Ponència presentada en: Mathematical Modelling in Engineering & Human Behaviour 2022 (MME&HB 2022). July 13th-15th, 202

Canela, Jordi

Pérez-Palau, Daniel

English

Repositori Institucional de la Universitat Jaume I

36Application of polynomial algebras to non-linearequation solversJ. Canela♭ and D. Pérez-Palau♮1(♭) Insitut Universitari de Matemàtiques i Aplicacions de Castelló and Departament deMatemàtiques,Universitat Jaume I,Av. Vicent Sos Baynat, s/n 12071 Castelló de la Plana, Spain.(♮) Escuela Superior de Ingenieŕıa y Tecnoloǵıa,Universidad Internacional de la Rioja (UNIR),Av. La Paz 137, 26006 Logroño, Spain.1 IntroductionPolynomial algebras emerged during the decade of the 1990 [3] as a tool to propagate efficiently setsof points around an initial condition x0. They were first used in the study of particle accelerators.The main idea of a polynomial algebra approach is to approximate a function g(x) by a poly-nomial P (ξ) in such a way that P (ξ) ≈ g(x0 + ξ). This technique is usually applied to functionslike Poincaré maps, flow maps, etc. The polynomial approximation of functions is equivalent tothe floating-point approximation of real values. As it is the case of the floating-point, where thereal numbers are truncated up to a given number of digits, the polynomials are truncated up to agiven order N . Therefore, Jets (or truncated polynomials) are used to approximate functions. Thelarger is the order N the better is the approximation of the function.The use of polynomial arithmetics has been widely extended for the last decades. Some examplesof its implementation can be found in [8, 9]. Using such arithmetics, given two truncated seriesSa and Sb one can compute the jet of all the classic operations, like products Sa · Sb, additionsSa +Sb, or divisions Sa/Sb (assuming that the independent term of Sb does not banish), as well asexponentials, logarithms or trigonometric operations, among other. For instance, if Sa =∑ni=0 aiξiand Sb =∑i=0 biξi. Then, the coefficients of Sc = Sa · Sb =∑i=0 ciξi and Sd = Sa/Sb =∑i=0 diξiare given by:ci =i∑j=0ai−jbj , cn =an −∑nj=1 cn−jbjb0. (1)These techniques have been used in several applications. The main one is in the propagation ofordinary differential equations. In [3] the bases of the technique were introduced. Later on, [1,2,10]used it to study and propagate trajectories and its uncertainties in the evolution of spacecrafts andasteroids. In [4, 5] the same ideas were applied in the modification of a Kalman filter and in theorbit determination with uncertainties. From a theoretical point of view, [6] used the jet transporttechnique to compute periodic orbits of delayed differential equations and [7] computed high orderexpansions of invariant manifolds around tori of high dimension.1daniel.perez@unir.netModelling for Engineering & Human Behaviour 2022As can be observed in the bibliography, the Jet Transport is widely used in the context of thepropagation of differential equations. However, there are other fields where it can be used. One ofthose applications is in the solution of non-linear equations with parameters.Let us consider a non-linear equation f(x) = 0. Usually, such equation can be solved numericallyby means of iterative methods such as Newton’s method:xk+1 = N(xk) = xk −f(xk)fx(xk).If the initial seed x0 is close enough to the solution x⋆ of f(x) = 0, the iteration converges to x⋆.In the following, let us consider a function f which depends on a parameter c. Hence, we wantto solve f(x, c) = 0. Therefore, we look for a solution x(c) such that f(x(c), c) = 0. It followsfrom the Implicit Function Theorem that x(c) exists under regularity conditions. We want to findx(c) = ∑ ai(c0 − c)i such that f(x(c), c) = 0. However, such solution x(c) may require an infinitesum. Hence, we will restrict to a finite order polynomial approximation xN (c) =∑Ni=0 ai(c0 − c)i.In Section 2 we show how to use the jet arithmetics to solve an example of such problem. Letc = c0 +ξ, and assume that x0 is a solution of f(x0, c0). The first step is to apply Newton’s methodto the function f(x, c0 + ξ) with the initial seed x0. We have thatxk+1(ξ) = xk(ξ)−f(xk(ξ), c+ ξ)fx(xk(ξ), c+ ξ).We show that at each step there is an increasing number of coefficients that correctly approximatex(c). In Section 3 we give the main result of the paper. There, we proof that at each iterate of theprocedure the number of vanishing coefficients of the Taylor expansion of f(xk(ξ), c0 + ξ) doubles.Finally, in Section 5 we give some conclusions.2 Preliminary exampleGiven the function f(x, c) = x2 + x + c, where c ∈ R, We want to find a parametric solution off(x, c) = 0 around c = c0 = 0. Notice that x = 0 is a solution of f(x, c0) = 0 (f(0, 0) = 0). Theexact solution is given by x(c) = −1 +√1− 4c2 . The Taylor series of the previous function can becomputed by derivation, obtainingx(c0 + δc) = −δc− δc2 − 2δc3 − 5δc4 − 14δc5 − 42δc6 − 132δc7 − 429δc8 − 1430δc9 + o(δc10).Instead of using derivation, we can obtain the Taylor series of xc (up to a given order) applyingthe Newton Polynomial Techniquexk+1 = xk −f(xk, 0 + ξ)fx(x, 0 + ξ).We look for a functional solution of f(x, c(ξ)) = 0, where c(ξ) = c0 + ξ = 0 + ξ. We take asinitial approximation of the constant solution x0(ξ) = 0, since f(x0(ξ), c(0)) = 0. After iteratingthe procedure once, we obtainx1(ξ) = x0(ξ)−f(x0(ξ), 0 + ξ)fx(x0(ξ), 0 + ξ)= 0 + 02 + 0 + 0 + ξ2 · 0 + 1 = −ξ.Repeating the procedure and applying the polynomial algebra formulas 1 to the fraction we getx2(ξ) = x1(ξ)−f(x1(ξ), 0 + ξ)fx(x1(ξ), 0 + ξ)= 0 + (−ξ)2 − ξ + 0 + ξ2(−ξ) + 1 = −ξ − ξ2 − 2ξ2 + o(ξ3)209Modelling for Engineering & Human Behaviour 2022Counting the constant term, the power series of x1(ξ) has 2 exact terms. After applying theprocedure, x2(ξ) has 4 exact terms. As we will see, at each step of new Newton PolynomialTechnique we double the number of exact terms. Indeed, for x3(ξ) we getx3(ξ) = −ξ − ξ2 − 2ξ3 − 5ξ4 − 14ξ5 − 42ξ6 − 132ξ7 + o(ξ8).3 Main TheoremIn this section we formalize the results discussed up to now. This is the content of Theorem A.Before stating the theorem, we define the concept of polynomial zero of a (sufficiently smooth)function f(x).Definition 1. A polynomial P (ξ) is a polynomial zero of order n of a function f iff(P (ξ)) = 0 +O(ξn).Theorem A. Let f(x, c) be a sufficiently smooth function depending on a parameter c ∈ R. LetN(x) denote the Newton method using polynomial iterates,Pk+1(ξ) = N(Pk(ξ)) = x−f(Pk(ξ), c0 + ξ)fx(Pk(ξ), c0 + ξ),and let P0(ξ) = x0 be the initial seed with f(x0, c0) = 0. Then, the iterate Pk(ξ) is a polynomialzero of order 2k of f(x, c+ ξ).Proof. We will prove the result by induction. Notice that the (constant) polynomial P0(ξ) = x0is a polynomial zero of order 1 = 20. Indeed, by hypothesis we have that f(x0, c0 + ξ) = 0 +O(ξ).For the sake of clarity, next prove that P1 is a polynomial zero of order 21. We have:P1(ξ) = P0(ξ)−f(P0(ξ), c0 + ξ)fx(P0(ξ), c0 + ξ)= x0 −f(x0, c0 + ξ)fx(x0, c0 + ξ).Evaluating f using its Taylor expansion with respect to x around x0 we get:f(P1(ξ), c0 + ξ) =f(x0 −f(x0, c0 + ξ)fx(x0, c0 + ξ), c0 + ξ)=f(x0, c0 + ξ)− fx(x0, c0 + ξ)(f(x0, c0 + ξ)fx(x0, c0 + ξ))+ 12fxx(x0, c0 + ξ)(− f(x0, c0 + ξ)fx(x0, c0 + ξ))2+O((− f(x0, c0 + ξ)fx(x0, c0 + ξ))3).Using that f(x0, c0 + ξ) = 0 + α1ξ + o(ξ) we get thatf(P1(ξ), c0 + ξ) = 0 +O(ξ2).We conclude that P1(ξ) is a polynomial zero of order 2 = 21.We will use induction to prove the general case. The induction hypothesis is that thepolynomial Pk is a polynomial zero of order 2k, that is:f(Pk(ξ), c0 + ξ) = O(ξ2k).210Modelling for Engineering & Human Behaviour 2022The polynomial Pk+1 is given byPk+1(ξ) = Pk(ξ)−f(Pk, c0 + ξ)fx(Pk, c0 + ξ)︸ ︷︷ ︸P̃k+1(ξ).Evaluating f(x, c0 + ξ) at Pk+1 and using the Taylor expansion of f(x, c0 + ξ) with respect tox around Pk(ξ) we getf(Pk+1(ξ), c0 + ξ) =f(Pk(ξ)− P̃k+1(ξ), c0 + ξ)=f(Pk(ξ), c0 + ξ)− fx(Pk(ξ), c0 + ξ)f(Pk(ξ), c0 + ξ)fx(Pk, c0 + ξ)+ 12fxx(Pk(ξ), c0 + ξ)(−P̃k+1(ξ))2 +O((P̃k+1(ξ))3).By induction hypothesis we have f(Pk, c0 + ξ) = O(ξ2k)and, thereforeP̃k+1(ξ) =f(Pk, c0 + ξ)fx(Pk, c0 + ξ)= O(ξ2k).Finally, we getf(Pk+1(ξ), c0 + ξ) =12fxx(Pk, c0 + ξ)(f(Pk(ξ), c0 + ξ)fx(Pk(ξ), c0 + ξ))2+O((P̃k+1(ξ))3)=12fxx(Pk(ξ), c0 + ξ)(O(ξ2k))2+O((O(ξ2k))3)=O(ξ2k+1).4 ConclusionsThis paper studies methods for solving non-linear equations using a polynomial algebra. Themethod computes the solution, x(c0 + ξ), of the non-linear equation f(x, c0 + ξ) = 0. This is doneby approximating x(c0 + ξ) as a jet, P (ξ), and working with polynomial arithmetics in a Newton’smethod scheme. The main theorem of the paper shows that each iterate of the procedure, Pk(ξ),multiplies by two the coefficients that vanish in f(Pk(ξ), c0 + ξ).References[1] Elisa Maria Alessi, Ariadna Farres, Arturo Vieiro, Angel Jorba, and Carles Simó. Jet trans-port and applications to neos. In Proceedings of the 1st IAA Planetary Defense Conference,Granada, Spain, pages 10–11, 2009.[2] Roberto Armellin and Pierluigi Di Lizia. Probabilistic optical and radar initial orbit determi-nation. Journal of Guidance, Control, and Dynamics, 41(1):101–118, 2018.[3] Martin Berz. Modern map methods in particle beam physics. Academic Press, 1999.[4] Jianlin Chen, Josep Masdemont, Gerard Gomez, and Jianping Yuan. Analysis of jet transport-based geostationary trajectory uncertainty propagation. Journal of Guidance, Control, andDynamics, 43, 04 2020.211Modelling for Engineering & Human Behaviour 2022[5] Jianlin Chen, Josep Masdemont, Gerard Gomez, and Jianping Yuan. An efficient statisti-cal adaptive order-switching methodology for kalman filters. Communications in NonlinearScience and Numerical Simulation, 93, 09 2020.[6] Joan Gimeno and Angel Jorba. Using automatic differentiation to compute periodic orbits ofdelay differential equations. Discrete & Continuous Dynamical Systems - B, 22, 01 2017.[7] Joan Gimeno, Angel Jorba, Begoña Nicolás, and Estrella Olmedo. Numerical computationof high-order expansions of invariant manifolds of high-dimensional tori. SIAM Journal onApplied Dynamical Systems, 21:1832–1861, 09 2022.[8] Àlex Haro, Marta Canadell, Jordi-Lluis Figueras, Alejandro Luque, and Josep Maŕıa Mon-delo. The Parameterization Method for Invariant Manifolds: From rigorous results to effectivecomputations. Springer, 2016.[9] Angel Jorba and Maorong Zou. A software package for the numerical integration of odes bymeans of high-order taylor methods. Experimental Mathematics, 14(1):99–117, 2005.[10] Monica Valli, Roberto Armellin, Pierluigi Di Lizia, and Michèle R Lavagna. Nonlinear filteringmethods for spacecraft navigation based on differential algebra. Acta Astronautica, 94(1):363–374, 2014.212

Application of polynomial algebras to non-linear equation solvers

https://repositori.uji.es/xmlui/bitstream/10234/202216/1/84031.pdf

Application of polynomial algebras to non-linear equation solvers

Abstract

Similar works

Full text

Available Versions

Repositori Institucional de la Universitat Jaume I