Showing papers on "Linear approximation published in 1978"
TL;DR: In this article, a review of analytical methods in electromagnetic scattering theory (i.e., geometrical and physical optics, perturbation, iteration, and integral-equation) which are applicable to the problems of remote sensing of the ocean is presented.
Abstract: This paper reviews analytical methods in electromagnetic scattering theory (i.e., geometrical and physical optics, perturbation, iteration, and integral-equation) which are applicable to the problems of remote sensing of the ocean. In dealing with Earth's surface (in this case, the weakly non-linear ocean), it is not possible to have a complete and exact description of its spatial and temporal statistics. Only the first few moments are generally available; and in the linear approximation the statistics are assumed homogeneous, stationary and Gaussian. For this case, the high-frequency methods (geometrical and physical optics) and perturbation (Rayleigh-Rice), or a combination of them, provide tractable analytical results (i.e., the specular-point, the slightly-rough Bragg scattering and the composite-surface models). The applicability and limitations of these models are discussed. At grazing incidence and for higher frequencies, other scattering mechanisms become significant; and shadowing, diffraction and trapping must be considered. The more exact methods (integral-equation and Green's function) have not been as successful in yielding tractable analytical solutions, although they have the potential to provide improved theoretical scattering results in the future.
1,003 citations
TL;DR: In this article, an infinite partial summation of the perturbation expansion for the so-called self-energy part of the Green's function is presented. And a spin-free formulation of the working equations for molecular applications is presented for a theoretical treatment of inner valence ionisation processes.
Abstract: An approximation scheme for the one-particle Green's function referred to as the two-particle-hole Tamm-Dancoff approximation (2ph-TDA) is introduced by means of a well defined infinite partial summation of the perturbation expansion for the so-called self-energy part. A spin-free formulation of the working equations is presented for molecular applications. It is demonstrated that the 2ph-TDA is a useful tool for a theoretical treatment of inner valence ionisation processes. A discussion of the physical content and the relationship to other approaches shows the central role of this approximation.
138 citations
TL;DR: The numerical results reported here, combined with the fact that in the absence of constraints the present algorithm reduces to the earlier unconstrained $l_1$ algorithm, indicate that this algorithm is very efficient.
Abstract: We describe an algorithm, based on the simplex method of linear programming, for solving the discrete $l_1$ approximation problem with any type of linear constraints. The numerical results reported here, combined with the fact that in the absence of constraints the present algorithm reduces to our earlier unconstrained $l_1$ algorithm, indicate that this algorithm is very efficient.
92 citations
61 citations
TL;DR: Convergence of the algorithm to an epsilon-optimal solution is proven and its effectiveness is demonstrated by solving a number of location problems and linear approximation problems.
Abstract: We present a subgradient algorithm for minimizing the maximum of a finite collection of functions It is assumed that each function is the sum of a finite collection of basic convex functions and that the number of different subgradient sets associated with nondifferentiable points of each basic function is finite on any bounded set Problems belonging to this class include the linear approximation problem and both the minimax and minisum problems of location theory Convergence of the algorithm to an epsilon-optimal solution is proven and its effectiveness is demonstrated by solving a number of location problems and linear approximation problems
47 citations
TL;DR: A mixed method is proposed that can present the main advantages of both classical methods in a particular case of mathematical programming methods that use a linear approximation of constraints expressed in the space of the inverse of the design variables.
Abstract: It is shown how existing methods based on optimality criteria can be generalized using a first order approximation for all the constraints. In this case, these methods can be presented as a particular case of mathematical programming methods that use a linear approximation of constraints expressed in the space of the inverse of the design variables. A mixed method is proposed that can present the main advantages of both classical methods.
37 citations
34 citations
TL;DR: It is shown that the Riccati equation for the linear approximation is solved by simple transformation of the averaging solution, and the numerical procedures are identical, except for the computation of an initial condition.
Abstract: A certain approximation scheme based on "piecewise linear" approximations of L 2 spaces is employed to formulate a numerical method for solving quadratic optimal control problems governed by linear retarded functional differential equations. This piecewise linear method is an extension of the so-called averaging technique. It is shown that the Riccati equation for the linear approximation is solved by simple transformation of the averaging solution. In fact, the numerical procedures are identical, except for the computation of an initial condition. Numerical examples are included.
14 citations
01 Jan 1978
TL;DR: This work considers problems arising in the determination of best approximations to complex-valued data and focuses on linear approximation in the l1 and l∞ norms.
Abstract: We consider problems arising in the determination of best approximations to complex-valued data The emphasis is on linear approximation in the l1 and l∞ norms, but some remarks on l∞ rational approximation are also included
4 citations
4 citations
TL;DR: In this paper, the authors proposed a linear approximation for a perturbed set of stochastic dynamic equations aimed at reducing computation time, which is based upon an assumption of small departures of the stochiastic dynamic solution from the usual deterministic forecast.
Abstract: The stochastic dynamic method for optimum weather forecasting and forecast variance estimation suffers from computational complexity sufficient to obviate its utility as an operational tool. The present paper suggests the use of a linear approximation for a perturbed set of stochastic dynamic equations aimed at reducing computation time. The linearization is based upon an assumption of small departures of the stochastic dynamic solution from the usual deterministic forecast. The resultant linear scheme was tested by evaluating its predictions for a simple mathematical model of the atmosphere, that described by Lorentz's “minimum hydrodynamic equations”. The linearization was found to be as effective as was the standard non-linear stochastic dynamics method, which approximates the exact stochastic dynamic solution by ignoring third-order moments. The problem of developing efficient numerical algorithms that take advantage of the simplification attendant upon linearization is not taken up in this paper, but will be reported upon as the work is pursued. DOI: 10.1111/j.2153-3490.1978.tb00856.x
TL;DR: The algorithm is applicable to linear approximation from Haar and weak Chebyshev families, and the nonlinear problems generated by certain $\gamma $-polynomials, including splines with free knots.
Abstract: We develop an algorithm, based on the continuation method, for computing an improved Chebyshev approximation to a function of class $C^2 [0,1]$. The algorithm is applicable to linear approximation from Haar and weak Chebyshev families, and the nonlinear problems generated by certain $\gamma $-polynomials, including splines with free knots. Convergence to a local best approximation is proved under appropriate conditions. This method is suggested as a candidate for an interactive algorithm for computing best nonlinear approximations, since it will produce improved approximations even if it cannot be iterated to convergence. Numerical results will be given in a subsequent paper [15].
TL;DR: Numerical results on two examples verify the utility of the present optimal mesh selection in the Hlavacek-Kubicek approximation to linear operators.
TL;DR: In this paper, the authors proposed a method of transient stability analysis based on the Runge-Kutta-Gill (RKG) model and the state transition method.
Abstract: In analyzing the transient stability of a complicated power system, it is effective to represent the part near the fault point by an exact model and the remaining part far from the fault point by a simplified model. The former part is called the target system and the latter, the external system. The target system is represented by a nonlinear model and the external system by a linear model. Features of the method of transient stability analysis proposed are as follows: the nonlinear target system is analyzed by the Runge-Kutta-Gill method and the linear external system by the state transition method; and the load flow of the external system is obtained by repeating simple multiplications using the transition matrix. A method is described to decompose a differential equation, making clear its features. The following problems are discussed: introduction of machine-to-machine admittance matrix; calculation errors caused by (a) linear approximation of external system, (b) constant node voltage assumption of external system during calculation time step interval, and (c) series expansion of state transition matrix; comparison between the proposed method, and the conventonal exact Runge-Kutta-Gill method; and application of the proposed method to the 39-node loop system and 61-node longitudinal system.
IBM1
TL;DR: This paper compares the representation of nonlinear functions of a single argument by approximations based on piecewise constant, piecewise adjacent, piece wise non-adjacent additional and piecewise non- adjacent segmented functions.
Abstract: A problem on mathematical programming is the linear approximation of nonlinearities in the constraints or in the objective function of a linear programming problem. In this paper, we compare the representation of nonlinear functions of a single argument by approximations based on piecewise constant, piecewise adjacent, piecewise non-adjacent additional and piecewise non-adjacent segmented functions. In each modelization we show the problem size and the results of the following techniques: separable programming, mixed integer programming with Special Order Sets of type 1, linear programming with Special Order Sets of type 2 and mixed integer programming.
TL;DR: In this article, it was shown that λ 1, λ 2, λ 3 are any non-zero real numbers not all of the same sign and that they are irrational.
Abstract: In this present paper we shall prove the following. Suppose that λ 1 , λ 2 , λ 3 are any non-zero real numbers not all of the same sign and that λ 1 /λ 2 is irrational. If η is any real number and, 0 p 1 , p 2 , p 3 ) for which
TL;DR: In this paper, a procedure is given which produces the optimal coefficients with no more effort than for linear approximations, and no initial guess is needed for the values of the coefficients nor for the maximum error of approximation.
Abstract: Rational fractions of the formR(x)/(c1 +c2x +c3x2 + ...) r are useful for approximating decay type functions over infinite and semi-infinite domains. A procedure is given which produces the optimal coefficients with no more effort than for linear approximations. No initial guess is needed for the values of the coefficients nor for the maximum error of approximation.
01 Apr 1978
TL;DR: A technique is described to approximate a linear time-invariant system by a linear m-adic invariant (LMI) system or, equivalently, approximate a circulant matrix by a supercircular matrix, which reduces the number of multiplies required for computing cyclic convolution.
Abstract: A technique is described to approximate a linear time-invariant (LTI) system by a linear m-adic invariant (LMI) system or, equivalently, approximate a circulant matrix by a supercirculant matrix. This approximation reduces the number of multiplies required for computing cyclic convolution. Furtermore, the concepts of LMI systems are presented in a tutorial fashion. Examples are included to illustrate the efficacy of the approximation technique.
TL;DR: In this paper, a logarithmic transform for the transmittance and a reciprocal one for the reflectance mode is used to obtain a linear approximation of the optical response of a homogeneous turbid medium.
Abstract: The optical response of a homogeneous turbid medium is often expressed analytically with the aid of the simplified theory of Kubelka and Munk. The equations derived from this theory yield the response A as an explicit function of absorption Kand scatter S. In practice, however, an inverted form is frequently needed, which would display absorption as an explicit function of A and S. No closed-form rigorous inversion of the response function A(S, K) is available, though approximate solutions exist. All direct inversions, analytical or empirical, are highly non-linear, even though for measurement and instrumentation purposes a linear characteristic is highly desirable. This paper shows that this can be achieved by using a logarithmic transform for the transmittance and a reciprocal one for the reflectance mode. Graphs are given for the coefficients of the linear approximation applicable over a wide range of the parameters S and K ; also shown is the mean error incurred in this range; it is nowhere in excess ...
TL;DR: In this article, the authors give a simplified derivation of the well-known equal ripple approximation to the modulus of a complex number as a linear combination of its real and imaginary parts, and also generalize the approximation to smaller sectors in phase, with corresponding smaller errors.
Abstract: We give a simplified derivation of the well-known equal ripple approximation to the modulus of a complex number as a linear combination of its real and imaginary parts. We also generalize the approximation to smaller sectors in phase, with corresponding smaller errors. We discuss the implementation of the generalized approximation, and indicate the number of computer operations required.
TL;DR: Some linear approximation formulae with very simple expressions of the remainder are given in this article, where a linear approximation for the remainder is given as a linear combination of the linear approximation and the remainder.
Abstract: Some linear approximation formulae with very simple expressions of the remainder are given.
01 Jan 1978
TL;DR: In this paper, the spectrum of short-range-order parameters is determined by the total set of nonzero ordering energies in the first three and the first eight coordination spheres, up to temperatures of order 3TC.
Abstract: The short-range-order parameters are calculated as a function of the temperature, in a linear approximation. The calculation is made in two versions: taking into account the ordering energy in the first three and the first eight coordination spheres. The temperature T varies in the range 1.11TC to 3.0TC. It is shown that up to temperatures of order 3TC the spectrum of short-range-order parameters is determined by the total set of nonzero ordering energies.
13 Apr 1978
TL;DR: This paper deals with a model designed to fit a curve to data subject to error using the Least-Squares criterion, which states that the best approximation is one where the A~'s are determined such that the sum of the squared difference of the true and approximating functions is made a minimum.
Abstract: This paper deals with a model designed to fit a curve to data subject to error. In approximating functions, the criterion of goodness of fit is to some degree arbitrary as there are several criterion which may be used. By letting f(~) denote the true functional value at xL, y(~) denote the approximating functional value at x6, and d~ denote (f(x~)-y(~)) in all cases, it is possible to list a few of these criterion as follows: a)Criterion i suggests making~.od ~ a minimum, where n is i less than the number of data points given. This is attractive because of its s~mplicity but is of little use in that it leads to ambiguous results. b)Criterion 2 suggests making~[d~ a minimum. This has some usage but can~ -allow one erroneous value to overly influence the evaluation of the summation value. c)The Mini-Max or Cnebychev criterion suggests that a boundary (d) be placed on the error (~) and one should strive to keep the error within the upper and lower limits of the boundaries. The approach used by this model is known as the Least-Squares criterion. The concept of linear approximation in the Least-Squares approach states that the best approximation in this sense is one where the A~'s are determined such that the sum of the squared difference of the true and approximating functions is made a minimum, where A~ are the coefficients of the approximating function It can also be stated~[f(x~-y(~)]~a minimum. As the title o~'this~=°'--paper implies, this model only deals with approximations of linear curves by Least Squares, ie. functions of the form: f(x)~A,@~(x) where n is the degree of the polynomial, AK is the coeffieient of the term K, and ~(x) is the argument of A~. For example, in the function y(x)=Ao+A,x+ A~x z , the following holds true: ¢o(X):],~,(x)=x, $ ¢~(x):x ~. The A~'s will be approximated by deriving a set of simultaneous equations using the following formula and then solving the.matrix: The equations given by this method are termed the Least-Squares equations. The ~} or aggregate notation is used for both discrete and continuous models. Depending upon the approximating function, one would substitute the i for a model having discrete data, and ~ for a model having continuous data. Deriving the LeastSquares normal equations for f(x)=Ao+Aox gives the following: