An improvement on precision of recursive function simulation in IEEE floating
point standard is presented. It is shown that the average of rounding towards
negative infinite and rounding towards positive infinite yields a better result
than the usual standard rounding to the nearest in the simulation of recursive
functions. In general, the method improves one digit of precision and it has
also been useful to avoid divergence from a correct stationary regime in the
logistic map. Numerical studies are presented to illustrate the method.Comment: DINCON 2017 - Conferencia Brasileira de Dinamica, Controle e
  Aplicacoes - Sao Jose do Rio Preto - Brazil. 8 page

Martins, Samir A. M.

Nepomuceno, Erivelton G.

Silva, Melanie R.

English

arXiv

An improvement on precision of recursive function simulation in IEEE floating

point standard is presented. It is shown that the average of rounding towards negative

infinite and rounding towards positive infinite yields a better result than the usual standard

rounding to the nearest in the simulation of recursive functions. In general, the method

improves one digit of precision and it has also been useful to avoid divergence from a correct

stationary regime in the logistic map. Numerical studies are presented to illustrate the

method

Rodrigues e Silva, Melanie

Nepomuceno, Erivelton

Angelo Milani Martins, Samir

NUI Maynooth Eprint Archive

Note on improvement precision of recursive function

simulation in floating point standard

MURAL - Maynooth University Research Archive Library

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/321511345Note on improvement precision of recursive function simulation in floatingpoint standardArticle · December 2017CITATIONS0READS803 authors:Some of the authors of this publication are also working on these related projects:Mathematical modelling of the transmission dynamics of SARS-CoV-2: Application of Data Science, Artificial Intelligence, Complex Networks and Control Theory ViewprojectPerformance of Computer Simulation of Recursive Functions View projectMelanie R. Silva4 PUBLICATIONS   5 CITATIONS   SEE PROFILEErivelton NepomucenoMaynooth University254 PUBLICATIONS   1,272 CITATIONS   SEE PROFILESamir A M MartinsFederal University of São João del-Rei70 PUBLICATIONS   321 CITATIONS   SEE PROFILEAll content following this page was uploaded by Erivelton Nepomuceno on 19 December 2017.The user has requested enhancement of the downloaded file.DINCON 2017CONFERÊNCIA BRASILEIRA DE DINÂMICA, CONTROLE E APLICAÇÕES30 de outubro a 01 de novembro de 2017 – São José do Rio Preto/SPNote on improvement precision of recursive functionsimulation in floating point standardMelanie Rodrigues e Silva1Group of Control and Modeling, UFSJ, São João del-Rei, MG, BrazilErivelton Geraldo Nepomuceno2Department of Electrical Engineering, UFSJ, São João del-Rei, MG, BrazilSamir Angelo Milani Martins3Department of Electrical Engineering, UFSJ, São João del-Rei, MG, BrazilAbstract. An improvement on precision of recursive function simulation in IEEE floatingpoint standard is presented. It is shown that the average of rounding towards negativeinfinite and rounding towards positive infinite yields a better result than the usual standardrounding to the nearest in the simulation of recursive functions. In general, the methodimproves one digit of precision and it has also been useful to avoid divergence from a correctstationary regime in the logistic map. Numerical studies are presented to illustrate themethod.Keywords. IEEE floating point, rounding mode, numerical simulation, recursive functionsand Logistic map.1 IntroductionThe mathematical implementation of functions and systems in computers enables thescientific investigation in many areas. According to [19], interferences and noises of realsystems are not incorporated into the simulations, though it is known that it is hardwareas a set of real physical systems then the noises are always present. However, limitationsof software and hardware are obstacles to the reliability of results [14, 15, 18, 22, 24, 26].Nevertheless, alternatives have been developed to circumvent hardware shortcomings andimprove results, such as in [3, 6, 7, 17, 25]. These alternatives that restricts the effectsof hardware limitations are usually called as rigorous computing, which is based on thedevelopment of refined methods for the implementation of algorithms [11, 14].Chaotic systems such as discrete maps and problems involving recursive functions havea differentiated degree of complexity, since the current iteration depends on the previous1me rodrigues silva@hotmail.com2nepomuceno@ufsj.edu.br3martins@ufsj.edu.brarXiv:1712.00530v1  [eess.SP]  2 Dec 20172ones [8]. The study of chaotic systems still receive attention and many of scientific conclu-sions in this area relies upon computer simulation. Chaotic behaviour has been associatedto many real applications, from electronic circuits [2] to ultrasonic cutting system, whichdynamic behavior is analyzed using a two-degree-of freedom Duffing oscillator model [21].Moreover, these iterations are usually evaluated as numerical ill-conditioned [5]. In gene-ral, the simulation process of these types of functions involves errors and its reliability hasbeen questioned in many works [14, 15]. In this sense, it is well known that in a floatingpoint environment, a computer simulation of a chaotic system may be limited to a shortnumber of iterates [26]. To deal with this problem, many tools have been investigated.We may summarize these tools in two categories. The first is focused on hardware perfor-mance using parallel and cluster computation [16]. The second major categories is devotedto improve the algorithm, using interval analysis, significance arithmetic or noisy-modecomputation [11]. In this paper, we focus our attention on the noisy-mode computation,but instead of adding pseudo-random digits, as suggested in [11], we exploit the randomnature of rounding error [4, 20]. Considering that the rounding errors are uniformly dis-tributed, we present the background necessary for the understanding of the guidelines andthe algorithm of a method that improves the precision of the computational simulationof recursive functions. In order to prove the efficiency of the proposed strategy, we bringexamples of the logistic map case.2 BackgroundThis section contains some definitions about recursive functions, orbits, and intervalpseudo-orbits. Let n ∈ N, be a metric space such that I ⊆ R, and f : I → R. Thus,recursive function can be defined as [18]:xn = f(xn−1), (1)or by composite functions:xn = f1(xn−1) = f2(xn−2) = . . . = fn(x0), (2)where f : I → I is a recursive function or a map of a state space into itself and xndenotes the state at the discrete time n. As suggested by [13], the sequence xn obtainedby iterating Eq. (1) starting from an initial condition x0 is called the orbit of x0.Definition 1. An orbit is a sequence of values of a map, represented by xn =[x0,x1,...,xn].The Definition 2. is suggested by [24] and highlights the presence of the differencebetween real and computational pseudo-orbits.Definition 2. Let i ∈ N represents a pseudo-orbit, which is defined by an initialcondition and a combination of software and hardware. A pseudo-orbit is an approximationof an orbit and can be represented as{x̂i,n} = [x̂i,0,x̂i,1, . . . ,x̂i,n] (3)3such that,| xn − x̂i,n |≤ ξi,n (4)where ξi,n ∈ R is the limit of error and ξi,n ≥ 0.There is not an unique pseudo-orbit, as there are different hardware, software, numeri-cal precision standard and discretization schemes, which can produce different results andconsequently different errors ξi,n.3 Method for error reductionIt is reasonable to assume that round-off errors are uniformly distributed [4, 9, 20, 23].According to IEEE 754-2008 [10], the arithmetic operations are also rounded. In this sense,we also assume here that a finite set of arithmetic function presents an error randomlydistributed.Rounding to the nearest is the most usual for computational arithmetic. However,there are software that enables the user to set the rounding mode towards positive infiniteor negative infinite. With these considerations in mind, we establish the main contributionof this letter in the following Lemma.Lemma 1. Let x̂−i,n and x̂+i,n be the calculated value by round towards positive infiniteand round towards negative infinite, respectively. The arithmetic average given byx̂j,n =x̂+i,n + x̂−i,n2(5)such as x̂j,n = f(x̂j,n−1)+δj,n, presents an round-off error smaller than than the round-offerror due the round to nearest as n→∞, therefore δj,n < ξi,n.Proof. Assuming Eq. (5), considering that x̂−i,n = f(x̂j,n−1)+δ−i,n and x̂+i,n = f(x̂j,n−1)+δ+i,n,then:x̂j,n =f(x̂j,n−1) + δ−i,n + f(x̂j,n−1) + δ+i,n2⇒ x̂j,n =2f(x̂j,n−1) + δ−i,n + δ+i,n2(6)andx̂j,n = f(x̂j,n−1) +δ−i,n + δ+i,n2(7)But, as we have been considered δ−i,n and δ+i,n uniformly distributed and it is well knownthat averaging a random variable leads to a reduction of the noise power in n, which is inthis case is 2. And that completes the proof. The Algorithm 1 presents a pseudo-code based on the Lemma 1. The round to thenearest is the standard mode. The operators round− and round+ stands for round towardsnegative infinite and round towards positive infinite. Where there is no indication ofthe round mode, one should consider as the round to the nearest. In this paper, weimplemented this algorithm in Matlab R2016a. The comparison is made with the highprecision values provided by the VPA toolbox of Matlab R2016a.4Algorithm 1: Simulation of recursive function based on Lemma 1.1 x̂i,0; %Initial condition2 N ; %Number of iterates3 x̂−i,0 ← round−(x̂i,0);4 x̂+i,0 ← round+(x̂i,0);5 x̂j,0 ← (x̂−i,0 + x̂+i,0)/2; %Average of the two round modes6 x̂−i,0 ← x̂j,0;7 x̂+i,0 ← x̂j,0;8 For n← 1 to N %Main loop9 x̂−i,n ← round−(f(x̂−j,n−1));10 x̂+i,n ← round+(f(x̂+j,n−1));11 x̂j,n ← (x̂−i,n + x̂+i,n)/2;12 x̂−i,n ← x̂j,n;13 x̂+i,n ← x̂j,n;14 EndFor4 Numerical ExamplesThis sections presents three numerical examples using the method proposed in thispaper. All the examples are based on the logistic map [1]xn+1 = rxn(1− xn), (8)where the initial condition and the value of bifurcation parameter r are indicated in eachexample.Example 4.1. Logistic map with x0 = 1/3.9 and r = 3.9.Only using the round to nearest, the pseudo-orbit presents a chaotic behaviour. Howe-ver, using the Algorithm 1, it is observed that the logistic map function converges to afixed point as shown in Table 1. As one can see, the x̂j,n reaches a fixed point, which isthe exact answer for this example, as verified in the following equations:x1 = 3.9(1/3.9)(1− (1/3.9)) = (1− 10/39) = 29/39 (9)x2 = 3.9(29/39)(1− 29/39) = 29/39 (10)and 29/39 ≈ 0.743589743589744 is a fixed point, which is the value that x̂j,n converges to.On the other hand, the pseudo-orbit x̂i,n diverges and presents a chaotic behaviour.Example 4.2. Logistic map with r = 3.9 and x0 = 0.01.The Figure 1 shows the logarithm (base 10) of the error for the first 20 iterates of thelogistic map, using the Algorithm 1 and a traditional method based on the round to thenearest. The Algorithm 1 starts with an error greater than the traditional method, butafter few iterates the error becomes smaller. The error presented in Figure 1 has beencalculated by means of VPA toolbox of Matlab with 1000-digit precision.5Tabela 1: Iterates of the logistic map with x0 = 1/3.9 and r = 3.9. x̂i,n is following therounding to the nearest and x̂j,n is following the Algorithm 1. The values are shown indecimal and hexadecimal notation. Notice that the third iterate is equal to the second forthe Algorithm 1. This can be seen only in the hex format.n x̂i,n (hex) x̂i,n (dec) x̂j,n (hex) x̂j,n (dec)1 3fd0690690690691 0.256410256410256 3fd0690690690691 0.2564102564102562 3fe7cb7cb7cb7cb8 0.743589743589744 3fe7cb7cb7cb7cb8 0.7435897435897443 3fe7cb7cb7cb7cb7 0.743589743589744 3fe7cb7cb7cb7cb8 0.7435897435897444 3fe7cb7cb7cb7cb9 0.743589743589744 3fe7cb7cb7cb7cb8 0.7435897435897445 3fe7cb7cb7cb7cb5 0.743589743589743 3fe7cb7cb7cb7cb8 0.7435897435897446 3fe7cb7cb7cb7cbd 0.743589743589744 3fe7cb7cb7cb7cb8 0.7435897435897447 3fe7cb7cb7cb7cae 0.743589743589743 3fe7cb7cb7cb7cb8 0.7435897435897448 3fe7cb7cb7cb7ccb 0.743589743589746 3fe7cb7cb7cb7cb8 0.7435897435897449 3fe7cb7cb7cb7c93 0.743589743589740 3fe7cb7cb7cb7cb8 0.74358974358974410 3fe7cb7cb7cb7cfd 0.743589743589751 3fe7cb7cb7cb7cb8 0.7435897435897440 5 10 15 20 25n-18-17-16-15-14-13-12-11ErrorFigura 1: Propagation error for the logistic map with r = 3.9 and x0 = 0.01. Traditionalmethod (blue). Algorithm 1 (red)Example 4.3. Logistic map with three sets of initial conditions and bifurcationparameter, given by (x0, r) = [(0.1,4.2); (0.2,4); (0.41,3.85)].Table 2 shows results of ξi,n and δj,n for the first ten iterations of the logistic map case,considering these three different set of initial conditions and bifurcation parameter. It ispossible to note a general decrease in the error that keeps along the iteration process. Wearbitrary chosen these parameters, but we have performed our tests with many other setswith similar results.6Tabela 2: Results of ξi,n and δj,n of the logistic map case, considering different initialconditions, x0 and r values.x0 = 0.1 and r = 4.2 x0 = 0.2 and r = 4 x0 = 0.41 and r = 3.85n ξi,n δj,n ξi,n δj,n ξi,n δj,n1 9.33254 e-18 9.33254 e-18 4.57088 e-16 1.20226 e-16 3.63078 e-16 2.69153 e-172 3.98107 e-17 3.31131 e-17 7.58577 e-16 9.12010 e-17 7.76247 e-16 6.76082 e-183 1.38038 e-16 8.31763 e-17 1.99526 e-15 2.08929 e-16 1.25893 e-15 4.67735 e-174 3.38844 e-16 1.17489 e-17 1.41253 e-15 6.76082 e-17 2.81838 e-15 1.28824 e-165 1.86208 e-15 7.58577 e-16 5.24807 e-15 2.69153 e-16 3.23593 e-15 2.51188 e-166 1.90546 e-15 6.91830 e-16 1.62181 e-14 8.51138 e-16 9.33254 e-15 7.07945 e-167 7.76247 e-15 2.81838 e-15 1.28824 e-14 6.76082 e-16 6.02559 e-15 4.67735 e-168 2.88403 e-14 1.04712 e-14 4.78630 e-14 2.51188 e-15 1.99526 e-14 1.54881 e-159 6.30957 e-14 2.29086 e-14 1.34896 e-13 7.07945 e-15 4.16869 e-14 3.23593 e-1510 1.41253 e-13 5.12861 e-14 4.16869 e-15 1.94984 e-16 5.88843 e-14 4.46683 e-155 ConclusionsThis paper presents a method based on the average of the round modes which promo-tes an better performance in the simulation of recursive function. The method exploitsthe stochastic nature of rounding. It consists on the average of two round mode whichpromotes a reduction of the power noise by a factor of 2. This is a good example of,although a simulation can be seen as a theoretical experiment, in fact, it presents somereal world dimension, as stochasticity, and therefore, usual tools presented in Engineeringcan be applied.We applied the method in three numerical examples. In the first case, we show that themethod is able to keep the correct fixed point, while the rounding to the nearest diverge tochaotic behaviour. In the second and third we show the general behaviour of the methodin reducing the error of the simulation.It is important to say that there is no significant increase of computational time andthis method can be easily extended to many other applications, such as numerical solutionof differential equations, as seen applied in [21]. In the future, we intend to investigatethe use of interval extensions and other rounding modes as further discussed in [24, 26].AcknowledgementsThis work has been supported by the CNPq and CAPES. A special thanks to Controland Modelling Group (GCOM).Referências[1] R. May Simple mathematical models with very complicated dynamics, Nature, vol.261, 459–467, (1976).[2] T. Matsumoto, A chaotic attractor from Chua’s circuit, Circuits and Systems, IEEETransactions on, vol. 31, 1055–1058, (1984).7[3] K. Sugihara and M. Iri, Two design principles of geometric algorithms in finite-precision arithmetic, Applied Mathematics Letters, vol. 2, 203–206, (1989).[4] K. Furuya, H. Date and K. Fukudome, Effects of finite register length in the fasthartley transform, Electronics and Communications in Japan (Part III: FundamentalElectronic Science, vol. 75, 76–86, (1992).[5] R. M. 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Nepomuceno, A Very Simple Method to Calculate the (Po-sitive) Largest Lyapunov Exponent Using Interval Extensions, International Journalof Bifurcation and Chaos, vol. 26, 1650226, (2016).[25] W. Shi and J. Wu and S. Lam and T. Srikanthan, Algorithms for bi-objective multiple-choice hardware/software partitioning, Computers & Electrical Engineering, vol. 50,127–142, (2016).[26] E. G. Nepomuceno and E. M. A. Mendes, On the analysis of pseudo-orbits of con-tinuous chaotic nonlinear systems simulated using discretization schemes in a digitalcomputer, Chaos, Solitons & Fractals, vol. 95, 21–32, (2017).View publication stats

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