comparative study of methods of various orders for finding repeated roots of nonlinear equations abstract in this paper we are considering 20 families of methods for finding repeated roots of a nonlinear equation the methods are of order up to 8 we use the idea of basin of attraction to compare the methods we found that 4 methods performed best based on 3 quantitative criteria

q analogs of distance matrices of 3 hypertrees abstract we consider the distance matrix of trees in 3 uniform hypergraphs which we call 3 hypertrees we give a formula for the inverse of a few q analogs of distance matrices of 3 hypertrees t some results are analogs of results by bapat et al for graphs we give an alternate proof of the result that the determinant of the distance matrix of a 3 hypertree t depends only on n the number of vertices of t further we give a pfaffian identity for a principal submatrix of some skew symmetrized distance matrices of 3 hypertrees when we fix an ordering of the vertices and assign signs appropriately a result of graham hoffman and hosoya relates the determinant of the distance matrix of a graph and the determinants of its two connected blocks when the graph has as blocks a fixed connected graph h which satisfy some conditions we give a formula for the inverse of its distance matrix this result generalises a result of graham and lovasz when each block of g is a fixed graph g we also give some corollaries about the sum of the entries of the inverse of the distance matrix of g and some of its analogs

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

arbitrarily oriented perfectly conducting wedge over a dielectric half space diffraction and total far field complex scattering targets are often made by structures constituted by wedges and penetrable substrates which may interact at near field in this paper we describe a complete procedure to study this problem with possible developments in radar technologies like gpr antenna development or electromagnetic compatibility tips near substrates the diffraction of an incident plane wave by a perfectly conducting pec wedge over a dielectric half space is studied using generalized wiener hopf equations gwhes and the solution is obtained using analytical and numerical analytical approaches that reduce the wiener hopf wh factorization to integral equations ies the mathematical aspects are described in a unified and consistent theory for angular and layered region problems the proposed procedure is valid for the general case and the paper focuses on e polarization at normal incidence the solutions are given in terms of geometrical uniform theory of diffraction gtd utd diffraction coefficients and total far fields for engineering applications this paper presents several numerical test cases that show the validity of the proposed method

on developing fourth order optimal families of methods for multiple roots and their dynamics there are few optimal fourth order methods for solving nonlinear equations when the multiplicity m of the required root is known in advance therefore the first focus of this paper is on developing new fourth order optimal families of iterative methods by a simple and elegant way computational and theoretical properties are fully studied along with a main theorem describing the convergence analysis another main focus of this paper is the dynamical analysis of the rational map associated with our proposed class for multiple roots as far as we know there are no deep study of this kind on iterative methods for multiple roots further using mathematica with its high precision compatibility a variety of concrete numerical experiments and relevant results are extensively treated to confirm the theoretical development

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 class of two point sixth order multiple zero finders of modified double newton type and their dynamics under the assumption of the known multiplicity of zeros of nonlinear equations a class of two point sextic order multiple zero finders and their dynamics are investigated in this paper by means of extensive analysis of modified double newton type of methods with the introduction of a bivariate weight function dependent on function to function and derivative to derivative ratios higher order convergence is obtained additional investigation is carried out for extraneous fixed points of the iterative maps associated with the proposed methods along with a comparison with typically selected cases through a variety of test equations numerical experiments strongly support the theory developed in this paper in addition relevant dynamics of the proposed methods is successfully explored for various polynomials with a number of illustrative basins of attraction

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

construction and efficiency of multipoint root ratio methods for finding multiple zeros abstract several root ratio multipoint methods for finding multiple zeros of univariate functions were recently presented the characteristic of these methods is that they deal with m th root of ratio of two functions hence the name root ratio methods where m is the multiplicity of the sought zero known in advance some of these methods were presented without any derivation and motivation it could be said out of the blue in this paper we present an easy and entirely natural way for constructing root ratio multipoint iterative methods starting from multipoint methods for finding simple zeros in this way a vast number of root ratio multipoint methods for multiple zeros existing as well new ones can be constructed for demonstration we derive four root ratio methods for multiple zeros besides we study computational cost of the considered methods and give a comparative analysis that involves cpu time needed for the extraction of the m th root this analysis shows that root ratio methods are pretty inefficient from the computational point of view and thus not suitable in practice a general discussion on a practical need for multipoint methods of very high order is also considered

modified jarratt method for computing multiple roots abstract in this paper we present a fourth order method for computing multiple roots of nonlinear equations the method is based on jarratt scheme for simple roots p jarratt some efficient fourth order multipoint methods for solving equations bit 9 1969 119 124 the method is optimal 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 it is observed that the present scheme is competitive with other similar robust methods

identities for minors of the laplacian resistance and distance matrices it is shown that if l and d are the laplacian and the distance matrix of a tree respectively then any minor of the laplacian equals the sum of the cofactors of the complementary submatrix of d up to sign and a power of 2 an analogous more general result is proved for the laplacian and the resistance matrix of any graph a similar identity is proved for graphs in which each block is a complete graph on r vertices and for q analogues of such matrices of a tree our main tool is an identity for the minors of a matrix and its inverse

on the convergence of an optimal fourth order family of methods and its dynamics in this paper we present the study of the semilocal and local convergence of an optimal fourth order family of methods moreover the dynamical behavior of this family of iterative methods applied to quadratic polynomials is studied some anomalies are found in this family be means of studying the dynamical behavior parameter spaces are shown and the study of the stability of all the fixed points is presented

a study of dynamics via mobius conjugacy map on a family of sixth order modified newton like multiple zero finders with bivariate polynomial weight functions abstract a generic family of sixth order modified newton like multiple zero finders have been proposed in geum et al 2016 among them we select a specific family of iterative methods with uniparametric bivariate polynomial weight functions and study their dynamics by investigating the relevant parameter spaces and dynamical planes via mobius conjugacy map applied to a prototype polynomial of the form z a m z b m the resulting dynamics is best illustrated through a variety of parameter spaces as well as dynamical planes

a biparametric extension of king s fourth order methods and their dynamics a class of two point quartic order simple zero finders and their dynamics are investigated in this paper by extending king s fourth order family of methods with the introduction of an error corrector having a weight function dependent on a function to function ratio higher order convergence is obtained through a variety of test equations numerical experiments strongly support the theory developed in this paper in addition relevant dynamics of the proposed methods is successfully explored for a prototype quadratic polynomial as well as parameter spaces and dynamical planes

on the solution of two coupled wiener hopf equations vector wiener hopi equations previously solved in closed form by using different techniques involve matrices having a particular structure a general procedure is given to factorize these matrices it leads to an approach which even if sometimes cumbersome is uniform and allows the solution of new problems in addition a new method is proposed which transforms simultaneous wiener hopf equations in the form of fredholm integral equation of the second kind

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

the wiener hopf technique for impenetrable wedges having arbitrary aperture angle diffraction by impenetrable wedges having arbitrary aperture angle is studied by means of the wiener hopf w h technique a system of functional equations called generalized wiener hopf equations gwhe is obtained only for certain values of the aperture angle are these equations recognizable as standard or classical wiener hopf equations cwhe however in all cases a mapping is found that reduces the gwhe to cwhe it means that the diffraction by an impenetrable wedge always reduces to a standard w h factorization the solution for the diffraction by a wedge with given face impedances the malyuzhinets problem is obtained in closed form by an explicit factorization of the kernel

wiener hopf solution for impenetrable wedges at skew incidence a new wiener hopf approach for the solution of impenetrable wedges at skew incidence is presented mathematical aspects are described in a unified and consistent theory for angular region problems solutions are obtained using analytical and numerical analytical approaches several numerical tests from the scientific literature validate the new technique and new solutions for anisotropic surface impedance wedges are solved at skew incidence the solutions are presented considering the geometrical and uniform theory of diffraction coefficients total fields and possible surface wave contributions

simple expressions for the long walk distance abstract the walk distances in graphs are defined as the result of appropriate transformations of the k 0 ta k proximity measures where a is the weighted adjacency matrix of a connected weighted graph and t is a sufficiently small positive parameter the walk distances are graph geodetic moreover they converge to the shortest path distance and to the so called long walk distance as the parameter t approaches its limiting values in this paper simple expressions for the long walk distance are obtained they involve the generalized inverse minors and inverses of submatrices of the symmetric irreducible singular m matrix l i a where is the perron root of a

a general multisegment artificial neural network architecture for the efficient evaluation of electromagnetic plane wave wedge diffraction a multisegment artificial neural network ann is proposed as an interpolation technique for the evaluation of the electromagnetic field diffracted at the edge of anisotropic impedance wedges under plane wave illumination at oblique incidence multisegmentation is needed as the high frequency wedge diffracted field is characterized by a number of discontinuities at the shadow boundaries of the geometrical optics and surface wave fields the proposed approach is applied as a test case to the problem of an anisotropic impedance right angled wedge illuminated by a skewly incident plane wave some exact analytical solutions valid for specific surface impedance tensors are used to obtain numerical data for the ann training phase as well as to show the interpolation capabilities of the implemented ann nevertheless the proposed ann structure is general and can be trained with data obtained from other available solutions analytical perturbative numerical valid for more general wedge configurations eventually leading to a single software tool encompassing all of them and providing accurate approximations of the wedge diffracted field in a relatively short time comparable to that of a closed form analytical solution

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

new families of nonlinear third order solvers for finding multiple roots in this paper we present two new families of iterative methods for multiple roots of nonlinear equations one of the families require one function and two derivative evaluation per step and the other family requires two function and one derivative evaluation it is shown that both are third order convergent for multiple roots numerical examples suggest that each family member can be competitive to other third order methods and newton s method for multiple roots in fact the second family is even better than the first

electromagnetic diffraction by arbitrary angle impedance wedges the problem of the diffraction of a plane electromagnetic wave incident at an oblique angle on a wedge of arbitrary angle with general tensor impedance boundary conditions is solved using a semi an

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 a general class of multipoint root finding methods of high computational efficiency a general class of n point iterative methods for solving nonlinear equations is constructed by combining methods of newton s type and an arbitrary two point method of the fourth order of convergence it is proved that these methods have the convergence order 2 n requiring only n 1 function evaluations per iteration in this way it is demonstrated that the proposed class of methods supports the kung traub hypothesis 1974 on the upper bound 2 n of the order of multipoint methods based on n 1 function evaluations consequently this class possess as high as possible computational efficiency in the sense of the kung traub hypothesis numerical examples are included to demonstrate exceptional convergence speed with only few function evaluations

embedding formulae for diffraction by rational wedge and angular geometries embedding is the process of taking the far field directivity from a diffraction problem or problems involving line sources or multipoles placed at a sharp edge and then constructing the far field for the same geometry for more general incidence using only this canonical problem s thus far embedding has been limited to planar parallel scattering surfaces for instance collections of parallel cracks or slits it appeared that there was a fundamental limitation to embedding disallowing its use for angular structures in this article we overcome this limitation and demonstrate the use of embedding upon wedge diffraction problems and upon a simple polygonal shape a limitation is that the final formulae are for rational wedge angles

on the non commutative factorization of wiener hopf kernels of khrapkov type there are a great many problems in mathematical physics and engineering which when modelled mathematically are reduced to wiener hopf functional equations defined in some region of a complex plane in simple models this equation is scalar but in more complicated situations inherent coupling etc can often give rise to coupled systems of equations and so the resulting wiener hopf kernels are of matrix form the crucial step in the solution of any wiener hopf equation is to decompose the kernel which can be a quite general function of the complex transform variable into a product of two factors which are regular in overlapping half planes this can be accomplished explicitly for scalar kernels and may be generalized to a particular class of matrix kernels namely those which permit a commutative decomposition although this class is wide and therefore contains some valuable physical problems it is found that many such kernels yield factorization elements with exponentially large behaviour at infinity in their half planes of analyticity this prevents a later step in the wiener hopf procedure being carried through and therefore introduces a serious limitation to the usefulness of khrapkov s approach previous researchers have offered schemes to eliminate the exponential behaviour of the khrapkov factors but for reasons discussed herein these have problems in their successful implementation this article offers a new procedure for quenching the exponential growth which is based on a method proposed recently by the author for fully coupled kernels which do not have commutative factors the approach is two step first a commutative factorization is obtained which is algebraic at infinity as required but which has spurious branch cuts in the regions of intended analyticity second an approximate procedure is used to eliminate the offending branch points by introduction of pade approximants for a certain scalar function and by inclusion of a regularizing meromorphic matrix for one particular example kernel in which an exact factorization happens to be known the explicit approximate kernel factors thus derived are shown to yield very close agreement to the exact factors the application of this work to problems in the area of fluid structural interaction is discussed

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

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

comparing the basins of attraction for kanwar bhatia kansal family to the best fourth order method there are relatively few optimal fourth order methods for solving nonlinear algebraic equations having roots of known multiplicity m in a previous paper we have compared 5 such methods two of which require the evaluation of the m 1 st root we have used the basin of attraction idea to recommend the best optimal fourth order method here we compare the family of methods developed by kanwar et al 2013 to the best known method we will also point out some mistake in deriving that family

dynamical behaviors of some iterative methods for multiple roots of nonlinear equations recently more and more higher order iterative methods for finding the multiple roots of nonlinear equations have been presented most of them require the information of the multiplicity of roots in this paper the conjugacy maps and the extraneous fixed points of some iterative methods for finding the multiple roots are discussed basins of attractions of them are also given to demonstrate their dynamical behaviors around the multiple roots for several polynomials

zeta functions of line middle total graphs of a graph and their coverings we consider the ihara zeta functions of line graphs middle graphs and total graphs of a regular graph and their regular or irregular covering graphs let l g m g and t g denote the line middle and total graph of g respectively we show that the line middle and total graph of a regular and irregular respectively covering of a graph g is a regular and irregular respectively covering of l g m g and t g respectively for a regular graph g we express the zeta functions of the line middle and total graph of any regular or irregular covering of g in terms of the characteristic polynomial of the covering also the complexities of the line middle and total graph of any regular or irregular covering of g are computed furthermore we discuss the l functions of the line middle and total graph of a regular graph g

new optimal class of higher order methods for multiple roots permitting f xn 0 finding multiple zeros of nonlinear functions pose many difficulties for many of the iterative methods a major difficulty in the application of iterative methods is the selection of initial guess such that neither guess is far from zero nor the derivative is small in the vicinity of the required root otherwise the methods would fail miserably finding a criterion for choosing initial guess is quite cumbersome and therefore more effective globally convergent algorithms for multiple roots are still needed therefore the aim of this paper is to present an improved optimal class of higher order methods having quartic convergence permitting f x 0 in the vicinity of the required root the present approach of deriving this optimal class is based on weight function approach all the methods considered here are found to be more effective and comparable to the similar robust methods available in literature

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

wave diffraction by wedges having arbitrary aperture angle abstract the problem of plane wave diffraction by a wedge sector having arbitrary aperture angle has a very long and interesting research background in fact we may recognize significant research on this topic for more than one century despite this fact up to now no clear unified approach was implemented to treat such a problem from a rigourous mathematical way and in a consequent appropriate sobolev space setting in the present paper we are considering the corresponding boundary value problems for the helmholtz equation with complex wave number admitting combinations of dirichlet and neumann boundary conditions the main ideas are based on a convenient combination of potential representation formulas associated with weighted mellin pseudo differential operators in appropriate sobolev spaces and a detailed fredholm analysis thus we prove that the problems have unique solutions with continuous dependence on the data which are represented by the single and double layer potentials where the densities are solutions of derived pseudo differential equations on the half line

on some modified families of multipoint iterative methods for multiple roots of nonlinear equations in this paper we propose a new one parameter family of schroder s method for finding the multiple roots of nonlinear equations numerically further we derive many new cubically convergent families of schroder type methods proposed families are derived from the modified newton s method for multiple roots and one parameter family of schroder s method furthermore we introduce new families of third order multipoint iterative methods for multiple roots free from second order derivative by semi discrete modifications of the above proposed methods one of the families requires two evaluations of the function and one evaluation of its first order derivative and the other family requires one evaluation of the function and two evaluations of its first order derivative per iteration numerical examples are also presented to demonstrate the performance of proposed iterative methods

finding the solution of nonlinear equations by a class of optimal methods this paper is devoted to the study of an iterative class for numerically approximating the solution of nonlinear equations in fact a general class of iterations using two evaluations of the first order derivative and one evaluation of the function per computing step is presented it is also proven that the class reaches the fourth order convergence therefore the novel methods from the class are jarratt type iterations which agree with the optimality hypothesis of kung traub the derived class is further extended for multiple roots that is to say a general optimal quartic class of iterations for multiple roots is contributed when the multiplicity of the roots is available numerical experiments are employed to support the theory developed in this work

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

optimal iterative methods for finding multiple roots of nonlinear equations using weight functions and dynamics abstract in this paper we propose a family of iterative methods for finding multiple roots with known multiplicity by means of the introduction of four univariate weight functions with the help of these weight functions that play an important role in the development of higher order convergent iterative techniques we are able to construct three point eight order optimal multiple root finders also numerical experiments have been applied to a number of test equations for different special schemes from this family satisfying the conditions given in the convergence analysis we have also compared the basins of attraction of some proposed and known methods in order to check the wideness of the sets of converging initial points for each problem

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 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

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

zeta functions of finite graphs and coverings iii abstract a graph theoretical analog of brauer siegel theory for zeta functions of number fields is developed using the theory of artin l functions for galois coverings of graphs from parts i and ii in the process we discuss possible versions of the riemann hypothesis for the ihara zeta function of an irregular graph

transformation methods for finding multiple roots of nonlinear equations abstract in this paper to estimate a multiple root p of an equation f x 0 we transform the function f x to a hyper tangent function combined with a simple difference formula whose value changes from 1 to 1 as x passes through the root p then we apply the so called numerical integration method to the transformed equation which may result in a specious approximate root furthermore in order to enhance the accuracy of the approximation we propose a steffensen type iterative method which does not require any derivatives of f x nor is quite affected by an initial approximation it is shown that the convergence order of the proposed method becomes cubic by simultaneous approximation to the root and its multiplicity results for some numerical examples show the efficiency of the new method

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

a q analogue of the distance matrix of a tree we consider a q analogue of the distance matrix called the q distance matrix of an unweighted tree and give formulae for the inverse and the determinant which generalize the existing formulae for the distance matrix we obtain the smith normal form of the q distance matrix of a tree the relationship of the q distance matrix with the laplacian matrix leads to q analogue of the laplacian matrix of a tree some of whose properties are also given we study another matrix related to the distance matrix the exponential distance matrix and show its relationship with the q laplacian and the q distance matrix a formula for the determinant of the q distance matrix of a weighted tree is also given

different anomalies in a jarratt family of iterative root finding methods abstract in this paper the behavior of a jarratt family of iterative methods applied to quadratic polynomials is studied some anomalies are found in this family be means of studying the dynamical behavior of this fourth order family of methods parameter spaces are shown and the study of the stability of all the fixed points is presented dynamical planes for members with good and bad dynamical behavior are also provided

on the dynamics of a triparametric family of optimal fourth order multiple zero finders with a weight function of the principal mth root of a function to function ratio abstract under the assumption of known root multiplicity m n a triparametric family of two point optimal quartic order methods locating multiple zeros are investigated in this paper by introducing a weight function dependent on a function to function ratio special cases of weight functions with selected free parameters are considered and studied through various test equations and numerical experiments to support the theory developed in this paper in addition we explore the relevant dynamics of proposed methods via mobius conjugacy map when applied to a prototype polynomial z a m z b m the results of such dynamics are visually illustrated through a variety of parameter spaces as well as dynamical planes

basin attractors for various methods abstract there are many methods for the solution of a nonlinear algebraic equation 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 several methods of various orders and present the basin of attraction for several examples it can be seen that not all higher order methods were created equal newton s halley s murakami s and neta johnson s methods are consistently better than the others in two of the examples neta s 16th order scheme was also as good

complex analytic dynamics on the riemann sphere dichotomie dynamique de fatou et julia points periodiques consequences du theoreme de montel l ensemble de julia est la fermeture de l ensemble des points periodiques repulseurs resultats classiques sur l ensemble de fatou classification de sullivan de l ensemble de fatou une condition pour le developpement sur l ensemble de julia la dynamique des polynomes l ensemble de mandelbrot et le travail de douady et hubbard le theoreme d application de riemann mesurable et la dynamique analytique

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

the wiener hopf solution of the isotropic penetrable wedge problem diffraction and total field the diffraction of an incident plane wave by an isotropic penetrable wedge is studied using generalized wiener hopf equations and the solution is obtained using analytical and numerical analytical approaches that reduce the wiener hopf factorization to fredholm integral equations of second kind mathematical aspects are described in a unified and consistent theory for angular region problems the formulation is presented in the general case of skew incidence and several numerical tests at normal incidence are reported to validate the new technique the solutions consider engineering applications in terms of gtd utd diffraction coefficients and total fields