Let Sn be the positive real symmetric matrix of order n with (i, j ) entry equal to ( i + j − 2 j − 1 ) , and let x be a positive real number. Eigenvalues of the Hadamard (or entry wise) power S n are considered. In particular for k a positive integer, it is shown that both the Perron root and the trace of S n are approximately equal to 4 4k−1 ( 2n− 2 n− 1 )k . © 2005 Elsevier Inc. All rights r...

Let φ(x) be an Eisenstein polynomial over a local field and α be a root of φ(x). We describe the Galois group of φ(x) in the case where the Newton polygon of φ(αx+ α)/x has only one side.

In this paper, we proposed a simple modification of McDougall and Wotherspoon [11] method for approximating the root of univariate function. Our modification is based on the approximating the derivative in the corrector step of the proposed McDougall and Wotherspoon Newton like method using secant method. Numerical examples demonstrate the efficiency of the proposed method.

The present paper deals with fifth order convergent Newton-type with and without derivative iterative methods for estimating a simple root of nonlinear equations. The error equations are used to establish the fifth order of convergence of the proposed iterative methods. Finally, various numerical comparisons are made using MATLAB to demonstrate the performence of the developed methods.

Goldschmidt’s Algorithms for division and square root are often characterized as being useful for hardware implementation, and lacking self-correction. A reexamination of these algorithms show that there are good software counterparts that retain the speed advantage of Goldschmidt’s Algorithm over the Newton-Raphson iteration. A final step is needed, however, to get the last bit rounded correctly.

We investigate the well known Newton method to find roots of entire holomorphic functions. Our main result is that the immediate basin of attraction for every root is simply connected and unbounded. We also introduce “virtual immediate basins” in which the dynamics converges to infinity; we prove that these are simply connected as well.

Given a degree n univariate polynomial f(x), the Budan-Fourier function Vf (x) counts the sign changes in the sequence of derivatives of f evaluated at x. The values at which this function jumps are called the virtual roots of f , these include the real roots of f and any multiple root of its derivatives. This concept was introduced (by an equivalent property) by Gonzales-Vega, Lombardi, Mahé i...

Abstract. Let f(x, y), g(x, y) denote either a pair of holomorphic function germs, or a pair of monic polynomials in x whose coefficients are Laurent series in y. A polar root is a Newton-Puiseux root, x = γ(y), of the Jacobian J = fygx − fxgy, but not a root of f · g. We define the tree-model, T (f, g), for the pair, using the set of contact orders of the Newton-Puiseux roots of f and g. Our m...

We consider the Newton iteration for computing the principal matrix pth root, which is rarely used in the application for the bad convergence and the poor stability. We analyze the convergence conditions. In particular it is proved that the method converges for any matrix A having eigenvalues with modulus less than 1 and with positive real part. Based on these results we provide a general algor...