There are several measures for comparing methods solving a single nonlinear equation. The first is the order of convergence, then cost to achieve such rate. This measured by counting number functions (and derivatives) evaluated at each step. After that, efficiency defined as function convergence and cost. Lately, idea basin attraction used. shows how far one can start still converge root. It al...

We revisit the quaternion Newton method for computing roots of a class of quaternion valued functions and propose modified algorithms for finding multiple roots of simple poly-nomials. We illustrate the performance of these new methods by presenting several numerical experiments.

There are several third order methods for solving a nonlinear algebraic equation having roots of a givenmultiplicitym. Here we compare a recent family of methods of order three to Euler– Cauchy’s method which is found to be the best in the previous work. There are fewer fourth order methods for multiple roots but we will not include them here. Published by Elsevier Inc.

We discuss the use of a single generic linear combination of multiplication matrices, and its reordered Schur factoriza-tion, to nd the roots of a system of multivariate polynomial equations. The principal contribution of the paper is to show how to reduce the multivariate problem to a univariate problem, even in the case of multiple roots, in a numerically stable way.

This lecture will show by example some of the problems that occur when the roots of a polynomial are computed using a standard polynomial root solver. In particular, polynomials of high degree with a large number of multiple roots will be considered, and it will be shown that even roundoff error due to floating point arithmetic, in the absence of data errors, is sufficient to cause totally inco...

We propose the numerical method defined by xn+1 = xn − λ f(xn − μh(xn)) f ′(xn) , n ∈ N, and determine the control parameter λ and μ to converge cubically. In addition, we derive the asymptotic error constant. Applying this proposed scheme to various test functions, numerical results show a good agreement with the theory analyzed in this paper and are proven using Mathematica with its high-prec...

Triune Ethics Theory (TET) is a psychological theory developed to meet three goals. First, it attempts to harvest critical findings from neurobiology, affective neuroscience, and cognitive science and to integrate them into moral psychology for the purpose of informing psychological research on the moral life of persons. In contrast to dominant theories that focus on top-down, deliberative reas...

The asymptotic distributions of Augmented-Dickey-Fuller (ADF) unit root tests for autoregressive processes with a unit or near-unit root are discussed in the presence of multiple stochastic level shifts of large size occurring independently in time. The distributions depend on a Brownian motion and a Poisson-type jump process. Due to the latter, tests based on standard critical values experienc...

In this paper, we study the algebraic equations over ring of p-complex numbers. Remainder division theorems and an analogue Bezout’s theorem for polynomials are represented. For 2nd 3rd degrees, conditions existence roots obtained, in some cases solutions given explicit form. arbitrary degree with invertible leading coefficient, on factorisation a unit coefficient proven where there simple root...

There are very few optimal fourth order methods for solving nonlinear algebraic equations having roots of multiplicity m. Here we compare 4 such methods, two of which require the evaluation of the ðm 1Þ root. We will show that such computation does not affect the overall cost of the method. Published by Elsevier Inc.