Convergence of recursive functions on computers

A theorem is presented which has applications in the numerical computation of fixed points of recursive functions. If a sequence of functions {fn} is convergent on a metric space I ⊆ ℝ, then it is possible to observe this behaviour on the set D , ℚ of all numbers represented in a computer. However, as D is not complete, the representation of fn on D is subject to an error. Then fn and fm are considered equal when its differences computed on D are equal or lower than the sum of error of each fn and fm. An example is given to illustrate the use of the theorem

Dinâmica, modelagem e controle de epidemias (Modelling and Control of Epidemics)

Epidemic is an alteration of one or more characteristics in a significant number of individuals of a population. Normally these characteristics are related to health. Individuals are considered as unique entities, for example, the human beings, ani- mals and even though machines or computers. The interaction between indivi- duals and environment consists in an epidemiological system. The classification of individuals in states is the most used approach to study an epidemiological system. Kermack and McKendrick developed the SIR model, which classifies the individuals in three states: susceptible, infectious and recovered. These three states are related by means of nonlinear differential equations. In this work the following aspects are investigated: i) influence of the vaccination and isolation on the dynamics of the SIR model; ii) models based on individuals (MBI); iii) use of optimal control and pulsed vaccination. The main contributions of this thesis are the following. First, it was verified that the vaccination and isolation consist in actions of control that modify the localization of transcritical bifurcation point. This change occurs proportionally to number of isolated individuals and inversely proportional to number of non-vaccinated individuals. The simultaneous use of the vaccination and isolation seems to be useful in certain circumstances. Secon- dly, a mathematical expression and an algorithm for the MBI was developed. It was evaluated that the MBI tends to present same results that the SIR model for very large populations and infinitesimal time intervals. An expression to calculate probability of eradication of an illness in a population was proposed. This proba- bility tends to increase with a reduction of population size. Finally, Pontryagin’s maximum principle was used to calculate an optimal control of vaccination using the SIR model. An analytical formula for the control law was obtained, which indicates a proportionality between population size and vaccination rate. After that, the intensity and the interval of pulsed vaccination using the SIR model were optimized by means of the Nelder-Mead’s algorithm. It was observed that an in- crease in the time interval of pulses can cause peaks in the number of infected individuals equivalent to the situation without vaccination. The proposed controls were applied in the MBI showing coherence with the results achieved in in the SIR model

Note on improvement precision of recursive function simulation in floating point standard

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

A lower bound error for free-run simulation of the polynomial NARMAX

A lower bound error for free-run simulation of the polynomial NARMAX (Nonlinear AutoRegressive Moving Average model with eXogenous input) is introduced. The ultimate goal of the polynomial NARMAX is to predict an arbitrary number of steps ahead. Free-run simulation is also used to validate the model. Although free-run simulation of the polynomial NARMAX is essential, little attention has been given to the error propagation to round off in digital computers. Our procedure is based on the comparison of two pseudo-orbits produced from two mathematical equivalent models, but different from the point of view of floating point representation. We apply successfully our technique for three identified models of the systems: sine map, Chua’s circuit and Duffing–Ueda oscillator. This technique may be used to reject a simulation, if a required precision is greater than the lower bound error, increasing the numerical reliability in free-run simulation of the polynomial NARMAX