High order iterative methods for approximating square roots

Given any natural numberm 2, we describe an iteration functiong _m (x) having the property that for any initial iterate \sqrt \alpha $$ " align="middle" border="0"> , the sequence of fixed-point iterationx _k +1 =g _m (x _k ) converges monotonically to having anm-th order rate of convergence. Form = 2 and 3,g _m (x) coincides with Newton's and Halley's iteration functions, respectively, as applied top(x) =x ² – .This research is supported in part by the National Science Foundation under Grant No. CCR-9208371.     

Finite and infinite nested square roots convergent to unity

The Ramanujan Journal - We investigate finite and infinite nested square root formulas convergent to unity.     

Ordering of nested square roots of 2 according to the Gray code

In this paper, we discuss some relations between zeros of Lucas–Lehmer polynomials and the Gray code. We study nested square roots of 2 applying a “binary code” that associates bits 0 and 1 to “plus” and “minus” signs in the nested form. This gives the possibility to obtain an ordering for the zeros of Lucas–Lehmer polynomials, which take the form of nested square roots of 2.     

On square roots of isometries

Abstract

In various normed spaces we answer the question of when a given isometry is a square of some isometry. In particular, we consider (real and complex) matrix spaces equipped with unitarily invariant norms and unitary congruence invariant norms, as well as some infinite dimensional spaces illustrating the difference between finite and infinite dimensions.     

On square roots of M-matrices

The question of the existence and uniqueness of an M-matrix which is a square root of an M-matrix is discussed. The results are then used to derive some new necessary and sufficient conditions for a real matrix with nonpositive off diagonal elements to be an M-matrix.     

On the square roots of strictly quasiaccretive complex matrices

The square roots of a complex n×n matrix A for which the real part of e^ixA, where $\text{χ?[}\text{?π}\text{2}\text{,}\text{π}\text{2}\text{]}$, is positive definite are investigated. It is shown, for example, that when $\text{χ=}\text{?π}\text{2}$ (i.e., A is strictly dissipative), A has a unique square root whose real and imaginary parts are both positive definite.     

On numerical ranges and roots

Existence of the fractional powers is established in Banach algebra setting, in terms of the numerical ranges of elements involved. The behavior of the spectra and (for Hermitian ∗-algebras satisfying some additional hypotheses) the ∗-numerical range under taking these powers also is investigated.     

On roots of random polynomials

We study the distribution of the complex roots of random polynomials of degree with i.i.d. coefficients. Using techniques related to Rice's treatment of the real roots question, we derive, under appropriate moment and regularity conditions, an exact formula for the average density of this distribution, which yields appropriate limit average densities. Further, using a different technique, we prove limit distribution results for coefficients in the domain of attraction of the stable law.

     

On computing complex square roots of real matrices

We present an idea for computing complex square roots of matrices using only real arithmetic.     

On the tensor square of non-abelian nilpotent finite-dimensional Lie algebras

For every finite p-group G of order p ⁿ with derived subgroup of order p ^m , Rocco [N.R. Rocco, On a construction related to the nonabelian tensor square of a group, Bol. Soc. Brasil. Mat. 1 (1991), pp. 63–79] proved that the order of tensor square of G is at most p ^n(n?m). This upper bound has been improved recently by the author [P. Niroomand, On the order of tensor square of non abelian prime power groups (submitted)]. The aim of this article is to obtain a similar result for a non-abelian nilpotent Lie algebra of finite dimension. More precisely, for any given n-dimensional non-abelian nilpotent Lie algebra L with derived subalgebra of dimension m we have dim(L???L)?≤?(n???m)(n???1)?+?2. Furthermore for m?=?1, the explicit structure of L is given when the equality holds.     

On Extensions of Chain Algebras and Atoms of the Lattice of Radicals

In this article we study the problem when a left distributive algebra over a field determines an atom of the lattice of all radicals of rings. We show that this problem is strictly connected with the problem of describing extensions of left chain algebras and give a characterization of left chain algebras determining atoms.     

On an unknown algorithm for computing square roots

A relatively unknown algorithm for computing square roots is presented. In a survey of more than 50 teachers of mathematics, not one of them recalled having ever encountered it before. The algorithm is first illustrated and then proved.     

On the roots of Wiener polynomials of graphs

The Wiener polynomial of a connected graph $G$ is defined as $W(G;x)=\sum x^{d(u,v)}$, where $d(u,v)$ denotes the distance between $u$ and $v$, and the sum is taken over all unordered pairs of distinct vertices of $G$. We examine the nature and location of the roots of Wiener polynomials of graphs, and in particular trees. We show that while the maximum modulus among all roots of Wiener polynomials of graphs of order $n$ is $\left(\genfrac{}{}{0ex}{}{n}{2}\right)?1$, the maximum modulus among all roots of Wiener polynomials of trees of order $n$ grows linearly in $n$. We prove that the closure of the collection of real roots of Wiener polynomials of all graphs is precisely $(?\infty ,0]$, while in the case of trees, it contains $(?\infty ,?1]$. Finally, we demonstrate that the imaginary parts and (positive) real parts of roots of Wiener polynomials can be arbitrarily large.     

Construction of nested orthogonal arrays

A (symmetric) nested orthogonal array is a symmetric orthogonal array OA(N,k,s,g) which contains an OA(M,k,r,g) as a subarray, where M<N and r<s. In this communication, some methods of construction of nested symmetric orthogonal arrays are given. Asymmetric nested orthogonal arrays are defined and a few methods of their construction are described.     

Wild radical square zero algebras

It is shown that a radical square zero algebra is wild, if and only if it is of Corner’s type, and it is strictly wild if and only if it is Endo-wild. This gives a negative answer to a problem posed by Simson.     

Doyen–Wilson theorem for nested Steiner triple systems

The obvious necessary conditions for the existence of a nested Steiner triple system of order v containing a nested subsystem of order w are v ≥ 3w + 4 and v ≡ w ≡ 1 (mod 6). We show that these conditions are also sufficient. © 2004 Wiley Periodicals, Inc.     

On completely multiplicative functions whose values are roots of unity

Summary We prove that if a complex valued completely multiplicative function F and a positive integer ℓ≦5 satisfy the condition F(N) = U_ℓ, where U_ℓ is the set of all ℓ-th roots of unity, then {F(n+1) F(n) ∣ nε N} = U_ℓ.