On an application of symbolic computation and computer graphics to root-finders: The case of multiple roots of unknown multiplicity

Abstract

a b s t r a c tThe contemporary powerful mathematical software enables a new approach to handling and manipulating complex mathematical expressions and other mathematical objects. Particularly, the use of symbolic computation leads to new contribution to constructing and analyzing numerical algorithms for solving very difficult problems in applied mathematics and other scientific disciplines. In this paper we are concerned with the problem of determining multiple zeros when the multiplicity is not known in advance, a task that is seldom considered in literature. By the use of computer algebra system Mathematica, we employ symbolic computation through several programs to construct and investigate algorithms which both determine a sought zero and its multiplicity. Applying a recurrent formula for generating iterative methods of higher order for solving nonlinear equations, we construct iterative methods that serve (i) for approximating a multiple zero of a given function f when the order of multiplicity is unknown and, simultaneously, (ii) for finding exact order of multiplicity. In particular, we state useful cubically convergent iterative sequences that find the exact multiplicity in a few iteration steps. Such approach, combined with a rapidly convergent method for multiple zeros, provides the construction of efficient composite algorithms for finding multiple zeros of very high accuracy. The properties of the proposed algorithms are illustrated by several numerical examples and basins of attraction.

On generalized Halley-like methods for solving nonlinear equations

“…An iterative method for solving nonlinear equations of the form f (x) = 0 can be accelerated using the following theorem, see [32,33]. Theorem 3.…”

On optimal parameter of Laguerre's family of zero-finding methods

“…[2,3,5,15,19,21,23,25,26]. We have performed the dynamic study of the methods (3)- (7) on PC with Intel processor i7-2600 working on 3.4 GHz.…”

Computer tools for the construction and analysis of some efficient root-finding simultaneous methods

“…When testing any root-finding method it is always useful to examine its convergence behavior in practical implementation and compare the obtained data with theoretical results. For this reason we have calculated the so-called computational order of convergence (COC, for brevity) ρ c using the formula 1) log e (ν−1) /e (ν−2)…”

“…r=4 e (1) 7.60(−3) 1.06(−3) 5.91(−4) 1.65(−4) e (2) 4.12(−12 All tested methods have started with the same initial approximations with e (0) ≈ 0.812. The error norms e (ν) (ν = 1, 2, 3) are displayed in Tables 1 and 2, where A(−q) means A × 10 −q .…”

“…Contemporary powerful computer algebra systems, developed during the last two decades, give a new boost and breakthrough to the development and study of the numerical analysis methods, see, e.g., [1]- [4] and references cited there. In this paper we focus on iterative methods for the simultaneous determination of (real or complex) zeros of algebraic equations, a powerful tool of numerical analysis that is often used in applied mathematics but also for solving various problems of a wide variety of areas of scientific disciplines such as physics, engineering, computer science, economics, biology, astronomy, chemistry, and so on, see, e.g., [5], [6].…”