Análise da solução de equações lineares com coeficientes intervalares em diferentes aritméticas intervalares

Abstract

When working with floating point numbers the result is only an approximation of a real value and errors generated by rounding or by instability of the algorithms that can lead to incorrect results. Using intervals for the representation of real numbers, it is possible to control this error propagation. The most widely used interval arithmetic in literature came in 1966, defined by Moore. However, recent papers show different interval arithmetic applied in place of the Moore’s arithmetic, since it has several faults, often returning incomplete results, including in simple equations, as in the case of linear equations. With several interval arithmetics present in the literature the main objective of the thesis is to analyze the solutions of linear equations with interval coefficients in the following arithmetic: Moore with Interval Approximation Theory, Markov, Affine, Constrained Interval Arithmetic, and Relative Distance Measure and investigate which provides a correct solution to solve these equations. The analyzes of the generic solution forms of the linear equations, together with the application of a numerical example demonstrate that, for the linear equations AX + B = C, AX + BX = C and AX + B = CX + D, only Moore’s arithmetic with Interval Approximation Theory, CIA and RDM returns complete solution for the four equations. In order to complement this result, we developed analysis of the complexity of each correct solution in arithmetic intervals of Moore with Interval Approximation Theory, CIA and RDM to investigate the computational effort to compute each solution, obtaining as a result for Moore’s arithmetic, constant order of complexity, and linear order for the CIA and RDM arithmetics. Moreover, it is important to note that the problem of calculating the correct solution for linear equations with interval coefficients is said treatable or computable. After analyzing the solutions and complexity it is verified that Moore’s arithmetic, whenever possible, must be used to obtain the complete solution. In cases where Moore does not return a complete solution, CIA and RDM are the arithmetic indicated for solution.