Accuracy of numerical solvers of non-linear equations based on Newton's and Halley's approximation using floating-point arithmetics

The paper describes particularities of using floating-point arithmetics for finding solutions of non-linear equations by the means of numerical approximation. Analysis of scientific literature shows scarcity of works studying these methods of numerical solvers in presence of limitations and particularities imposed by algebras of floating-point numbers despite well-known significance of these aspects when it comes to overall accuracy, predictability and usability of numerical solvers. Therefore, the paper describes some interesting results of theoretical and experimental study of these well-known Newton's and Halley's methods from the point of view of their implementability and problems that arise when floating-point arithmetic is used. The analysis is conducted both theoretically and experimentally using floating-point machines. On one hand, the experiments demonstrate correspondence between the predicted efficiency factors and ones that are measured, but on the other hand these measurements contradict intuitively predicted behavior of solvers if no floating-point specifics are taken into account.

Authors: А. A. Chusov, Yu. I. Efimova

Direction: Informatics, Computer Technologies And Control

Keywords: Householder methods, Newton's method, Halley's method, numerical solvers of non-linear equations, floating-point arithmetics, IEEE-754

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