The History of Approximation Theory

Anastassiou, George A.

doi:10.1007/0-8176-4475-x

Cited by 6 publications

(5 citation statements)

References 19 publications

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“…Application of the Chebyshev norm leads to the least maximum absolute deviation of errors as the approximation criterion, and dates back to Laplace's classical work [8] (book 3, chap. V, §39) (see also [11,27]).…”

Section: Introductionmentioning

confidence: 99%

Using Parameter Elimination to Solve Discrete Linear Chebyshev Approximation Problems

Krivulin

2020

Mathematics

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We consider discrete linear Chebyshev approximation problems in which the unknown parameters of linear function are fitted by minimizing the least maximum absolute deviation of errors. Such problems find application in the solution of overdetermined systems of linear equations that appear in many practical contexts. The least maximum absolute deviation estimator is used in regression analysis in statistics when the distribution of errors has bounded support. To derive a direct solution of the problem, we propose an algebraic approach based on a parameter elimination technique. As a key component of the approach, an elimination lemma is proved to handle the problem by reducing it to a problem with one parameter eliminated, together with a box constraint imposed on this parameter. We demonstrate the application of the lemma to the direct solution of linear regression problems with one and two parameters. We develop a procedure to solve multidimensional approximation (multiple linear regression) problems in a finite number of steps. The procedure follows a method that comprises two phases: backward elimination and forward substitution of parameters. We describe the main components of the procedure and estimate its computational complexity. We implement symbolic computations in MATLAB to obtain exact solutions for two numerical examples.

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Section: Introductionmentioning

confidence: 99%

Using Parameter Elimination to Solve Discrete Linear Chebyshev Approximation Problems

Krivulin

2020

Mathematics

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“…Given a set of points and corresponding values of the function, the goal is to find a continuous function of a particular form, which minimizes approximation error in the sense of a certain metric. In many cases, the approximating functions are assumed to be piecewise polynomials (including piecewise linear functions), whereas the error is measured by the Chebyshev metric, which leads to Chebyshev best approximation problems (Mhaskar and Pai, 2000;Steffens, 2006) that date back to Laplace's classical work de Laplace (1832, Book 3, Chapter V, §39).…”

Section: Introductionmentioning

confidence: 99%

On Solution of Tropical Discrete Best Approximation Problems

Krivulin

2024

Preprint

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We consider a discrete best approximation problem formulated in the framework of tropical algebra, which deals with the theory and applications of algebraic systems with idempotent operations. Given a set of samples of input and output of an unknown function, the problem is to construct a generalized tropical Puiseux polynomial that best approximates the function in the sense of a tropical distance function. The construction of an approximate polynomial involves the evaluation of both unknown coefficient and exponent of each monomial in the polynomial. To solve the approximation problem, we first reduce the problem to an equation in unknown vector of coefficients, which is given by a matrix with entries parameterized by unknown exponents. We derive a best approximate solution of the equation, which yields both vector of coefficients and approximation error parameterized by the exponents. Optimal values of exponents are found by minimization of the approximation error, which is transformed into minimization of a function of exponents over all partitions of a finite set. We solve this minimization problem in terms of max-plus algebra (where addition is defined as maximum and multiplication as arithmetic addition) by using a computational procedure based on the agglomerative clustering technique. This solution is extended to the minimization problem of finding optimal exponents in the polynomial in terms of max-algebra (where addition is defined as maximum). The results obtained are applied to develop new solutions for conventional problems of discrete best Chebyshev approximation of real functions by piecewise linear functions and piecewise Puiseux polynomials. We discuss computational complexity of the proposed solution and estimate upper bounds on the computational time. We demonstrate examples of approximation problems solved in terms of max-plus and max-algebra, and give graphical illustrations. MSC Classification: 15A80 , 90C24 , 41A50 , 41A65 , 65D15

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“…The minimax optimization problem at (1) is commonly formulated in terms of the distance between two reals a and b, given by the absolute value d(a, b) = |a − b|, which leads to the Chebyshev metric to measure the error. The problem is then referred to as the best Chebyshev approximation problem [1][2][3], and it has drawn much attention over the centuries since Laplace's classical work [4] (Book 3, Chapter V, §39).…”

Section: Introductionmentioning

confidence: 99%

Algebraic Solution of Tropical Best Approximation Problems

Krivulin

2023

Mathematics

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We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and output of an unknown function defined on an idempotent semifield, the problem is to find a best approximation of the function, by tropical Puiseux polynomial and rational functions. A new solution approach is proposed, which involves the reduction of the problem of polynomial approximation to the best approximate solution of a tropical linear vector equation with an unknown vector on one side (a one-sided equation). We derive a best approximate solution to the one-sided equation, and we evaluate the inherent approximation error in a direct analytical form. Furthermore, we reduce the rational approximation problem to the best approximate solution of an equation with unknown vectors on both sides (a two-sided equation). A best approximate solution to the two-sided equation is obtained in numerical form, by using an iterative alternating algorithm. To illustrate the new technique developed, we solve example approximation problems in terms of a real semifield, where addition is defined as maximum and multiplication as arithmetic addition (max-plus algebra), which corresponds to the best Chebyshev approximation by piecewise linear functions.

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The History of Approximation Theory

Cited by 6 publications

References 19 publications

Using Parameter Elimination to Solve Discrete Linear Chebyshev Approximation Problems

Using Parameter Elimination to Solve Discrete Linear Chebyshev Approximation Problems

On Solution of Tropical Discrete Best Approximation Problems

Algebraic Solution of Tropical Best Approximation Problems

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