on the global convergence of schroder s iteration formula for real zeros of entire functions abstract schroder s formula of the second kind of order m of convergence s2 m formula is a generalization of newton s m 2 and halley s m 3 iterative formulae for finding zeros of functions the authors showed that the s2 m formula of every odd order m 3 converges globally and monotonically to real zeros of polynomials on the real line for halley s formula such the convergence property for real zeros of entire functions had been shown by davies and dawson in this paper we extend both results by showing that the s2 m formula of every odd order m 5 has the same convergence property for real zeros of entire functions by numerical examples we illustrate the monotonic convergence of the formula of odd order m 3 5 and 7 and the non monotonic convergence of even order m 2 4 and 6 further we compare several formulae of both first and second kinds in performance

on halley iteration this paper proposes some modified halley iterations for finding the zeros of polynomials we investigate the non overshoot properties of the modified halley iterations and other important properties that play key roles in solving symmetric eigenproblems we also extend halley iteration to systems of polynomial equations in several variables

nonstandard analysis and jump conditions for converging shock waves nonstandard analysis is an area of modern mathematics that studies abstract number systems containing both infinitesimal and infinite numbers this article applies nonstandard analysis to derive jump conditions for one dimensional converging shock waves in a compressible inviscid perfect gas it is assumed that the shock thickness occurs on an infinitesimal interval and the jump functions in the thermodynamic and fluid dynamic parameters occur smoothly across this interval predistributions of the heaviside function and the dirac delta measure are introduced to model the flow parameters across a shock wave the equations of motion expressed in nonconservative form are then applied to derive unambiguous relationships between the jump functions for the flow parameters

the quasi laguerre iteration the quasi laguerre iteration has been successfully established by the same authors in the spirit of laguerre s iteration for solving the eigenvalues of symmetric tridiagonal matrices the improvement in efficiency over laguerre s iteration is drastic this paper supplements the theoretical background of this new iteration including the proofs of the convergence properties

on the global convergence of schroder s iterative formulae for real roots of algebraic equations abstract schroder s formulae of the first s1 and second s2 kind of order m of convergence are generalizations of newton s m 2 and halley s s2 m 3 iterative formulae for finding zeros of functions davies and dawson show that for entire functions with only real zeros halley s formula converges globally and monotonically to their zeros independently of the initial value on the real line we show that the s2 formulae of odd order 5 enjoy the same convergence feature for polynomials with only real zeros numerical examples illustrate this we illustrate no monotonic convergence of the s1 formulae and of the s2 formulae of even order

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

a non standard analysis approach to systems involving friction many physical systems involve discontinuous forces the conventional numerical discretization approach does not always yield a consistent prediction of the behavior of these systems here as an example of a system involving discontinuous forces a simple frictional system subjected to continuous as well as discontinuous excitations is considered a set of orthogonal tracking functions is introduced and the discontinuous excitation is expanded in a series of these tracking functions using non standard analysis the discontinuous functions are rigorously represented by continuous hyperreal approximations the transformation of discontinuous forces to continuous ones allows consistent prediction of the system s behavior thus providing a reliable technique for simulation of systems involving discontinuous forces

generalizations of laguerre s method higher order methods laguerre s method is an efficient and reliable method for finding zeros of polynomials and certain other functions a new derivation and motivation of laguerre s method is given which allows it to be included in a class of methods as general as methods of order three or more based on direct generalized hermite or hyperosculatory interpolation members of this new class share with laguerre s method the property of being globally convergent to zeros of polynomials with only real zeros and have the same order of convergence at simple zeros as the classic methods based on generalized hermite interpolation methods of order 4 and 3 303 are investigated and numerical results indicate that for large 100 times 100 eigenvalue problems the method of order 3 303 is as efficient and reliable as laguerre s method

a one parameter family of locally quartically convergent zero finding methods a one parameter family of iteration functions for finding simple and multiple zeros of analytic functions is derived the family includes as a special case traub s quartic square root method and as limiting cases the kiss method of order 4 the halley and the newton methods all the methods of the family are locally quartically convergent for a simple or multiple zero with known multiplicity the asymptotic error constants for the methods of the family are given the decreasing ratio at infinity of iteration functions is discussed the optimum parameter of the family for polynomials is given

fixed point and newton s methods for solving a nonlinear equation from linear to high order convergence in this paper we revisit the necessary and sufficient conditions for linear and high order convergence of fixed point and newton s methods based on these conditions we extend schroder s process of the first kind to increase the order of convergence of the fixed point method we also obtain two processes to increase the order of convergence of newton s method one of which is schroder s process of the second kind for which several forms are also presented a link between schroder s two processes is given examples and numerical experiments are included

extraneous fixed points basin boundaries and chaotic dynamics for schro der and ko nig rational iteration functions the schroder and konig iteration schemes to find the zeros of a polynomial functiong z represent generalizations of newton s method in both schemes iteration functionsf m z are constructed so that sequencesz n 1 f m z n converge locally to a rootz ofg z aso z n z m it is well known that attractive cycles other than the zerosz may exist for newton s method m 2 asm increases the iteration functions add extraneous fixed points and cycles whether attractive or repulsive they affect the julia set basin boundaries the konig functionsk m z appear to minimize such perturbations in the case of two roots e g g z z 2 1 cayley s classical result for the basins of attraction of newton s method is extended for allk m z the existence of chaotic z n sequences is also demonstrated for these iteration methods

numerical algorithms based on analytic function values at roots of unity let f z be an analytic or meromorphic function in the closed unit disk sampled at the n th roots of unity based on these data how can we approximately evaluate f z or f m z at a point z in the disk how can we calculate the zeros or poles of f in the disk these questions exhibit in the purest form certain algorithmic issues that arise across computational science in areas including integral equations partial differential equations and large scale linear algebra we analyze some of the possibilities and emphasize the distinction between algorithms based on polynomial or rational interpolation and those based on trapezoidal rule approximations of cauchy integrals we then show how these developments apply to the problem of computing the eigenvalues in the disk of a matrix of large dimension finally we highlight the power of rational in comparison with polynomial approximations for some of these problems

geometric constructions of iterative functions to solve nonlinear equations in this paper we present the geometrical interpretation of several iterative methods to solve a nonlinear scalar equation in addition we also review the extension to general banach spaces and some computational aspects of these methods

the lax solution to a hamilton jacobi equation and its generalizations part 2 this article is a continuation of j math sci 99 no 5 1541 1547 2000 devoted to the validity of the lax formula cited in the article of crandall ishii and lions bull ams 27 no 1 1 67 2000 u left x t right mathop inf limits y in mathbb r n left user1 v left y right frac 1 2t left x y right 2 right for a solution to the hamilton jacobi nonlinear partial differential equation frac partial u partial t frac 1 2 left nabla u right 2 0 u left t 0 right user1 v where the cauchy data user1 v mathbb r user1 n to mathbb r are now a function semicontinuous from below left cdot right left langle cdot cdot right rangle is the usual norm in mathbb r user1 n n in mathbb z and t in mathbb r is a positive evolution parameter we proved that the lax formula solves the cauchy problem 2 at all points x in mathbb r n t in mathbb r fixed save for an exceptional set of points r of the f type having zero lebesgue measure in addition we formulate a similar lax type formula without proof for a solution to a new nonlinear equation of the hamilton jacobi type frac partial u partial t frac 1 2 left nabla u right 2 frac beta u 2 left x right 2 frac 1 2 left langle jx x right rangle 0 where user1 j mathbb r user1 n to mathbb r user1 n is a diagonal positive definite matrix mentioned in part i and having interesting applications in modern mathematical physics

generalized computation of schroder iteration functions to motivate families of julia and mandelbrot like sets schroder iteration functions a generalization of the newton raphson method to determine roots of equations are generally rational functions which possess some critical points free to converge to attracting cycles these free critical points however satisfy some higher degree polynomial equations we present a new algorithmic construction to compute in general all of the schroder functions terms as well as to maximize the computational efficiency of these functions associated with a one parameter family of cubic polynomials finally we examine the julia sets of the schroder functions constructed to converge to the nth roots of unity these roots basins of attraction and the orbits of all free critical points of these functions for order higher than four as applied to the one parameter family of cubic polynomials mentioned above

a nonstandard analysis of a simple discontinuous force equation modelling continuous motion modern mathematical techniques have been applied to analyze a simple discontinuous force equation the results of which are applicable to general friction problems containing discontinuities

on schroder s families of root finding methods schroder s methods of the first and second kind for solving a nonlinear equation f x 0 originally derived in 1870 are of great importance in the theory and practice of iteration processes they were rediscovered several times and expressed in different forms during the last 130 years it was proved in the paper of petkovic and herceg 1999 7 that even seven families of iteration methods for solving nonlinear equations are mutually equivalent in this paper we show that these families are also equivalent to another four families of iteration methods and find that all of them have the origin in schroder s generalized method of the second kind presented in 1870 in the continuation we consider smale s open problem from 1994 about possible link between schroder s methods of the first and second kind and state the link in a simple way

analysis of solutions of a noncanonical hamilton jacobi equation using the generalized characteristics method and the hopf lax representations abstract the cauchy problem for a noncanonical nonlinear hamilton jacobi equation is studied using the method of generalized characteristics the hopf lax representation and a recent generalization of the leray schauder fixed point theorem also are used to analyze the solutions

the schwarzian newton method for solving nonlinear equations with applications the author acknowledges financial support from ministerio de economia y competitividad project mtm2012 34787