On an application of symbolic computation and computer graphics to root-finders: The case of multiple roots of unknown multiplicity
Abstract
The 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.
Description
The article of record as published may be found at http://dx.doi.org/10.1016/j.cam.2016.06.008
Rights
This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.Collections
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