constructing higher order methods for obtaining the multiple roots of nonlinear equations this paper concentrates on iterative methods for obtaining the multiple roots of nonlinear equations using the computer algebra system mathematica we construct an iterative scheme and discuss the conditions to obtain fourth order methods from it all the presented fourth order methods require one function and two derivative evaluation per iteration and are optimal higher order iterative methods for obtaining multiple roots we present some special methods from the iterative scheme including some known already numerical examples are also given to show their performance

new third order nonlinear solvers for multiple roots abstract two third order methods for finding multiple zeros of nonlinear functions are developed one method is based on chebyshev s third order scheme for simple roots and the other is a family based on a variant of chebyshev s which does not require the second derivative two other more efficient methods of lower order are also given these last two methods are variants of chebyshev s and osada s schemes the informational efficiency of the methods is discussed all these methods require the knowledge of the multiplicity

a third order modification of newton s method for multiple roots in this paper we present a new third order modification of newton s method for multiple roots which is based on existing third order multiple root finding methods numerical examples show that the new method is competitive to other methods for multiple roots

a general family of third order method for finding multiple roots abstract in this paper we describe a general family of iterative methods for approximating a multiple root z with multiplicity m of a complex defined function almost of the family of the methods existing in the literature that use two function and one derivative evaluations are a special choice of this general method we give some conditions to have the third order of convergence and we discuss how to choose a small asymptotic error constant which may be affect the speed of the convergence using mathematica with its high precision compatibility we present some numerical examples to confirm the theoretical results

analogies between finite dimensional irreducible representations of so 2n and infinite dimensional irreducible representations of sp 2n r ii plethysms the basic spin difference character of so 2n is a useful device in dealing with characters of irreducible spinor representations of so 2n it is shown here that its kth fold symmetrized powers or plethysms associated with partitions of k factorize in such a way that r where r is the frobenius rank of the analogy between so 2n and sp 2n r is shown to be such that the plethysms of the basic harmonic or metaplectic character of sp 2n r factorize in the same way to give r moreover the analogy is shown to extend to the explicit decompositions into characters of irreducible representations of so 2n and sp 2n r not only for the plethysms themselves but also for their factors and explicit formulas are derived for each of these decompositions expressed in terms of various group subgroup branching rule multiplicities particularly those defined by the restriction from o k to the symmetric group sk illustrative examples are included as well as a

letter to the editor new higher order methods for solving nonlinear equations with multiple roots for a nonlinear equation f x 0 having a multiple root we consider steffensen s transformation t using the transformation say f q x t qf x for integer q 2 repeatedly we develop higher order iterative methods which require neither derivatives of f x nor the multiplicity of the root it is proved that the convergence order of the proposed iterative method is 1 2 q 2 for any equation having a multiple root of multiplicity m 2 the efficiency of the new method is shown by the results for some numerical examples

a new third order family of nonlinear solvers for multiple roots in this paper a new family of third order methods for finding multiple roots of nonlinear equations has been introduced this family requires one function and two derivative evaluation per iteration the family contains several known third order methods as special cases some examples are presented to show the performance of the presented family

on an application of symbolic computation and computer graphics to root finders the contemporary powerful mathematical software enables a new approach to handling and manipulating complex mathematical expressions and other mathematical objects particularly the use of symbolic computation leads to new contribution to constructing and analyzing numerical algorithms for solving very difficult problems in applied mathematics and other scientific disciplines in this paper we are concerned with the problem of determining multiple zeros when the multiplicity is not known in advance a task that is seldom considered in literature by the use of computer algebra system mathematica we employ symbolic computation through several programs to construct and investigate algorithms which both determine a sought zero and its multiplicity applying a recurrent formula for generating iterative methods of higher order for solving nonlinear equations we construct iterative methods that serve i for approximating a multiple zero of a given function f when the order of multiplicity is unknown and simultaneously ii for finding exact order of multiplicity in particular we state useful cubically convergent iterative sequences that find the exact multiplicity in a few iteration steps such approach combined with a rapidly convergent method for multiple zeros provides the construction of efficient composite algorithms for finding multiple zeros of very high accuracy the properties of the proposed algorithms are illustrated by several numerical examples and basins of attraction

a new fourth order iterative method for finding multiple roots of nonlinear equations in this paper we present a new fourth order method for finding multiple roots of nonlinear equations it requires one evaluation of the function and two of its first derivative per iteration finally some numerical examples are given to show the performance of the presented method compared with some known third order methods

an optimal reconstruction of chebyshev halley type methods for nonlinear equations having multiple zeros abstract establishment of a new optimal higher order iterative scheme for multiple zeros with known multiplicity m 1 of univariate function is one of the hard and demanding task in the area of computational mathematics and numerical analysis we write this paper with the aim to propose a new higher order reconstruction of chebyshev halley type iteration functions for multiple zeros in a simple way minimum order of convergence of our scheme is seven and maximum reaches to eight without using any extra functional evaluation with regard to computational cost each member of our scheme needs four functional evaluations at each step so the maximum efficiency index of our scheme is 1 6818 for 1 in addition we thoroughly study the convergence analysis which confirms the theoretical order of convergence moreover we check the efficiency and compare them to the existing ones on four real life problems namely fractional conversion chemical engineering continuous stirred tank reactor and eigen value problems and two well known academic problems finally we confirm on the basis of obtained results that our methods have better residual error errors between two consecutive iterations and stable computational order of convergence as compared to the other existing ones

on the development of iterative methods for multiple roots 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 m 1 s t root we will show that such computation does not affect the overall cost of the method

families of third and fourth order methods for multiple roots of nonlinear equations this paper presents two families of higher order iterative methods for solving multiple roots of nonlinear equations one is of order three and the other is of order four the presented iterative families all require two evaluations of the function and one evaluation of its first derivative thus the latter is of optimal order the third order family contains several iterative methods known already and different from the optimal fourth order methods for multiple roots known already the presented fourth order family use the modified newton s method as its first step local convergence analyses and some special cases of the presented families are given we also carry out some numerical examples to show their performance

some second derivative free variants of halley s method for multiple roots in this paper we present two new families of third order methods for finding multiple roots of nonlinear equations each of them is based on a variant of the halley s method for simple roots free from second derivative one of the families requires one evaluation of the function and two of its first derivative per iteration and the other family requires two evaluations of the function and one of its first derivative several numerical examples are given to illustrate the performance of the presented methods

some fourth order nonlinear solvers with closed formulae for multiple roots in this paper we present six new fourth order methods with closed formulae for finding multiple roots of nonlinear equations the first four of them require one function and three derivative evaluation per iteration the last two require one function and two derivative evaluation per iteration several numerical examples are given to show the performance of the presented methods compared with some known methods

basin attractors for various methods for multiple roots abstract there are several methods for approximating the multiple zeros of a nonlinear function when the multiplicity is known the methods are classified by the order informational efficiency and efficiency index here we consider other criteria namely the basin of attraction of the method and its dependence on the order we discuss all known methods of orders two to four and present the basin of attraction for several examples

high order nonlinear solver for multiple roots a method of order four for finding multiple zeros of nonlinear functions is developed the method is based on jarratt s fifth order method for simple roots and it requires one evaluation of the function and three evaluations of the derivative the informational efficiency of the method is the same as previously developed schemes of lower order for the special case of double root we found a family of fourth order methods requiring one less derivative thus this family is more efficient than all others all these methods require the knowledge of the multiplicity

new third and fourth order nonlinear solvers for computing multiple roots abstract we present a new third order method for finding multiple roots of nonlinear equations based on the scheme for simple roots developed by kou et al j kou y li x wang a family of fourth order methods for solving non linear equations appl math comput 188 2007 1031 1036 further investigation gives rise to new third and fourth order families of methods which do not require second derivative the fourth order family has optimal order since it requires three evaluations per step namely one evaluation of function and two evaluations of first derivative the efficacy is tested on a number of relevant numerical problems computational results ascertain that the present methods are competitive with other similar robust methods

a new family of methods for single and multiple roots abstract we present a new family of iterative methods for multiple and single roots of nonlinear equations this family contains as a special case the authors family for finding simple roots from herceg and herceg 2015 some well known classical methods for simple roots for example newton potra ptak newton steffensen king and ostrowski s methods belong to this family which implies that our new family contains modifications of these methods suitable for finding multiple roots convergence analysis shows that our family contains methods of convergence order from 2 to 4 the new methods require two function evaluations and one evaluation of the first derivative per iteration so all our fourth order methods are optimal in terms of the kung and traub conjecture several examples are presented and compared through various test equations relevant numerical experiments strongly support the claimed theory in this paper extraneous fixed points of the iterative maps associated with the proposed methods are also investigated their dynamics is explored along with illustrated basins of attraction for various polynomials

a sixth order family of three point modified newton like multiple root finders and the dynamics behind their extraneous fixed points a class of three point sixth order multiple root finders and the dynamics behind their extraneous fixed points are investigated by extending modified newton like methods with the introduction of the multivariate weight functions in the intermediate steps the multivariate weight functions dependent on function to function ratios play a key role in constructing higher order iterative methods extensive investigation of extraneous fixed points of the proposed iterative methods is carried out for the study of the dynamics associated with corresponding basins of attraction numerical experiments applied to a number of test equations strongly support the underlying theory pursued in this paper relevant dynamics of the proposed methods is well presented with a variety of illustrative basins of attraction applied to various test polynomials

iterative methods for solving nonlinear equations with finitely many roots in an interval in this paper we consider a nonlinear equation f x 0 having finitely many roots in a bounded interval based on the so called numerical integration method b i yun a non iterative method for solving non linear equations appl math comput 198 2008 691 699 without any initial guess we propose iterative methods to obtain all the roots of the nonlinear equation in the result an algorithm to find all of the simple roots and multiple ones as well as the extrema of f x is developed moreover criteria for distinguishing zeros and extrema are included in the algorithm availability of the proposed method is demonstrated by some numerical examples

convergence radius of halley s method for multiple roots under center holder continuous condition recently a new treatment based on taylor s expansion to give the estimate of the convergence radius of iterative method for multiple roots has been presented it has been successfully applied to enlarge the convergence radius of the modified newton s method and osada s method for multiple roots this paper re investigates the convergence radius of halley s method under the condition that the derivative f m 1 of function f satisfies the center holder continuous condition we show that our result can be obtained under much weaker condition and has a wider range of application than that given by bi et al 2011 21

constructing a family of optimal eighth order modified newton type multiple zero finders along with the dynamics behind their purely imaginary extraneous fixed points abstract an optimal family of eighth order multiple zero finders and the dynamics behind their basins of attraction are proposed by considering modified newton type methods with multivariate weight functions extensive investigation of purely imaginary extraneous fixed points of the proposed iterative methods is carried out for the study of the dynamics associated with corresponding basins of attraction numerical experiments strongly support the underlying theory pursued in this paper an exploration of the relevant dynamics of the proposed methods is presented along with illustrative basins of attraction for various polynomials

convergence radius of osada s method under center holder continuous condition recently a new treatment based on taylor s expansion to give the estimate of the convergence radius of iterative method for multiple roots has been presented it has been successfully applied to enlarge the estimate of the convergence radius of the modified newton s method for multiple roots using the similarly treatment this paper investigates the convergence radius of the osada s method under the condition that the derivative f m 1 of function f satisfies the center holder continuous condition by some examples we show the treatment is simpler and efficient once again the uniqueness ball of solution is also discussed