Problem Of Rational Approximation Research Articles

Algebraic Solution of Tropical Best Approximation Problems

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.

Classical Problems of Rational Approximation

Classical Problems of Rational Approximation

An Algorithm for Best Generalised Rational Approximation of Continuous Functions

In this paper we introduce an algorithm for solving variational inequality problems when the operator is pseudomonotone and point-to-set (therefore not relying on continuity assumptions). Our motivation is the development of a method for solving optimisation problems appearing in Chebyshev rational and generalised rational approximation problems, where the approximations are constructed as ratios of linear forms (linear combinations of basis functions). The coefficients of the linear forms are subject to optimisation and the basis functions are continuous functions. It is known that the objective functions in generalised rational approximation problems are quasiconvex. In this paper we prove a stronger result, the objective functions are pseudoconvex in the sense of Penot and Quang. Then we develop numerical methods, that are efficient for a wide range of pseudoconvex functions and test them on generalised rational approximation problems.

The extension of the linear inequality method for generalized rational Chebyshev approximation to approximation by general quasilinear functions

In this paper, we demonstrate that a well-known linear inequality method developed for rational Chebyshev approximation is equivalent to the application of the bisection method used in quasiconvex optimization. Although this correspondence is not surprising, it naturally connects rational and generalized rational Chebyshev approximation problems with modern developments in the area of quasiconvex functions and, therefore, offers more theoretical and computational tools for solving this problem. The second important contribution of this paper is the extension of the linear inequality method to a broader class of Chebyshev approximation problems, where the corresponding objective functions remain quasiconvex. In this broader class of functions, the inequalities are no longer required to be linear: it is enough for each inequality to define a convex set and the computational challenge is in solving the corresponding convex feasibility problems. Therefore, we propose a more systematic and general approach for treating Chebyshev approximation problems. In particular, we are looking at the problems where the approximations are quasilinear functions with respect to their parameters that are also the decision variables in the corresponding optimization problems.

On geometry of numbers and uniform rational approximation to the Veronese curve

Consider the classical problem of rational simultaneous approximation to a point in \({\mathbb {R}}^{n}\). The optimal lower bound on the gap between the induced ordinary and uniform approximation exponents has been established by Marnat and Moshchevitin in 2018. Recently Nguyen, Poels and Roy provided information on the best approximating rational vectors to the points where the gap is close to this minimal value. Combining the latter result with parametric geometry of numbers, we effectively bound the dual linear form exponents in the described situation. As an application, we slightly improve the upper bound for the classical exponent of uniform Diophantine approximation \({\widehat{\lambda }}_{n}(\xi )\), for even \(n\ge 4\). Unfortunately our improvements are small, for \(n=4\) only in the fifth decimal digit. However, the underlying method in principle can be improved with more effort to provide better bounds. We indeed establish reasonably stronger results for numbers which almost satisfy equality in the estimate by Marnat and Moshchevitin. We conclude with consequences on the classical problem of approximation to real numbers by algebraic numbers/integers of uniformly bounded degree.

Rational Function Approximation of the Relay with Hysteresis Nonlinear Element

Abstract This paper considers the problem of rational function approximation of the relay with hysteresis characteristic. The relay with hysteresis element has differential-based characteristic with memory, which is rate dependent and two-valued, involving the input signal velocity in its description. The presented model includes two ideal relay characteristics, which selection is depending on the input behavior. The rational function description is implemented for analytical approximation of the relay switching behavior. Both relay discontinuous jumps are approximated in terms of a hyperbolic tangent function, which on its own turn is replaced by continued fraction and Pade series expressions. The nearness of approximation is achieved by using one parameter descriptions for the hyperbolic tangent function. The errors of approximation of the ideal relay characteristic are also evaluated.

Distribution of the logarithmic derivative of a rational function on the line

We prove that for an arbitrary real rational function r of degree n, a measure of the set $$\{x\in \mathbb{R}: |r'(x)/r(x)|\ge n\}$$ is at most $$2\pi\Theta$$ ( $$\Theta\approx 1.347$$ is the weak $$(1,1)$$ -norm of the Hilbert transform), and this bound is extremal. A problem of rational approximations on the whole real line is also considered.

$\mathcal{H}_2$-Optimal Model Reduction Using Projected Nonlinear Least Squares

In many applications throughout science and engineering, model reduction plays an important role replacing expensive large-scale linear dynamical systems by inexpensive reduced order models that capture key features of the original, full order model. One approach to model reduction finds reduced order models that are locally optimal approximations in the $\mathcal{H}_2$ norm, an approach taken by the Iterative Rational Krylov Algorithm (IRKA) among others. Here we introduce a new approach for $\mathcal{H}_2$-optimal model reduction using the projected nonlinear least squares framework previously introduced in [J. M. Hokanson, SIAM J. Sci. Comput. 39 (2017), pp. A3107--A3128]. At each iteration, we project the $\mathcal{H}_2$ optimization problem onto a finite-dimensional subspace yielding a weighted least squares rational approximation problem. Subsequent iterations append this subspace such that the least squares rational approximant asymptotically satisfies the first order necessary conditions of the original, $\mathcal{H}_2$ optimization problem. This enables us to build reduced order models with similar error in the $\mathcal{H}_2$ norm but using far fewer evaluations of the expensive, full order model compared to competing methods. Moreover, our new algorithm only requires access to the transfer function of the full order model, unlike IRKA which requires a state-space representation or TF-IRKA which requires both the transfer function and its derivative. Applying the projected nonlinear least squares framework to the $\mathcal{H}_2$-optimal model reduction problem open new avenues for related model reduction problems.

On Infinitely Many Rational Approximants to ζ(3)

A set of second order holonomic difference equations was deduced from a set of simultaneous rational approximation problems. Some orthogonal forms involved in the approximation were used to compute the Casorati determinants for its linearly independent solutions. These solutions constitute the numerator and denominator sequences of rational approximants to ζ ( 3 ) . A correspondence from the set of parameters involved in the holonomic difference equation to the set of holonomic bi-sequences formed by these numerators and denominators appears. Infinitely many rational approximants can be generated.

Approximation to Constant Functions by Electrostatic Fields due to Electrons and Positrons

We study a uniform approximation to constant functions f(z) = const on compact subsets K of complex plane by logarithmic derivatives of rational functions with free poles. This problem can be treated in terms of electrostatics: we construct on K the constant electrostatic field due to electrons and positrons at points ∉ K. If K is a disk or an interval, we get the approximation, which close to the best. Also we get the new identity for generalized Laguerre polynomials. Our results related to the classical problem of rational approximation to the exponential function.

Continued fractions and Parametric geometry of numbers

Recently, W. M. Schmidt and L. Summerer developed a new theory called Parametric Geometry of Numbers which approximates the behaviour of the successive minima of a family of convex bodies in ℝ n related to the problem of simultaneous rational approximation to given real numbers. In the case of one number, we show that the qualitative behaviour of the minima reflects the continued fraction expansion of the smallest distance from this number to an integer.

Carlson iterating and rational approximation of arbitrary order fractional calculus operator

With the development of factional calculus theory and applications in different fields in recent years, the rational approximation problem of fractional calculus operator has become a hot spot of research. In the early 1950s and 1960s, Carlson and Halijak proposed regular Newton iterating method to implement rational approximation of the one-nth calculus operator. Carlson regular Newton iterating method has a great sense of innovation for the rational approximation of fractional calculus operator, however, it has been used only for certain calculus operators. The aim of this paper is to achieve rational approximation of arbitrary order fractional calculus operator. The realization is achieved via the generalization of Carlson regular Newton iterating method. To construct a rational function sequence which is convergent to irrational fractional calculus operator function, the rational approximation problem of fractional calculus operator is transformed into the algebra iterating solution of arithmetic root of binomial equation. To speed up the convergence, the pre-distortion function is introduced. And the Newton iterating formula is used to solve arithmetic root. Then the approximated rational impedance function of arbitrary order fractional calculus operator is obtained. For nine different operational orders with n changing from 2 to 5, the impedance functions are calculated respectively through choosing eight different initial impedances for a certain operational order. Considering fractional order operation characteristics of the impedance function and the physical realization of network synthesis, the impedance function should satisfy these basic properties simultaneously: computational rationality, positive reality principle and operational validity. In other words, there exists only rational computation of operational variable s in the expression of impedance function. All the zeros and poles of impedance function are located on the negative real axis of s complex plane or the left-half plane of s complex plane in conjugate pairs. The frequency-domain characteristics of impedance function approximate to those of ideal fractional calculus operator over a certain frequency range. Given suitable initial impedance and for an arbitrary operational order, it is proved that the impedance function could meet all properties above by studying the zero-pole distribution and analyzing frequency-domain characteristics of the impedance function. Therefore, the impedance function could take on operational performance of the ideal fractional calculus operator and achieve the physical realization. It is of great effectiveness in the generalization of this kind of method in both theory and experiment. The results educed in this paper are the basis for further theoretic research and engineering application in constructing the arbitrary order fractional circuits and systems.

On rational approximation of Markov functions on finite sets

In this article we study questions related to the approximation of Markov functions on a finite set of points on the real line by rational functions with real coefficients. The main results include formulas for the error in best rational approximation. A connection of rational approximation problems with the discrete Hankel operator is also investigated.

Existence of compact sets with minimum capacity in problems of rational approximation of multivalued analytic functions

Existence of compact sets with minimum capacity in problems of rational approximation of multivalued analytic functions

Rational Approximation of Transfer Functions for Non-Negative EPT Densities

An Exponential-Polynomial-Trigonometric (EPT) function is defined on [0, ∞) by a minimal realization (A, b, c). A stable non-negative EPT function of a fixed degree is fitted to the histogram of a large set of data using an L2 criterion. If we neglect the non-negativity constraint this is shown to be equivalent to a rational approximation problem which is approached using the RARL2 software. We show how, under the additional assumption of the existence of a strictly dominant real pole of the rational function, the non-negativity constraint on the EPT function can be imposed by performing a constraint convex optimization on b at each stage at which an (A, c) pair is determined. In this convex optimization step a recent generalized Budan-Fourier sequence approach to determine non-negativity of an EPT function on a finite interval plays a major role.

Müntz Rational Approximation in Orlicz Spaces

Using K-functional and modulus of continuity and some technique of inequalities,studied M(u|¨)ntz rational approximation problems in Orlicz spaces,and obtained three kinds estimate of degree of rational approximation.

Variation of the equilibrium energy and the S-property of stationary compact sets

This paper studies a variation of the equilibrium energy for a certain fairly general functional which appears naturally in the solution of many rational approximation problems of multi-valued analytic functions.The main result of this work states that for the energy functional under consideration and a certain class of admissible compact sets, related to the function to be approximated, the corresponding stationary compact set is fully characterized by the so-called S-property.Bibliography: 38 titles.

Open mushrooms: stickiness revisited

We investigate mushroom billiards, a class of dynamical systems with sharply divided phase space. For typical values of the control parameter of the system ρ, an infinite number of marginally unstable periodic orbits (MUPOs) exist making the system sticky in the sense that unstable orbits approach regular regions in phase space and thus exhibit quasi-regular behaviour for long periods of time. The problem of finding these MUPOs is expressed as the well-known problem of finding optimal rational approximations of a real number, subject to some system-specific constraints. By introducing a generalized mushroom and using properties of continued fractions, we describe for the first time a zero measure set of control parameter values ρ ∊ (0, 1) for which all MUPOs are destroyed and therefore the system is less sticky, leading to a different power law exponent for the Poincaré recurrence time distribution statistics. The open mushroom (billiard with a hole) is then considered in order to quantify the stickiness exhibited due to MUPOs and exact leading order expressions for the algebraic decay of the survival probability function are calculated for mushrooms with triangular and rectangular stems. Numerical simulations are also performed which confirm our predictions for both sticky and less sticky mushrooms.

Efficient order basis computation (abstract only)

Order basis (also known as 3/4-basis, minimal approximant basis), originally developed by Beckermann and Labahn in 1994 for rational approximation and interpolation problems, has recently been applied to many other important problems in polynomial matrix computation. These include column reduction, matrix inverse, determinant, and null space basis computation. Storjohann, in 2006, provided an efficient way to compute a part of order basis that is within a given degree bound. We extend Storjohann's result by providing a way to compute a complete order basis with a similar computational cos.

A multidimensional continued fraction based on a high-order recurrence relation

The paper describes and studies an iterative algorithm for finding of a set of linear forms over vectors of integers. The algorithm small values of a set of linear forms over vectors of integers. The algorithm uses a linear recurrence relation to generate a vector sequence, the basic idea being to choose the integral coefficients in the recurrence relation in such a way that the linear forms take small values, subject to the requirement that the integers should not become too large. The problem of choosing good coefficients for the recurrence relation is thus related to the problem of finding a good approximation of a given vector by a vector in a certain one-parameter family of lattices; the novel feature of our approach is that practical formulae for the coefficients are obtained by considering the limit as the parameter tends to zero. The paper discusses two procedures to solve the underlying inhomogeneous Diophantine approximation problem: the first, which we call naive rounding leads to a multidimensional continued fraction algorithm with suboptimal asymptotic convergence properties; in particular, when it is applied to the familiar problem of simultaneous rational approximation, the algorithm reduces to the classical Jacobi-Perron algorithm. The second procedure is Babai's nearest-plane procedure. We compare the two procedures numerically; our experiments suggest that the multidimensional continued fraction corresponding to nearest-plane converges at an optimal asymptotic rate.