4,169 research outputs found
An optimal three-point eighth-order iterative method without memory for solving nonlinear equations with its dynamics
We present a three-point iterative method without memory for solving
nonlinear equations in one variable. The proposed method provides convergence
order eight with four function evaluations per iteration. Hence, it possesses a
very high computational efficiency and supports Kung and Traub's conjecture.
The construction, the convergence analysis, and the numerical implementation of
the method will be presented. Using several test problems, the proposed method
will be compared with existing methods of convergence order eight concerning
accuracy and basin of attraction. Furthermore, some measures are used to judge
methods with respect to their performance in finding the basin of attraction.Comment: arXiv admin note: substantial text overlap with arXiv:1508.0174
A Collection of Challenging Optimization Problems in Science, Engineering and Economics
Function optimization and finding simultaneous solutions of a system of
nonlinear equations (SNE) are two closely related and important optimization
problems. However, unlike in the case of function optimization in which one is
required to find the global minimum and sometimes local minima, a database of
challenging SNEs where one is required to find stationary points (extrama and
saddle points) is not readily available. In this article, we initiate building
such a database of important SNE (which also includes related function
optimization problems), arising from Science, Engineering and Economics. After
providing a short review of the most commonly used mathematical and
computational approaches to find solutions of such systems, we provide a
preliminary list of challenging problems by writing the Mathematical
formulation down, briefly explaning the origin and importance of the problem
and giving a short account on the currently known results, for each of the
problems. We anticipate that this database will not only help benchmarking
novel numerical methods for solving SNEs and function optimization problems but
also will help advancing the corresponding research areas.Comment: Accepted as an invited contribution to the special session on
Evolutionary Computation for Nonlinear Equation Systems at the 2015 IEEE
Congress on Evolutionary Computation (at Sendai International Center, Sendai,
Japan, from 25th to 28th May, 2015.
Simple and Nearly Optimal Polynomial Root-finding by Means of Root Radii Approximation
We propose a new simple but nearly optimal algorithm for the approximation of
all sufficiently well isolated complex roots and root clusters of a univariate
polynomial. Quite typically the known root-finders at first compute some crude
but reasonably good approximations to well-conditioned roots (that is, those
isolated from the other roots) and then refine the approximations very fast, by
using Boolean time which is nearly optimal, up to a polylogarithmic factor. By
combining and extending some old root-finding techniques, the geometry of the
complex plane, and randomized parametrization, we accelerate the initial stage
of obtaining crude to all well-conditioned simple and multiple roots as well as
isolated root clusters. Our algorithm performs this stage at a Boolean cost
dominated by the nearly optimal cost of subsequent refinement of these
approximations, which we can perform concurrently, with minimum processor
communication and synchronization. Our techniques are quite simple and
elementary; their power and application range may increase in their combination
with the known efficient root-finding methods.Comment: 12 pages, 1 figur
Deflation for semismooth equations
Variational inequalities can in general support distinct solutions. In this
paper we study an algorithm for computing distinct solutions of a variational
inequality, without varying the initial guess supplied to the solver. The
central idea is the combination of a semismooth Newton method with a deflation
operator that eliminates known solutions from consideration. Given one root of
a semismooth residual, deflation constructs a new problem for which a
semismooth Newton method will not converge to the known root, even from the
same initial guess. This enables the discovery of other roots. We prove the
effectiveness of the deflation technique under the same assumptions that
guarantee locally superlinear convergence of a semismooth Newton method. We
demonstrate its utility on various finite- and infinite-dimensional examples
drawn from constrained optimization, game theory, economics and solid
mechanics.Comment: 24 pages, 3 figure
Computing Puiseux series : a fast divide and conquer algorithm
Let be a polynomial of total degree defined over
a perfect field of characteristic zero or greater than .
Assuming separable with respect to , we provide an algorithm that
computes the singular parts of all Puiseux series of above in less
than operations in , where
is the valuation of the resultant of and its partial derivative with
respect to . To this aim, we use a divide and conquer strategy and replace
univariate factorization by dynamic evaluation. As a first main corollary, we
compute the irreducible factors of in up to an
arbitrary precision with arithmetic
operations. As a second main corollary, we compute the genus of the plane curve
defined by with arithmetic operations and, if
, with bit operations
using a probabilistic algorithm, where is the logarithmic heigth of .Comment: 27 pages, 2 figure
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