higher order efficient class of chebyshev halley type methods construction of two point sixth order methods for simple root is an ambitious and challenging task in numerical analysis therefore the main aim of this paper is to introduce a new highly efficient two point sixth order class of chebyshev halley type methods free from second order derivative for the first time each member of the proposed class requires only four functional evaluations viz two evaluations of function f and two of first order derivative f per full iteration a variety of concrete numerical examples illustrate that our proposed methods are more efficient and perform better than existing two point three point sixth order methods available in the literature from their dynamical study it has been observed that our proposed methods have better stability and robustness as compared to the other existing methods

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

panconnectivity and pancyclicity of hypercube like interconnection networks with faulty elements in this paper we deal with the graph g email protected g 1 obtained from merging two graphs g 0 and g 1 with n vertices each by n pairwise nonadjacent edges joining vertices in g 0 and vertices in g 1 the main problems studied are how fault panconnectivity and fault pancyclicity of g 0 and g 1 are translated into fault panconnectivity and fault pancyclicity of g email protected g 1 respectively many interconnection networks such as hypercube like interconnection networks can be represented in the form of g email protected g 1 connecting two lower dimensional networks g 0 and g 1 applying our results to a class of hypercube like interconnection networks called restricted hl graphs we show that in a restricted hl graph g of degree m 3 each pair of vertices are joined by a path in email protected f of every length from 2m 3 to v email protected f 1 for any set f of faulty elements vertices and or edges with f m 3 and there exists a cycle of every length from 4 to v email protected f for any fault set f with f m 2

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

hyper hamiltonian laceability on edge fault star graph the star graph possess many nice topological properties edge fault tolerance is an important issue for a network since the edges in the network may fail sometimes in this paper we show that the n dimensional star graph is n 3 edge fault tolerant hamiltonian laceable n 3 edge fault tolerant strongly hamiltonian laceable and n 4 edge fault tolerant hyper hamiltonian laceable all these results are optimal in a sense described in this paper

attracting cycles for the relaxed newton s method we study the relaxed newton s method applied to polynomials in particular we give a technique such that for any n 2 we may construct a polynomial so that when the method is applied to a polynomial the resulting rational function has an attracting cycle of period n we show that when we use the method to extract radicals the set consisting of the points at which the method fails to converge to the roots of the polynomial p z z m c this set includes the julia set has zero lebesgue measure consequently iterate sequences under the relaxed newton s method converge to the roots of the preceding polynomial with probability one

on a family of halley like methods to find simple roots of nonlinear equations there are many methods for solving a nonlinear algebraic equation here we introduce a family of halley like methods and show that euler chebyshev and bsc are just members of the family we discuss the conjugacy maps and the effect of the extraneous roots on the basins of attraction

constructing attracting periodic orbits of three third order iterative methods in this paper we shall use three third order root finding iterative methods that do not require the use of second derivatives for their formulation and present a procedure to construct polynomials in such a way that superattracting periodic orbits of any prescribed period will occur when these iterative methods are applied

attractive cycles in the iteration of meromorphic functions the existence of attractive cycles constitutes a serious impediment to the solution of nonlinear equations by iterative methods this problem is illustrated in the case of the solution of the equationz tanz c for complex values ofc by newton s method relevant results from the theory of the iteration of rational functions are cited and extended to the analysis of this case in which a meromorphic function is iterated extensive numerical results including many attractive cycles are summarized

conditional edge fault tolerant edge bipancyclicity of hypercubes the hypercube has been one of the most popular interconnection networks for parallel computer communication systems in this paper we assume that each node is incident with at least two fault free links under this assumption we show that every fault free edge lies on a fault free cycle of every even length from 6 to 2 n inclusive even if it has up to 2n 5 link faults the result is optimal with respect to the number of link faults tolerated

dynamics of a family of third order iterative methods that do not require using second derivatives the purpose of this article is to present results that amount to a description of the conjugacy classes of three third order root finding iterative methods that do not require the use of second derivatives for their formulation for complex polynomials of degrees two three and four for degrees two and three a full description of the conjugacy classes is accomplished in each case by a one parameter family of polynomials this is done in such a way that when one applies one of these three root finding iterative methods to the elements of these parametrized families a family of iterative methods is obtained in such a way that its dynamics represents up to conjugacy the dynamics of the corresponding iterative root finding method applied to any complex polynomial having the same degree for degree four analogous partial results are obtained

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

cycle embedding of augmented cubes a graph g v e is vertex pancyclic if for every vertex u and any integer l ranging from a positive constant l to v g contains a cycle c of length l such that u is in c in this paper we show that an n dimensional augmented cube where n 2 is vertex pancyclic with l 3

constructing edge disjoint spanning trees in locally twisted cubes the use of edge disjoint spanning trees for data broadcasting and scattering problem in networks provides a number of advantages including the increase of bandwidth and fault tolerance in this paper we present an algorithm for constructing n edge disjoint spanning trees in an n dimensional locally twisted cube since the n dimensional locally twisted cube is regular with the common degree n the number of constructed trees is optimal

an analysis of a family of maheshwari based optimal eighth order methods in this paper we analyze an optimal eighth order family of methods based on maheshwari s fourth order method this family of methods uses a weight function we analyze the family using the information on the extraneous fixed points two measures of closeness of an extraneous points set to the imaginary axis are considered and applied to the members of the family to find its best performer the results are compared to a modified version of wang liu method

on embedding cycles into faulty twisted cubes the twisted cube tq n is an alternative to the popular hypercube network recently some interesting properties of tq n were investigated in this paper we study the pancycle problem on faulty twisted cubes let f e and f v be the numbers of faulty edges and faulty vertices in tq n respectively we show that with f e f v 3

a construction of attracting periodic orbits for some classical third order iterative methods we use a family of root finding iterative methods for finding roots of nonlinear equations we present a procedure for constructing polynomials so that superattracting periodic orbits of any prescribed period occur when these methods are applied this family includes chebyshev s method halley s method the super halley method and the c methods as particular cases

basins of attraction for optimal eighth order methods to find simple roots of nonlinear equations several optimal eighth order methods to obtain simple roots are analyzed the methods are based on two step fourth order optimal methods and a third step of modified newton the modification is performed by taking an interpolating polynomial to replace either f z n or f z n in six of the eight methods we have used a hermite interpolating polynomial the other two schemes use inverse interpolation we discovered that the eighth order methods based on jarratt s optimal fourth order methods perform well and those based on king s or kung traub s methods do not in all cases tested the replacement of f z by hermite interpolation is better than the replacement of the derivative f z

accelerating generators of iterative methods for finding multiple roots of nonlinear equations two accelerating generators that produce iterative root finding methods of arbitrary order of convergence are presented primary attention is paid to algorithms for finding multiple roots of nonlinear functions and in particular of algebraic polynomials first two classes of algorithms for solving nonlinear equations are studied those with a known order of multiplicity and others with no information on multiplicity we also demonstrate the acceleration of iterative methods for the simultaneous approximations of multiple roots of algebraic polynomials a discussion about the computational efficiency of the root solvers considered and three numerical examples are given

dynamical aspects of some convex acceleration methods as purely iterative algorithm for newton s maps in this paper we define purely iterative algorithm for newton s maps which is a slight modification of the concept of purely iterative algorithm due to smale for this we use a characterization of rational maps which arise from newton s method applied to complex polynomials we prove the scaling theorem for purely iterative algorithm for newton s map then we focus our study in dynamical aspects of three root finding iterative methods viewed as a purely iterative algorithm for newton s map whittaker s iterative method the super halley iterative method and a modification of the latter we give a characterization of the attracting fixed points which correspond to the roots of a polynomial also numerical examples are included in order to show how to use the characterization of fixed points finally we give a description of the parameter spaces of the methods under study applied to a one parameter family of generic cubic polynomials

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

basins of attraction for several methods to find simple roots of nonlinear equations there are many methods for solving 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 third and fourth order methods to find simple zeros the relationship between the basins of attraction and the corresponding conjugacy maps will be discussed in numerical experiments the effect of the extraneous roots on the basins is also discussed

on developing a higher order family of double newton methods with a bivariate weighting function a high order family of two point methods costing two derivatives and two functions are developed by introducing a two variable weighting function in the second step of the classical double newton method their theoretical and computational properties are fully investigated along with a main theorem describing the order of convergence and the asymptotic error constant as well as proper choices of special cases a variety of concrete numerical examples and relevant results are extensively treated to verify the underlying theoretical development in addition this paper investigates the dynamics of rational iterative maps associated with the proposed method and an existing method based on illustrated description of basins of attraction for various polynomials

edge fault tolerant bipanconnectivity of hypercubes this paper shows that for any two distinct vertices u and v with distance d in the hypercube q n n 3 with at most 2n 5 faulty edges and each vertex incident with least two fault free edges there exist fault free uv paths of length in q n for every with d 4 2 n 1 and d 0 mod2 this result improves some known results on edge fault bipanconnectivity of hypercubes the proof is based on the recursive structure of q n

a first overview on the real dynamics of chebyshev s method in this paper we explore some properties of the well known root finding chebyshev s method applied to polynomials defined on the real field in particular we are interested in showing the existence of extraneous fixed points that is fixed points of the iteration map that are not root of the considered polynomial the existence of such extraneous fixed points is a specific property in the dynamical study of chebyshev s method that does not happen in other known iterative methods as newton s or halley s methods in addition in this work we consider other dynamical aspects of the method as for instance the feigenbaum bifurcation diagrams or the parameter plane

long paths in hypercubes with conditional node faults let f be a set of f 2 n 5 faulty nodes in an n cube q n such that every node of q n still has at least two fault free neighbors then we show that q n f contains a path of length at least 2 n 2 f 1 respectively 2 n 2 f 2 between any two nodes of odd respectively even distance since the n cube is bipartite the path of length 2 n 2 f 1 or 2 n 2 f 2 turns out to be the longest if all faulty nodes belong to the same partite set as a contribution our study improves upon the previous result presented by j s fu longest fault free paths in hypercubes with vertex faults information sciences 176 2006 759 771 where only n 2 faulty nodes are considered

an optimal family of eighth order simple root finders with weight functions dependent on function to function ratios and their dynamics underlying extraneous fixed points we extend in this paper an optimal family of three step eighth order methods developed by duni etal 2011 with higher order weight functions employed in the second and third sub steps and investigate their dynamics under the relevant extraneous fixed points among which purely imaginary ones are specially treated for the analysis of the rich dynamics their theoretical and computational properties are fully investigated along with a main theorem describing the order of convergence and the asymptotic error constant as well as proper choices of special cases a wide variety of relevant numerical examples are illustrated to confirm the underlying theoretical development in addition this paper investigates the dynamics of selected existing optimal eighth order iterative maps with the help of illustrative basins of attraction for various polynomials

choosing weight functions in iterative methods for simple roots weight functions with a parameter are introduced into an iteration process to increase the order of the convergence and enhance the behavior of the iteration process the parameter can be chosen to restrict extraneous fixed points to the imaginary axis and provide the best basin of attraction the process is demonstrated on several examples

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

some new topological properties of the triangular pyramid networks abstract the triangular pyramid proposed by razavi and sarbazi azad the triangular pyramid routing and topological properties information sciences 180 2010 2328 2339 is a hierarchical structure based on triangular meshes in this paper we show that the triangular pyramid shares some nice symmetry properties of the pyramid we also determine the connectivity of the triangular pyramid and prove that it is 1 fault tolerant vertex pancyclic

on the dynamics of the euler iterative function abstract the dynamics of euler s third order iterative method which is used to find roots of non linear equations applied to complex polynomials of degrees three and four is studied the conjugacy classes of this method are found explicitly

a fast diagnosis algorithm for locally twisted cube multiprocessor systems under the mm model comparison based diagnosis is a practical approach to the system level fault diagnosis of multiprocessors the locally twisted cube is a newly introduced hypercube variant which not only possesses lower diameter and better graph embedding capability as compared with a hypercube of the same size but retains some nice properties of hypercubes this paper addresses the fault diagnosis of locally twisted cubes under the mm comparison model by utilizing the existence of abundant cycles within a locally twisted cube we present a new diagnosis algorithm with elaborately organized data this algorithm can run in o nlog 2 2n time where n stands for the total number of nodes in comparison the classical sengupta dahbura diagnosis algorithm takes as much as o n 5 time to achieve the same goal as a consequence the proposed algorithm is remarkably superior to the sengupta dahbura algorithm in terms of the time overhead

symmetries of the julia sets of konig s methods for polynomials abstract let r be the group of symmetries of the julia set of a rational map r and k f n be the konig s method for polynomial f of order n 2 for any given integer n 2 we prove that if f is in normal form then f is a subgroup of k f n we also obtain a necessary and sufficient condition for the julia set of k f n to be a line

prescribed cycles of konig s method for polynomials abstract we consider konig s method for finding roots of polynomials let k f n be the function defined in konig s method for polynomial f of order n n 2 for any given complex number c and any given set z 1 z 2 z k of k k 2 distinct complex numbers we present a procedure of constructing a polynomial f n for which is a k cycle of k f n n with multiplier

hamiltonian laceability of bubble sort graphs with edge faults it is known that the n dimensional bubble sort graph b n is bipartite n 1 regular and has n vertices we first show that for any vertex v b n v has a hamiltonian path between any two vertices in the same partite set without v let f be a subset of edges of b n we next show that b n f has a hamiltonian path between any two vertices of different partite sets if f is at most n 3 then we also prove that b n f has a path of length n 2 between any pair of vertices in the same partite set

on the iteration of a rational function computer experiments with newton s method using newton s method to look for roots of a polynomial in the complex plane amounts to iterating a certain rational function this article describes the behavior of newton iteration for cubic polynomials after a change of variables these polynomials can be parametrized by a single complex parameter and the newton transformation has a single critical point other than its fixed points at the roots of the polynomial we describe the behavior of the orbit of the free critical point as the parameter is varied the julia set points where newton s method fail to converge is also pictured these sets exhibit an unexpected stability of their gross structure while the changes in small scale structure are intricate and subtle

an infinite family of bounds on zeros of analytic functions and relationship to smale s bound smale s analysis of newton s iteration function induce a lower bound on the gap between two distinct zeros of a given complex valued analytic function f z in this paper we make use of a fundamental family of iteration functions b m z m 2 to derive an infinite family of lower bounds on the above gap however even for m 2 where b 2 z coincides with newton s our lower bound is more than twice as good as smale s bound or its improved version given by blum cucker shub and smale when f z is a complex polynomial of degree n for small m the corresponding bound is computable in o n log n arithmetic operations for quadratic polynomials as m increases the lower bounds converge to the actual gap we show how to use these bounds to compute lower bounds on the distance between an arbitrary point and the nearest root of f z in particular using the latter result we show that given a complex polynomial f z a n z n a 0 a n a o 0 for each m 2 we can compute upper and lower bounds u m and l m such that the roots of f z lie in the annulus z l m z u m in particular l 2 1 2 max a k a 0 1 k k 1 n u 2 2 max a n k a n 1 k k 1 n and l3 5 1 2 max a 1 a k 1 a 0 a k a 2 0 1 k k 2 n 1 u 3 5 1 2 max a n 1 a n k 1 a n o n k a 2 n 1 k k 2 n 1 where a 1 a n 1 0 an application of the latter bounds is within weyl s classical quad tree algorithm for computing all roots of a given complex polynomial

extended obstruction tensors and renormalized volume coefficients abstract the behavior is investigated of the renormalized volume coefficients associated to a pseudo riemannian metric under conformal change of the metric it is shown that the renormalized volume coefficients define second order fully nonlinear operators in the conformal factor whose algebraic structure is elucidated via the introduction of extended obstruction tensors these together with the schouten tensor constitute building blocks for the coefficients in the ambient metric expansion the renormalized volume coefficients have recently been considered by chang and fang motivated by comparison with the elementary symmetric functions of the eigenvalues of the schouten tensor

julia sets and mandelbrot like sets associated with higher order schro der rational iteration functions a computer assisted study schroder iteration functions sm z a generalization of newton s method for which m 2 are constructed so that the sequence z i sm zn converges locally to a root z of g z 0 as o iz z in for g z a polynomial this involves the iteration of rational functions over the complex riemann sphere which is described by the classical theory of julia and fatou and subsequent developments the julia sets for the sm z as applied to the simple cases g z z 1 are examined for increasing m with the help of microcomputer plots the possible types of behavior of zn iteration sequences are catalogued by examining the orbits of free critical points of the sm z as applied to a one parameter family of cubic polynomials

a kind of conditional vertex connectivity of cayley graphs generated by 2 trees let g be a graph then t v g is called an r k vertex cut if g t is disconnected and each vertex in v g t has at least k neighbors in g t the size of a smallest r k vertex cut is the r k vertex connectivity of g and is denoted by k k g in this paper we determine the numbers k 1 and k 2 for cayley graphs generated by 2 trees including the popular alternating group graphs

fault tolerant edge pancyclicity of locally twisted cubes the n dimensional locally twisted cube ltq n is a new variant of the hypercube which possesses some properties superior to the hypercube this paper investigates the fault tolerant edge pancyclicity of ltq n and shows that if ltq n n 3 contains at most n 3 faulty vertices and or edges then for any fault free edge e and any integer with 6 2 n f v there is a fault free cycle of length containing the edge e where f v is the number of faulty vertices the result is optimal in some senses the proof is based on the recursive structure of ltq n

a kind of conditional connectivity of cayley graphs generated by wheel graphs for a connected graph g v e a subset fv is called an rk vertex cut of g if gf is disconnected and each vertex in vf has at least k neighbors in gf the cardinality of the minimum rk vertex cut is the rk vertex connectivity of g and is denoted by k g the conditional connectivity is a measure to explore the structure of networks beyond the vertex connectivity let sym n be the symmetric group on 1 2 n and t be a set of transpositions of sym n denote by g t the graph with vertex set 1 2 n and edge set ij ij t if g t is a wheel graph then simply denote the cayley graph cay sym n t by wgn in this paper we determine the values of 1 and 2 for cayley graphs generated by wheel graphs and prove that 1 wgn 4n6 and 2 wgn 8n18

flexible cycle embedding in the locally twisted cube with nodes positioned at any prescribed distance abstract a hamiltonian graph g is panpositionably hamiltonian if for any two distinct vertices x and y of g it contains a hamiltonian cycle c such that d c x y l for any integer l satisfying d g x y l v g 2 where d g x y respectively d c x y denotes the distance between vertices x and y in g respectively on c and v g is the total number of vertices in g as the importance of hamiltonian properties for data communication between units in parallel and distributed systems the panpositionable hamiltonicity involves more flexible cycle embedding for message transmission this paper shows that for two arbitrary nodes x and y of the n dimensional locally twisted cube ltq n n 4 and for any integer l d d 2 d 3 d 4 2 n 1 where d d ltq n x y there exists a hamiltonian cycle c of ltq n such that d c x y l

on general convergence in extracting radicals via a fundamental family of iteration functions newton s method is well known to be generally convergent for solving x n c 0 in this paper we first extend this result to the next two members of an infinite family of high order methods referred to here as the basic family which starts with newton s method while computing roots of unity numerically is a trivial task studying the general convergence of the basic family in this simple case is an important first step toward the understanding of the global behavior of this fundamental family with the aid of polynomiography techniques for the visualization of polynomial root finding we further conjecture the general convergence of all members of the basic family when extracting radicals using the computer algebra system maple we obtain some partial results toward the proof of our conjecture

embedding meshes into locally twisted cubes as a newly introduced interconnection network for parallel computing the locally twisted cube possesses many desirable properties in this paper mesh embeddings in locally twisted cubes are studied let ltq n v e denote the n dimensional locally twisted cube we present three major results in this paper 1 for any integer n 1 a 2x2 n 1 mesh can be embedded in ltq n with dilation 1 and expansion 1 2 for any integer n 4 two node disjoint 4x2 n 3 meshes can be embedded in ltq n with dilation 1 and expansion 2 3 for any integer n 3 a 4 x 2 n 2 1 mesh can be embedded in ltq n with dilation 2 the first two results are optimal in the sense that the dilations of all embeddings are 1 the embedding of the 2x2 n 1 mesh is also optimal in terms of expansion we also present the analysis of 2px2q mesh embedding in locally twisted cubes

superattracting cycles of the relaxed newton s method for entire functions abstract in this paper we study certain dynamical properties of the relaxed newton s method for entire functions we present a procedure for constructing entire functions so that superattracting cycles of any prescribed period occur when the relaxed newton s method is applied based on the above conclusion we give a model of any prescribed type cycles for four disjoint subclasses of meromorphic functions including polynomials transcendental entire functions rational functions but not polynomials and transcendental meromorphic functions but not entire functions

complete path embeddings in crossed cubes crossed cubes are popular variants of hypercubes in this paper we study path embeddings between any two distinct nodes in crossed cubes we prove two important results in the n dimensional crossed cube a for any two nodes all paths whose lengths are greater than or equal to the distance between the two nodes plus 2 can be embedded between the two nodes with dilation 1 b for any two integers n 2 and l with 1

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

hamiltonian connectivity of restricted hypercube like networks under the conditional fault model restricted hypercube like networks rhlns are an important class of interconnection networks for parallel computing systems which include most popular variants of the hypercubes such as crossed cubes mobius cubes twisted cubes and locally twisted cubes this paper deals with the fault tolerant hamiltonian connectivity of rhlns under the conditional fault model let g be an n dimensional rhln and email protected v g e g where n 7 and f 2n 10 we prove that for any two nodes u email protected v g f satisfying a simple necessary condition on neighbors of u and v there exists a hamiltonian or near hamiltonian path between u and v in g f the result extends further the fault tolerant graph embedding capability of rhlns

conditional connectivity of star graph networks under embedding restriction the n dimensional star graph s n is one of the most attractive interconnection networks of large scale multiprocessor systems the k embedding restricted edge connectivity h k s n z k s n of s n is defined to be the cardinality of a minimum subset of edges nodes if any whose deletion disconnects s n and each node of the remaining components lies in an undamaged k dimensional substar s k in this paper we investigate the k embedding restricted edge connectivity of s n and determine the values of h k s n and z k s n for some k s

a systematic approach for embedding of hamiltonian cycles through a prescribed edge in locally twisted cubes the locally twisted cube interconnection network has been recognized as an attractive alternative to the hypercube network previously the locally twisted cube has been shown to contain a hamiltonian cycle the main contribution of this paper is to provide the necessary and sufficient conditions for determining a characterization of permutations of link dimensions constructing hamiltonian cycles in a locally twisted cube for those permutations we propose a linear algorithm for finding a hamiltonian cycle through a given edge as a result we obtain a lower bound for the number of hamiltonian cycles through a given edge in an n dimensional locally twisted cube

holographic formula for q curvature ii we extend the holographic formula for the critical q curvature in graham and juhl 2007 9 to all q curvatures moreover we confirm a conjecture of juhl 2009 11

an analysis of a khattri s 4th order family of methods in this paper we analyze an optimal fourth order family of methods suggested by khattri and babajee 2013 we analyze the family using the information on the extraneous fixed points two measures of closeness to the imaginary axis of the set of extraneous points are considered and applied to the members of the family to find its best performer the results are compared to three best members of king s family of methods

julia sets for the standard newton s method halley s method and schroder s method abstract we analyze the julia sets theory for the standard newton s method halley s method and schroder s method study structural characteristics of julia sets show the properties and conditions of fixed points for the standard newton s method halley s method and schroder s method we observe the relation between roots of polynomial and julia sets structure namely if keeping the relative position of the roots of a polynomial invariable then the topological structure is also invariable if there is an extraneous fixed point then the extraneous fixed point is also invariable otherwise the structures of julia sets and fixed points would be changed

hamiltonian properties of honeycomb meshes meshes are widely used topologies for networks on chip noc honeycomb meshes have better topological properties than meshes in order to communicate efficiently in a linear or cyclic manner it is benefited that there is a hamiltonian path or hamiltonian cycle in noc in this paper we give a necessary and sufficient condition for the existence of hamiltonian path between any pair of vertices in a honeycomb mesh and for the existence of hamiltonian path in a honeycomb mesh with one faulty vertex besides we give a systematic method to construct a hamiltonian path in honeycomb meshes

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

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

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

on a family of laguerre methods to find multiple roots of nonlinear equations there are several methods for solving a nonlinear algebraic equation having roots of a given multiplicity m here we compare a family of laguerre methods of order three as well as two others of the same order and show that euler cauchy s method is best we discuss the conjugacy maps and the effect of the extraneous roots on the basins of attraction