A Remarkable Chord Iterative Method for Roots of Uncertain Multiplicity

In this note we at first briefly review iterative methods for effectively approaching a root of an unknown multiplicity. We describe a first order, then a second order estimate for the multiplicity index m of the approached root. Next we present a second order, two-step method for iteratively nearing a root of an unknown multiplicity. Subsequently, we introduce a novel chord, or a two- step method, not requiring beforehand knowledge of the multiplicity index m of the sought root, nor requiring higher order derivatives of the equilibrium function, which is quadratically convergent for any , and then reverts to superlinear.

Computationally efficient direction finding using polynomial rooting with reduced-order and real-valued computations

The root multiple signal classification（root-MUSIC） algorithm is one of the most important techniques for direction of arrival（DOA） estimation. Using a uniform linear array（ULA） composed of M sensors, this method usually estimates L signal DOAs by finding roots that lie closest to the unit circle of a（2M-1）-order polynomial, where L 〈 M. A novel efficient root-MUSIC-based method for direction estimation is presented, in which the order of polynomial is efficiently reduced to 2L. Compared with the unitary root-MUSIC（U-root-MUSIC） approach which involves real-valued computations only in the subspace decomposition stage, both tasks of subspace decomposition and polynomial rooting are implemented with real-valued computations in the new technique,which hence shows a significant efficiency advantage over most state-of-the-art techniques. Numerical simulations are conducted to verify the correctness and efficiency of the new estimator.

High-Order Iterative Methods Repeating Roots a Constructive Recapitulation

This paper considers practical, high-order methods for the iterative location of the roots of nonlinear equations, one at a time. Special attention is being paid to algorithms also applicable to multiple roots of initially known and unknown multiplicity. Efficient methods are presented in this note for the evaluation of the multiplicity index of the root being sought. Also reviewed here are super-linear and super-cubic methods that converge contrarily or alternatingly, enabling us, not only to approach the root briskly and confidently but also to actually bound and bracket it as we progress.

Numerical realization of the conditions of Max Nther's residual intersection theorem

The aim of this paper is to study numerical realization of the conditions of Max Nother＇s residual intersection theorem. The numerical realization relies on obtaining the inter- section of two algebraic curves by homotopy continuation method, computing the approximate places of an algebraic curve, getting the exact orders of a polynomial at the places, and determin- ing the multiplicity and character of a point of an algebraic curve. The numerical experiments show that our method is accurate, effective and robust without using multiprecision arithmetic, even if the coefficients of algebraic curves are inexact. We also conclude that the computational complexity of the numerical realization is polynomial time.

Nonlinear Equations Solving with Intelligent Optimization Algorithms: A Survey

Nonlinear Equations(NEs),which may usually have multiple roots,are ubiquitous in diverse fields.One of the main purposes of solving NEs is to locate as many roots as possible simultaneously in a single run,however,it is a difficult and challenging task in numerical computation.In recent years,Intelligent Optimization Algorithms(IOAs)have shown to be particularly effective in solving NEs.This paper provides a comprehensive survey on IOAs that have been exploited to locate multiple roots of NEs.This paper first revisits the fundamental definition of NEs and reviews the most recent development of the transformation techniques.Then,solving NEs with IOAs is reviewed,followed by the benchmark functions and the performance comparison of several state-of-the-art algorithms.Finally,this paper points out the challenges and some possible open issues for solving NEs.

Solving Nonlinear Equations Systems with an Enhanced Reinforcement Learning Based Differential Evolution

Nonlinear equations systems(NESs)arise in a wide range of domains.Solving NESs requires the algorithm to locate multiple roots simultaneously.To deal with NESs efficiently,this study presents an enhanced reinforcement learning based differential evolution with the following major characteristics:(1)the design of state function uses the information on the fitness alternation action;(2)different neighborhood sizes and mutation strategies are combined as optional actions;and(3)the unbalanced assignment method is adopted to change the reward value to select the optimal actions.To evaluate the performance of our approach,30 NESs test problems and 18 test instances with different features are selected as the test suite.The experimental results indicate that the proposed approach can improve the performance in solving NESs,and outperform several state-of-the-art methods.