsequent instability in the magnetohydrodynamic approximation. A method for solving the perturbation equations in the linear approximation is discussed and attention is given to the results in the high conductivity limit.
7 citations
TL;DR: A linear approximation to the second-derivative matrix in terms of first derivatives of residuals is found to be very accurate in the neighborhood of minima; it provides a way of introducing second-Derivative information that is significantly superior to the use of variable-metric algorithms.
6 citations
TL;DR: In this paper, the angular distribution of the molecules leaving the channel is calculated by using the wall collision rate as an intermediate result, and the distribution of transmitted reflected molecules with respect to the number of wall collisions in the channel was calculated.
Abstract: Molecular flow in a cylindrical channel is investigated with a Monte-Carlo method by tracing the random walk of the molecules in the channel. The angular distribution of the molecules leaving the channel is calculated by using the wall collision rate as an intermediate result. The Monte-Carlo wall collision rate is compared with a linear and a cubic approximation to wall collision rate, calculated by DeMarcus and Neudachin with a variational method. For short channels the linear approximation gives a satisfactory description of the Monte-Carlo results for the angular distribution, for long channels deviations of a few percent occur at small angles. Using the cubic approximation will decrease these deviations to less than one percent. The distribution of the transmitted reflected molecules with respect to the number of wall collisions in the channel is calculated. These collision number distributions help to achieve insight in the perturbation of the flow by nonideal conditions as adsorbing walls.
6 citations
TL;DR: In this article, the stability of two-and three-dimensional higher-order modes of a nonlinear wave equation against exponentially increasing perturbations was studied and the conditions under which higher order modes can be observed were derived.
Abstract: The linear approximation is used to study the stability of two- and three-dimensional higher-order modes of a nonlinear wave equation against exponentially increasing perturbations. For all the nonlinear models considered the higher modes are unstable; the number of exponentially increasing perturbations and their growth rate are determined by the mode number and the form of the nonlinear relationship. Numerical tests are described in the parabolic approximation on the stability of the first axially symmetric mode against small amplitude perturbations and the conditions are determined under which higher-order modes can be observed.
5 citations
TL;DR: In this paper, it is shown how the scattering angle of an ion by a Moliere potential can be calculated by linear interpolation, and the maximum error introduced by the linear approximation is calculated as a function of the interval width used for interpolation.
Abstract: In the context of a Monte Carlo simulation of channeling, it is shown how the scattering angle of an ion by a Moliere potential can be calculated by linear interpolation. In the case of impulse approximation, this reduces the computing time to ≃1/17 of that needed by the usual procedure. The maximum error introduced by the linear approximation is calculated as a function of the interval width used for interpolation. Using Firsov's inversion formula, the corresponding approximation to the atomic potential is also evaluted. It is shown that such approximation is very good, if compared with the present knowledge of atomic potential.
TL;DR: This paper describes how the method of inequalities may be applied to the design of multi input multi output feedback systems which contain a set of single valued bounded non linear elements.
01 Jan 1976
TL;DR: When the nonlinear approximation problem is treated by using Newton's method, at each iteration step the solution of a linear approximation problems is required, so for generating a more effective algorithm the latter problem is solved only on a finite point set.
Abstract: When the nonlinear approximation problem is treated by using Newton's method, at each iteration step the solution of a linear approximation problem is required. If we are concerned with nonlinear Chebyshev approximation, the (linear) auxiliary problem is also non trivial. Thus for generating a more effective algorithm the latter problem is solved only on a finite point set. However, then we must not only choose reference points like in Remez-type algorithmus; the reference set has to be augmented in order to take care of numerical stability.
TL;DR: An algorithm for computing a linear Chebyshev approximation to a function defined on a finite set of points is presented, based upon the simplex method of linear programming.
Abstract: An algorithm for computing a linear Chebyshev approximation to a function defined on a finite set of points is presented. The method requires the accuracy of the approximation to be specified, and determines the least degree approximation which achieves this accuracy. The algorithm is based upon the simplex method of linear programming. A FORTRAN program is supplied in the Appendix.
TL;DR: The alternative quadratic approximation which is proposed does not suffer this deficiency, but still enjoys the advantages of higher order, and obeys very well an h 4 Richardson extrapolation, whereas the old one does so rather poorly.
Journal Article•
01 Jan 1976
TL;DR: The authors discusses the global convergence behavior of iterative algorithms solving nonlinear approximation problems via a sequence of linear approximation problems and provides sufficient conditions for global convergence with respect to critical points as accumulation points.
Abstract: This contribution discusses the global convergence behavior of iterative algorithms solving nonlinear approximation problems via a sequence of linear approximation problems. Essentially there are two sufficient conditions for global convergence:
1)
The algorithm should have only critical points as accumulation points.
2)
The parametrization should imply that the parameters of refined approximations have accumulation points.
TL;DR: In this paper, a generalization of hypo-elasticity to include thermal and memory effects is investigated theoretically, and the set of the constitutive equations are regarded as a linear transformation between seven-dimensional spaces composed of stretching and temperature rate, and of stress rate and entropy density rate.
Abstract: A generalization of hypo-elasticity to include thermal and memory effects is investigated theoretically. The set of the constitutive equations are regarded as a linear transformation between seven-dimensional spaces composed of stretching and temperature rate, and of stress rate and entropy density rate. The failure conditions are defined by the linear operator. The linear approximation of the constitutive equations for strain and temperature histories are presented. A special case of the dependence on the histories of deformation and temperature, called the function type, is proposed.
Dissertation•
01 Jan 1976
TL;DR: Weinstein and Henry as mentioned in this paper considered uniform extensions of the Freud's theorem for both the generalized, polynomial and the rational settings, and showed that both methods of product approximation possess desirable properties analogous to the classical theory for univariate Tchebycheff approximability.
Abstract: Let (cp^,. . . ,cp̂ } and { be riand m-dimensional Haar sets defined on the intervals I = [a,b ] and. J = [c,d], respectively. The (linear) product approximation of F e C(D), D = I x J 5 is defined, to be (PF) (x5y) „ n v m. „ n = L 1=1 L 1=1 Z1^fAyh1(X) where 2, i=1 fi (y)cpi (x) is the IjVj best approximation of F ^ (x) F(x5y) from 0 = span (cp̂ 5. . . 5cpn ] . with respect to the uniform norm and. Zj loeŝf uniform approximation of f± (y) over J from I = span [flS...3fm] 5 i = I5...5n. The rational product approximation of F e . C(D) is defined in a similar fashion. It is shown that both methods of product approximation possess desirable properties analogous to the classical theory for univariate Tchebycheff approxima tion. This study contributes to the recent developments in multivariate approximation theory initiated by 8 . E. Weinstein and. M. S . Henry. As the above considerations involve univariate Tchebycheff approximation, some results in this area are also established. In particular, uniform extensions of the classical Freud's theorem are proven for both the generalized, polynomial and. the rational settings.
TL;DR: In this paper, the authors used a linear approximation to solve the problem of the frictional resistance of an unbounded plate moving in an arbitrary manner along the boundary of a half-space occupied by a viscoelastic liquid of general type.
Abstract: The theory of linear viscoelastic liquids is developed by analogy with the theory of linear hereditary media in electrodynamics [1]. In a simplified description of the reduction of turbulent friction resistance in viscoelastic polymer solutions an important place is occupied by the problem of a plane moving parallel to itself [2–10]. Thus, in order to determine the damping of turbulence near a wall according to the Van Drist model, harmonic oscillations of a plate are considered [2]. In the Einstein-Lee model and the Dankwerts-Hanratty model the solution of the problem of flow near a suddenly starting plane is used. In the present study, using a linear approximation, we solve the problem of the frictional resistance of an unbounded plate moving in an arbitrary manner along the boundary of a half-space occupied by a viscoelastic liquid of general type. We find the consequences of the causality of the initial determining equations of the liquid and show that the resistance in viscoelastic liquids is less than in a viscous liquid. We give the results of calculations for the simplest models of a liquid.
TL;DR: In this paper, an explicit formula expressing the error of a linear approximation to a given function is given, which gives a simple method of choosing the breakpoints of a piecewise linear approximation.
TL;DR: This paper presents a number of auxiliary problems with non-linear approximation of the constraints, whose solution is obtained analytically.
Abstract: IT IS proposed to use the conjugate gradient method in linear and non-linear programming problems and to estimate the Lagrange multipliers from the analytic solution of subsidiary problems. Recently iterative methods of solving non-linear programming problems using various auxiliary problems [1–5] to construct the vector step, have become more and more popular. As a rule these problems are obtained by linear [1] or non-linear [2,3] approximation of the target function and by linear approximation of the constraints. These problems are usually solved by applying the corresponding algorithms, which considerably reduces their domain of rational application [2]. In this paper we present a number of auxiliary problems with non-linear approximation of the constraints, whose solution is obtained analytically.