T. C. T. Kotiah · International Journal of Mathematical Education in Science and Technology · 1996
The best linear approximation of a function f (x)near a point cis the tangent line at (c, f (c)).In spite of the well‐known measure of the error in the approximation in elementary calculus, it is not clear for how wide an interval about cis the tangent line the best. A fairly strong competitor is the least squares linear approximation of the function that can be defined over any interval (β, α) containing c.General formulae are derived for the slope and y‐intercept of the least squares line. A comparison is made with the tangent line approximations in the special cases of the sine function and the cube root function for selected values of c, βand α. Numerical results indicate that the least squares approximation is generally better if xis relatively far from c.Values of care subject to the common restriction that f(c) must be readily computed. It is shown that, for given x,it is not always possible to find an appropriate csuch that the resulting tangent line approximation is guaranteed to be superior to the comparable least squares solution.
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A. Nedić, D. P. Bertsekas · 2002
We consider policy evaluation algorithms within the context of infinite-horizon dynamic programming problems with discounted cost. We focus on discrete-time dynamic systems with a large number of states, and we discuss two methods, which use simulation, temporal di#erences, and linear cost function approximation. The first method is a new gradient-like algorithm involving least-squares subproblems and a diminishing stepsize, which is based on the #-policy iteration method of Bertsekas and Io#e. The second method is the LSTD(#) algorithm recently proposed by Boyan, which for # =0coincides with the linear least-squares temporal-di#erence algorithm of Bradtke and Barto. At present, there is only a convergence result by Bradtke and Barto for the LSTD(0) algorithm. Here, we strengthen this result by showing the convergence of LSTD(#), with probability 1, for every # [0, 1].
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Buhmann, M, Cambridge Univ. (United Kingdom). Dept. of Applied Mathematics and Theoretical Physics · OpenGrey (Institut de l'Information Scientifique et Technique) · 1993
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Martin Buhmann · Mathematical Proceedings of the Cambridge Philosophical Society · 1993
Abstract In this article we study the convergence behaviour of least squares approximations of various types by radial basis functions, i.e. least squares approximations from spaces spanned by radially symmetric functions φ (‖· − x j ‖). Here the x j are given ‘centres’ in ℝ n which we assume to lie on a grid. The inner products with respect to which the least squares problem is considered are discrete and Sobolev, i.e. may involve derivative information. Favourable estimates for the least squares errors are found that are shown to decrease as powers of the gridspacing. The work on discrete least squares approximations also gives rise to the construction of prewavelets from radial function spaces with respect to a discrete Sobolev inner product, which are discussed in the paper as well.
Cited by 9
Mohammad Karimnejad Esfahani, Stefano De Marchı, Francesco Marchetti · Constructive Mathematical Analysis · 2023
Functions with discontinuities appear in many applications such as image reconstruction, signal processing, optimal control problems, interface problems, engineering applications and so on. Accurate approximation and interpolation of these functions are therefore of great importance. In this paper, we design a moving least-squares approach for scattered data approximation that incorporates the discontinuities in the weight functions. The idea is to control the influence of the data sites on the approximant, not only with regards to their distance from the evaluation point, but also with respect to the discontinuity of the underlying function. We also provide an error estimate on a suitable piecewise Sobolev Space. The numerical experiments are in compliance with the convergence rate derived theoretically.
Cited by 7Open access
Ouboti Djaneye-Boundjou, Raúl Ordóñez · Automatica · 2025
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Bryce D. Wilkins, Theodore V. Hromadka · International Journal of Computational Methods and Experimental Measurements · 2022
Two approaches for formulating a computational Complex Variable Boundary Element Method (CVBEM) model are examined. In particular, this paper considers a collocation approach as well as a least squares approach. Both techniques are used to fit the CVBEM approximation function to given boundary conditions of benchmark boundary value problems (BVPs). Both modeling techniques provide satisfactory computational results, when applied to the demonstration problems, but differ in specific outcomes depending on the number of nodes used and the type of BVP being examined. Historically, the CVBEM has been implemented using the collocation approach. Therefore, the novelty of this work is in formulating the least squares approach and applying the least squares formulation to a Dirichlet BVP as well as a mixed BVP. This work does not claim that one technique should always be used over the other, but rather it seeks to demonstrate the viability of the least squares approach and assert that both techniques for determining the coefficients of the CVBEM approximation function should be considered during the modeling process.
Cited by 1Open access
L.C. Calvez, P. Vilbé, P. Glouannec · IEEE Transactions on Circuits and Systems · 1985
This correspondence describes a method for finding a least squares approximation for a function subject to linear equality constraints. Provided the number of constraints is reasonably small, the proposed technique overcomes the usual numerical difficulties associated with the inversion of ill-conditioned matrices. The dimensionality of a class of problems has been reduced by introducing some generalized concepts and using complex Lagrange multipliers.
Cited by 6
Serhii Lupenko, Maciej Wiatr, Andrzej Metelski · EURASIP Journal on Advances in Signal Processing · 2025
The work is devoted to a problem of the rhythm function estimation of a cyclic random process, which is based on the least squares approximation methods instead of well-known interpolation approach. Analytical dependencies between errors of estimation of a discrete rhythm function and errors of segmentation of a cyclic random process into cycles and zones were constructed. This made it possible to develop a procedure for calculating and controlling errors of estimating rhythm function of a cyclic random process as certain functions of errors of the segmentation method. The general problem of least squares approximation of the rhythm function of a cyclic random process is formulated as a problem of optimal selection of a parametric function derived from a predetermined class of functions that satisfy the necessary and sufficient conditions of the rhythm function of a cyclic random process. New parametric classes of rhythm characteristics of cyclic random processes such as parametric monomials of degree k, parametric logarithmic functions and parametric exponential functions have been built. The advantage of considered method over well-known interpolation approach refers to the improvement of accuracy of rhythm function estimation and reduction of the rhythm function estimation parameters’ number. For example, in presented computer simulation experiment for the parametric class of monomials of degree 2, average value of the mean square errors for 500 simulations in the case of the interpolation is over 40 times higher than the corresponding value for approximation. Moreover, for that parametric class, the number of estimated parameters is almost equal to doubled number of considered cycles in the case of piecewise linear interpolation and is reduced to 1 for least square approximation. The results obtained in the work constitute the basis for improvement of rhythm-adaptive methods and spectral analysis of cyclic random processes, including the area of statistical methods for detecting hidden cyclic structures of the investigated cyclic stochastic signals with an irregular rhythm.
Cited by 1Open access
Sanpeng Zheng, Renzhong Feng, Aitong Huang · Mathematics · 2020
The radial basis function (RBF) is a class of approximation functions commonly used in interpolation and least squares. The RBF is especially suitable for scattered data approximation and high dimensional function approximation. The smoothness and approximation accuracy of the RBF are affected by its shape parameter. There has been some research on the shape parameter, but the research on the optimal shape parameter of the least squares based on the RBF is scarce. This paper proposes a way for the measurement of the optimal shape parameter of the least squares approximation based on the RBF and an algorithm to solve the corresponding optimal parameter. The method consists of considering the shape parameter as an optimization variable of the least squares problem, such that the linear least squares problem becomes nonlinear. A dimensionality reduction is applied to the nonlinear least squares problem in order to simplify the objective function. To solve the optimization problem efficiently after the dimensional reduction, the derivative-free optimization is adopted. The numerical experiments indicate that the proposed method is efficient and reliable. Multiple kinds of RBFs are tested for their effects and compared. It is found through the experiments that the RBF least squares with the optimal shape parameter is much better than the polynomial least squares. The method is successfully applied to the fitting of real data.
Cited by 11Open access
Md. Faijul Amin, Savitha Ramasamy, Muhammad Ilias Amin et al. · 2011
This paper presents a fully complex-valued functional link network (CFLN). The CFLN is a single-layered neural network, which introduces nonlinearity in the input layer using nonlinear functions of the original input variables. In this study, we consider multivariate polynomials as the nonlinear functions. Unlike multilayer neural networks, the CFLN is free from local minima problem, and it offers very fast learning in parameters because of its linear structure. In the complex domain, polynomial based CFLN has an additional advantage of not requiring activation functions, which is a major concern in the complex-valued neural networks. However, it is important to select a smaller subset of polynomial terms (monomials) for faster and better performance, since the number of all possible monomials may be quite large. In this paper, we use the orthogonal least squares method in a constructive fashion (starting from lower degree to higher) for the selection of a parsimonious subset of monomials. Simulation results demonstrate that computing CFLN in purely complex domain is advantageous than in double-dimensional real domain, in terms of number of connection parameters, faster design, and possibly generalization performance. Moreover, our proposed CFLN compares favorably with several other multilayer networks in the complex domain.
Cited by 13
Zeleznik, Frank J., Gordon, Sanford., United States. National Aeronautics and Space Administration. · The Journal of the Abraham Lincoln Association
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Effendy Tanojo · SHILAP Revista de lepidopterología · 2007
In recent years, meshless methods have gained their popularity, mainly due to the fact that absolutely no elements are required to discretize the problem domain. This is possible due to the nature of the approximation functions used in this method. Approximation functions used to form the shape functions use only the so-called ânodal selectionâ procedure without the need of elements definition. The most popular approximation function used is the moving least-squares shape functions. Published works in meshless methods, however, present only the basic formulas of the moving least-squares shape functions. This paper presents the complete and detailed derivations of not only the moving least-squares shape functions, but also their derivatives (up to the second order derivatives), using the exponential weight function. The derivations are then programmed and verified.
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Andrii Horbono · ГРААЛЬ НАУКИ · 2025
The article compares different options for approximating the deflection function of a circular plate under different types of load using the least squares method. The usual least squares method, the least squares method that passes exactly through the first point and the least squares method that passes exactly through first and last points were considered. The approximation equation of the function was obtained, and the corresponding graphs were constructed. Optimal parameters for the exact approximation of the function were determined.
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Shuying Ma, Min Lv, Fangfang Deng et al. · Journal of Hazardous Materials · 2014
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Y. Garault, Pierre Jarry · 1973
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Gökhan Kiper, Barış Bağdadioğlu · Mechanisms and machine science · 2014
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Tomomi Uchiyama, Kiyoshi MINEMURA · TRANSACTIONS OF THE JAPAN SOCIETY OF MECHANICAL ENGINEERS Series B · 1992
With regard to inverse design problems, numerical integration of the first-order partial derivatives given on discrete points in n-dimensional spaces is demanded occasionally, for example, to obtain a velocity field satisfying the velocity gradients given in the field. In this paper, such numerical quadratures are proposed by the use of the shape function for finite element methods or the finite difference approximation, based on a least squares method. Applicability of these methods is confirmed by applying them to a two-dimensional problem, and the characteristics of computational accuracy and the CPU time necessitated are discussed.
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Zhekai Liu, Yuan Zhu, Ling Meng · 2025
The classical overmodulation methods exhibit limited fidelity in approximating the output voltage waveform to the ideal sinusoidal waveform, resulting in high total harmonic distortion and suboptimal transition performance at high modulation indices. To address these issues, this paper proposes an approximation function to ensure that the actual output voltage closely approximates the reference voltage. The approximation function is derived based on the law of conservation of energy: by deriving the area equivalence rules for the voltage vector trajectory, comparing the output voltage vector with the sinusoidal reference voltage vector, the problem is reformulated as a least squares function approximation. Specifically, the voltage projection onto real-axis is modeled as the least squares approximation function with respect to the angle and amplitude. The sine wave is to be approximated and the least squares approximation error is applied to optimize angle and amplitude. Finally, simulation analysis and bench experiment results validate the effectiveness of the proposed methods.
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P. S. Malachivskyy, L. S. Melnychok, Ya. V. Pizyur et al. · Kibernetyka ta Systemnyi Analiz · 2024
A method of constructing the Chebyshev approximation with the smallest relative error of multivariable functions by a power expression is proposed. It consists in constructing an intermediate Chebyshev approximation of the function, which is the root of the corresponding power of the given function by a polynomial. The parameters of the polynomial approximation are calculated as the limiting mean-power approximation according to the iterative scheme using the least squares method with a variable weight function. Test examples are given that confirm the fast convergence of the method of constructing the Chebyshev approximation using the power expression of functions of one, two, and three variables. Keywords: Chebyshev approximation of the multivariable functions, power expression, mean-power approximation, least squares method, variable weight function.
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