transformation groups for soliton equations iv a new hierarchy of soliton equations of kp type abstract a new approach to soliton equations based on functions or hirota s dependent variables vertex operators and the clifford algebra of free fermions is applied to study a new hierarchy of kadomtsev petviashvili type equations the bkp hierarchy the infinite dimensional orthogonal group acts on the space of bkp functions the sawada kotera equation is obtained as a reduction of bkp its infinitesimal transformations constitute the euclidean lie algebra a2 2

randomized local search evolutionary algorithms and the minimum spanning tree problem randomized search heuristics among them randomized local search and evolutionary algorithms are applied to problems whose structure is not well understood as well as to problems in combinatorial optimization the analysis of these randomized search heuristics has been started for some well known problems and this approach is followed here for the minimum spanning tree problem after motivating this line of research it is shown that randomized search heuristics find minimum spanning trees in expected polynomial time without employing the global technique of greedy algorithms

reversibility of general 1d linear cellular automata over the binary field z 2 under null boundary conditions the reversibility of cellular automata ca has long been studied but there are still many problems untouched this paper tackles the reversibility problem of the general case concerning 1d ca under null boundary conditions defined by linear rules over the binary field z 2 although transition matrix has been widely used for some special linear ca lca rules it has severe limitations its complexity depends on the number of cells and determining its reversibility becomes another unsolved tough problem for general rules by constructing deterministic finite automata dfa of the de bruijn graph presentation of ca we conclude the reversibility problem of all 1d linear rules over z 2 under null boundary conditions it turns out that any 1d lca except for some unilateral rules over z 2 under null boundary conditions can be reversible once having a proper number of cells and dfa provides an efficient way to find those numbers moreover the complexity of the dfa solution is independent of the number of cells in addition the relative low complexity of dfa can be further significantly reduced by using only a small part of it

on a higher order secant method in this paper we carry out a modification of the classical secant method our goal is to accelerate the convergence until order two but without using any derivative two convergence theorems are presented finally some numerical experiments are analyzed

extended sufficient semilocal convergence for the secant method we establish new sufficient convergence conditions for the secant method to a locally unique solution of a nonlinear equation in a banach space using our new concept of recurrent functions and combining lipschitz and center lipschitz conditions on the divided difference operator we obtain a new semilocal convergence analysis of the secant method moreover our sufficient convergence conditions expand the applicability of the secant method in cases not covered before dennis 1971 9 hernandez et al 2005 8 laasonen 1969 15 ortega and rheinboldt 1970 11 potra 1982 5 potra 1985 7 schmidt 1978 18 yamamoto 1987 12 wolfe 1978 19 numerical examples are also provided in this study

on efficient weighted newton methods for solving systems of nonlinear equations in this study we present iterative methods of convergence order four and six for solving systems of nonlinear equations the fourth order scheme is composed of two steps namely newton iteration as the first step and weighted newton iteration as the second step the sixth order scheme is composed of three steps the first two steps are same as that of fourth order scheme whereas the third step is again based on weighted newton iteration computational efficiency in its general form is discussed comparison between the efficiencies of proposed techniques and existing techniques is made it is proved that for large systems the new methods are more efficient numerical tests are performed which confirm the theoretical results moreover theoretical order of convergence is verified in the examples

analysing the efficiency of some modifications of the secant method some modifications of the secant method for solving nonlinear equations are revisited and the local order of convergence is found in a direct symbolic computation to do this a development of the inverse of the first order divided differences of a function of several variables in two points is presented a generalisation of the efficiency index used in the scalar case to several variables is also analysed in order to use the most competitive algorithm

a method for finding sharp error bounds for newton s method under the kantorovic assumptions this paper gives a method for finding sharpa posteriori error bounds for newton s method under the assumptions of kantorovich s theorem on the basis of this method new error bounds are derived and comparison is made among the known bounds of dennis 2 doring 4 gragg tapia 5 kantorovich 6 7 kornstaedt 9 lancaster 10 miel 11 13 moret 14 ostrowski 17 18 potra 19 and potra ptak 20

efficient exhaustive listings of reversible one dimensional cellular automata this paper looks at an algebraic formulation of one dimensional cellular automata using the formulation connections to combinatorial structures and graph theory become clear strong results about uniqueness and isomorphism allows us to outline effective algorithms for the generation of exhaustive lists of reversible one dimensional cellular automata and to count the number of distinct examples that exist these algorithms use the orderly algorithm methods to avoid the pitfalls of brute force searches

third order iterative methods with applications to hammerstein equations a unified approach the geometrical interpretation of a family of higher order iterative methods for solving nonlinear scalar equations was presented in s amat s busquier j m gutierrez geometric constructions of iterative functions to solve nonlinear equations j comput appl math 157 1 2003 197 205 this family includes as particular cases some of the most famous third order iterative methods chebyshev methods halley methods super halley methods c methods and newton type two step methods the aim of the present paper is to analyze the convergence of this family for equations defined between two banach spaces by using a technique developed in j a ezquerro m a hernandez halley s method for operators with unbounded second derivative appl numer math 57 3 2007 354 360 this technique allows us to obtain a general semilocal convergence result for these methods where the usual conditions on the second derivative are relaxed on the other hand the main practical difficulty related to the classical third order iterative methods is the evaluation of bilinear operators typically second order frechet derivatives however in some cases the second derivative is easy to evaluate a clear example is provided by the approximation of hammerstein equations where it is diagonal by blocks we finish the paper by applying our methods to some nonlinear integral equations of this type

stability of king s family of iterative methods with memory in the literature exist many iterative methods with memory for solving nonlinear equations the most of them designed in the last years as they use the information of at least the two previous iterates to generate the new one usual techniques of complex dynamics are not useful in this case in this paper we present some real multidimensional dynamical tools to undertake this task applied on a very well known family of iterative schemes king s class it is showed that the most of elements of this class present a very stable behavior visualized in different dynamical planes however pathological cases as attracting strange fixed points or periodic orbits can also be found

local convergence of inexact newton methods a simple affine invariant condition which ensures local convergence of inexact newton methods is provided this condition is closely linked with the characterization of superlinear convergence given by dennis and more radius of convergence and rate of convergence results are derived and we show the theory is in some sense sharp we apply the theory to various inexact newton methods including difference newton like methods and the general newton like method in which the iterates are perturbed by rounding errors in the computation

ostrowski type methods for solving systems of nonlinear equations abstract four generalized algorithms builded up from ostrowski s method for solving systems of nonlinear equations are written and analyzed a development of an inverse first order divided difference operator for functions of several variables is presented as well as a direct computation of the local order of convergence for these variants of ostrowski s method furthermore a sequence that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced

secant like methods for solving nonlinear integral equations of the hammerstein type we consider a one parametric family of secant type iterations for solving nonlinear equations in banach spaces we establish a semilocal convergence result for these iterations by means of a technique based on a new system of recurrence relations this result is then applied to obtain existence and uniqueness results for nonlinear integral equations of the hammerstein type we also present a numerical example where the solution of a particular hammerstein integral equation is approximated by different secant type methods

on the local convergence of a family of two step iterative methods for solving nonlinear equations a local convergence analysis for a generalized family of two step secant like methods with frozen operator for solving nonlinear equations is presented unifying earlier methods such as secant s newton chebyshev like steffensen and other new variants the family of iterative schemes is built up where a profound and clear study of the computational efficiency is also carried out numerical examples and an application using multiple precision and a stopping criterion are implemented without using any known root finally a study comparing the order efficiency and elapsed time of the methods suggested supports the theoretical results claimed

a variant of the newton kantorovich theorem for nonlinear integral equations of mixed hammerstein type abstract we study nonlinear integral equations of mixed hammerstein type using newton s method as follows we investigate the theoretical significance of newton s method to draw conclusions about the existence and uniqueness of solutions of these equations after that we approximate the solutions of a particular nonlinear integral equation by newton s method for this we use the majorant principle which is based on the concept of majorizing sequence given by kantorovich and milder convergence conditions than those of kantorovich actually we prove a semilocal convergence theorem which is applicable to situations where kantorovich s theorem is not

positivity of legendrian thom polynomials we study legendrian singularities arising in complex contact geometry we define a one parameter family of bases in the ring of legendrian characteristic classes such that any legendrian thom polynomial has nonnegative coefficients when expanded in these bases the method uses a suitable lagrange grassmann bundle on the product of projective spaces this is an extension of a nonnegativity result for lagrangian thom polynomials obtained by the authors previously for a fixed specialization other specializations of the parameter lead to upper bounds for the coefficients of the given basis one gets also upper bounds of the coefficients from the positivity of classical thom polynomials for mappings obtained previously by the last two authors

high order family of multivariate iterative methods convergence and stability abstract in this manuscript we design an efficient sixth order scheme for solving nonlinear systems of equations with only two steps in its iterative expression moreover it belongs to a new parametric class of methods whose order of convergence is at least four in this family the most stable members have been selected by using techniques of real multidimensional dynamics also some members with undesirable chaotic behavior have been found and rejected for practical purposes finally all these high order schemes have been numerically checked and compared with other existing procedures of the same order of convergence showing good and stable performance

a simple and efficient method with high order convergence for solving systems of nonlinear equations we propose an m 1 step modified newton method of convergence order m 2 to solve systems of nonlinear equations which are third frechet differentiable in a convex set containing the zero computational efficiency in the general form for a positive integer m is discussed which shows that the efficiency increases with m when applied to large systems moreover a comparison between the efficiency of this technique and some existing efficient methods is made which implies that the present method is more efficient particularly for solving large systems of equations theoretical results about order of convergence and computational efficiency are largely verified in numerical examples

a semilocal convergence analysis for directional newton methods a semilocal convergence analysis for directional newton methods in n variables is provided in this study using weaker hypotheses than in the elegant related work by y levin and a ben israel and introducing the center lipschitz condition we provide under the same computational cost as in levin and ben israel a semilocal convergence analysis with the following advantages weaker convergence conditions larger convergence domain finer error estimates on the distances involved and an at least as precise information on the location of the zero of the function a numerical example where our results apply to solve an equation but not the ones in levin and ben israel is also provided in this study

on the semilocal convergence of efficient chebyshev secant type methods we introduce a three step chebyshev secant type method cstm with high efficiency index for solving nonlinear equations in a banach space setting we provide a semilocal convergence analysis for cstm using recurrence relations numerical examples validating our theoretical results are also provided in this study

a new class of methods with higher order of convergence for solving systems of nonlinear equations by studying the commonness of some fifth order methods modified from third order ones for solving systems of nonlinear equations we propose a new class of three step methods of convergence order five by modifying a class of two step methods with cubic convergence next for a given method of order p 2 which uses the extended newton iteration yk xk af xk 1f xk as a predictor a new method of order p 2 is proposed for example we construct a class of m 2 step methods of convergence order 2m 3 by introducing only one evaluation of the function to each of the last m steps for any positive integer m in this paper we mainly focus on the class of fifth order methods when m 1 computational efficiency in the general form is considered several examples for numerical tests are given to show the asymptotic behavior and the computational efficiency of these higher order methods

a family of kurchatov type methods and its stability we present a parametric family of iterative methods with memory for solving nonlinear equations that includes kurchatov s scheme preserving its second order convergence by using the tools of multidimensional real dynamics the stability of members of this family is analyzed on low degree polynomials showing that some elements of this class have more stable behavior than the original kurchatov s method we extend this family to multidimensional case and present different numerical tests for several members of the class on nonlinear systems the numerical results obtained confirm the dynamical analysis made

quantum cohomology rings of lagrangian and orthogonal grassmannians and total positivity we verify in an elementary way a result of peterson for the maximal orthogonal and lagrangian grassmannians and then find vafa intriligator type formulas which compute their 3 point genus zero gromov witten invariants finally we study the total positivity of the related peterson s varieties and show that rietsch s conjecture about the total positivity holds for these cases

letter to the editor frozen divided difference scheme for solving systems of nonlinear equations the development of an inverse first order divided difference operator for functions of several variables as well as a direct computation of the local order of convergence of an iterative method is presented a generalized algorithm of the secant method for solving a system of nonlinear equations is studied and the maximum computational efficiency is computed furthermore a sequence that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced

extending the newton kantorovich hypothesis for solving equations the famous newton kantorovich hypothesis kantorovich and akilov 1982 3 argyros 2007 2 argyros and hilout 2009 7 has been used for a long time as a sufficient condition for the convergence of newton s method to a solution of an equation in connection with the lipschitz continuity of the frechet derivative of the operator involved here using lipschitz and center lipschitz conditions and our new idea of recurrent functions we show that the newton kantorovich hypothesis can be weakened under the same information moreover the error bounds are tighter than the corresponding ones given by the dominating newton kantorovich theorem argyros 1998 1 2 7 ezquerro and hernandez 2002 11 3 proinov 2009 2010 16 17 numerical examples including a nonlinear integral equation of chandrasekhar type chandrasekhar 1960 9 as well as a two boundary value problem with a green s kernel argyros 2007 2 are also provided in this study

kantorovich s type theorems for systems of equations with constant rank derivatives the famous newton kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of newton s method to a solution of an equation here we present a kantorovich type convergence analysis for the gauss newton s method which improves the result in w m hauszler a kantorovich type convergence analysis for the gauss newton method numer math 48 1986 119 125 and extends the main theorem in i k argyros on the newton kantorovich hypothesis for solving equations j comput appl math 169 2004 315 332 furthermore the radius of convergence ball is also obtained

an efficient derivative free family of fourth order methods for solving systems of nonlinear equations we present a derivative free two step family of fourth order methods for solving systems of nonlinear equations using the well known traub steffensen method in the first step in order to determine the local convergence order we apply the first order divided difference operator for functions of several variables and direct computation by taylor s expansion computational efficiencies of the methods of new family are considered and compared with existing methods of similar structure it is showed that the new family is especially efficient in solving large systems four numerical examples are given to compare the proposed methods with existing methods and to confirm the theoretical results

on the newton kantorovich hypothesis for solving equations the famous newton kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of newton s method to a solution of an equation in connection with the lipschitz continuity of the frechet derivative of the operator involved here using lipschitz and center lipschitz conditions we show that the newton kantorovich hypothesis can be weakened the error bounds obtained under our semilocal convergence result are more precise than the corresponding ones given by the dominating newton kantorovich theorem

on an iterative algorithm with superquadratic convergence for solving nonlinear operator equations we study an iterative method with order 1 2 for solving nonlinear operator equations in banach spaces algorithms for specific operator equations are built up we present the received new results of the local and semilocal convergence in case when the first order divided differences of a nonlinear operator are holder continuous moreover a quadratic nonlinear majorant for a nonlinear operator according to the conditions laid upon it is built a priori and a posteriori estimations of the method s error are received the method needs almost the same number of computations as the classical secant method but has a higher order of convergence we apply our results to the numerical solving of a nonlinear boundary value problem of second order and to the systems of nonlinear equations of large dimension

chaotic quantum behaved particle swarm optimization algorithm for solving nonlinear system of equations abstract this study proposes a novel chaotic quantum behaved particle swarm optimization algorithm for solving nonlinear system of equations different chaotic maps are introduced to enhance the effectiveness and robustness of the algorithm several benchmark studies are carried out logistic map gives the best results and is utilized in solving nonlinear equation sets nine well known problems are solved with our algorithm and results are compared with quantum behaved particle swarm optimization intelligent tuned harmony search gravitational search algorithm and literature studies comparison results reveal that the proposed algorithm can cope with the highly non linear problems and outperforms many algorithms which exist in the literature

a two step steffensen s method under modified convergence conditions a simplification of a third order iterative method is proposed the main advantage of this method is that it does not need to evaluate neither any frechet derivative nor any bilinear operator a semilocal convergence theorem in banach spaces under modified kantorovich conditions is analyzed a local convergence analysis is also performed finally some numerical results are presented

steffensen type methods for solving nonlinear equations in the present paper by approximating the derivatives in the well known fourth order ostrowski s method and in a sixth order improved ostrowski s method by central difference quotients we obtain new modifications of these methods free from derivatives we prove the important fact that the methods obtained preserve their convergence orders 4 and 6 respectively without calculating any derivatives finally numerical tests confirm the theoretical results and allow us to compare these variants with the corresponding methods that make use of derivatives and with the classical newton s method

cmmse a novel scheme having seventh order convergence for nonlinear systems abstract in this paper we present a new three point iterative scheme for obtaining the solution of nonlinear system having seventh order convergence the beauty of our scheme is that we obtained the seventh order convergence with minimal computational cost as compare to the existing ones in addition we also analyze the theoretical convergence properties of the propose scheme moreover we show its applicability of them on a total six numbers of nonlinear models first three of them are boundary value hammerstein integral and 2d bratu s problems the last three are standard academic large systems of nonlinear equations of order 50 50 100 100 and 120 120 respectively finally we concluded on the basis of obtained numerical experiments that our iterative method performs better in terms of residual error computational efficiency error between the two consecutive iterations cpu time asymptotic error constant term and approximated root

a novel family of composite newton traub methods for solving systems of nonlinear equations we present a family of three step iterative methods of convergence order five for solving systems of nonlinear equations the methodology is based on the two step traub s method with cubic convergence for solving scalar equations computational efficiency of the new methods is considered and compared with some well known existing methods numerical tests are performed on some problems of different nature which confirm robust and efficient convergence behavior of the proposed methods moreover theoretical results concerning order of convergence and computational efficiency are verified in the numerical problems stability of the methods is tested by drawing basins of attraction in a two dimensional polynomial system

newton methods on banach spaces with a convergence structure and applications abstract in this study we use inexact newton like methods to find solutions of nonlinear operator equations on banach spaces with a convergence structure our technique involves the introduction of a generalized norm as an operator from a linear space into a partially ordered banach space in this way the metric properties of the examined problem can be analyzed more precisely moreover this approach allows us to derive from the same theorem on the one hand semilocal results of kantorovich type and on the other hand global results based on monotonicity considerations by imposing very general lipschitz like conditions on the operators involved on the one hand we cover a wider range of problems and on the other hand by choosing our operators appropriately we can find sharper error bounds on the distances involved than before finally our methods are used to solve integral equations that cannot be solved with existing methods

invertible behavior in elementary cellular automata with memory elementary cellular automata ecas have been studied for their ability to generate complex global behavior despite their simplicity one variation of ecas is obtained by adding memory to each cell in a neighborhood this process generates a provisional configuration in which the application of an evolution rule establishes the dynamics of the system this version is known as an eca with memory ecam most previous work on ecams analyzed the complex behavior taking chaotic ecas however the present paper investigates reversible ecams as obtained from reversible and permutative ecas these ecas have at least one ancestor for every configuration thus the correct permutation of states may specify the memory function to obtain reversible ecams for permutative ecas which are often irreversible we demonstrate that the use of a quiescent state and the correct manipulation of de bruijn blocks produce reversible ecams

variants of a classic traub s result we present a generalization of a modification of a classic traub s result which allows constructing from a given iterative method with order of convergence p efficient iterative methods with order of convergence at least pq 1 q p 1 q n

on the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods a development of an inverse first order divided difference operator for functions of several variables is presented two generalized derivative free algorithms built up from ostrowski s method for solving systems of nonlinear equations are written and analyzed a direct computation of the local order of convergence for these variants of ostrowski s method is given in order to preserve the local order of convergence any divided difference operator is not valid two counterexamples of computation of a classical divided difference operator without preserving the order are presented a rigorous study to know a priori if the new method will preserve the order of the original modified method is presented the conclusion is that this fact does not depend on the method but on the systems of equations and if the associated divided difference verifies a particular condition a new divided difference operator solving this problem is proposed furthermore a computation that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced this study can be applied directly to other newton s type methods where derivatives are approximated by divided differences

a technique to choose the most efficient method between secant method and some variants abstract a few variants of the secant method for solving nonlinear equations are analyzed and studied in order to compute the local order of convergence of these iterative methods a development of the inverse operator of the first order divided differences of a function of several variables in two points is presented using a direct symbolic computation the computational efficiency and the approximated computational order of convergence are introduced and computed choosing the most efficient method among the presented ones furthermore we give a technique in order to estimate the computational cost of any iterative method and this measure allows us to choose the most efficient among them

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

a newton kantorvich theorem for equations involving m frechet differentiable operators and applications in radiative transfer abstract in this study we approximate a locally unique solution of a nonlinear operator equation in banach space using newton s method a complete error analysis showing the quadratic convergence of our method is also given our new theorem uses lipschitz or holder continuity assumptions on m frechet differentiable operators where m 2 is a positive integer a numerical example is given to show that our results provide a better information on the location of the solution as well as finer error bounds on the distances involved than earlier results a second numerical example shows how to solve a nonlinear integral equation appearing in radiative transfer

a new family of eighth order iterative methods for solving nonlinear equations a family of eighth order iterative methods with four evaluations for the solution of nonlinear equations is presented kung and traub conjectured that an iteration method without memory based on n evaluations could achieve optimal convergence order 2 n 1 the new family of eighth order methods agrees with the conjecture of kung traub for the case n 4 therefore this family of methods has efficiency index equal to 1 682 numerical comparisons are made with several other existing methods to show the performance of the presented methods

an improvement to ostrowski root finding method an improvement to the iterative method based on the ostrowski one to compute nonlinear equation solutions which increases the local order of convergence is suggested the adaptation of a strategy presented here gives a new iteration function with an additional evaluation of the function it also shows a smaller cost if we use adaptive multi precision arithmetic the numerical results computed using this system with a floating point system representing 200 decimal digits support this theory

third order convergence theorem by using majorizing function for a modified newton method in banach space abstract following the ideas of frontini and sormani we present a modified newton method in banach space which is used to solve the nonlinear operator equation we establish the newton kantorovich convergence theorem for the modified newton method with third order convergence in banach space by using majorizing function we also get the error estimate finally two examples are provided to show the application of our theorem

welch sets for random generation and representation of reversible one dimensional cellular automata abstract reversible one dimensional cellular automata are studied from the perspective of welch sets this paper presents an algorithm to generate random welch sets that define a reversible cellular automaton then properties of welch sets are used in order to establish two bipartite graphs describing the evolution rule of reversible cellular automata the first graph gives an alternative representation for the dynamics of these systems as block mappings and shifts the second graph offers a compact representation for the evolution rule of reversible cellular automata both graphs and their matrix representations are illustrated by the generation of random reversible cellular automata with 6 and 18 states

on the computational efficiency index and some iterative methods for solving systems of nonlinear equations in this paper two new iterative methods are built up and analyzed a generalization of the efficiency index used in the scalar case to several variables in iterative methods for solving systems of nonlinear equations is revisited analytic proofs of the local order of convergence based on developments of multilineal functions and numerical concepts that will be used to illustrate the analytic results are given an approximation of the computational order of convergence is computed independently of the knowledge of the root and the necessary time to get one correct decimal is studied in our examples

weak convergence conditions for inexact newton type methods abstract we establish a new semilocal convergence results for inexact newton type methods for approximating a locally unique solution of a nonlinear equation in a banach spaces setting we show that our sufficient convergence conditions are weaker and the estimates of error bounds are tighter in some cases than in earlier works 15 31 special cases and numerical examples are also provided in this study

majorizing sequences for iterative methods we provide convergence results for very general majorizing sequences of iterative methods using our new concept of recurrent functions we unify the semilocal convergence analysis of newton type methods ntm under more general lipschitz type conditions we present two very general majorizing sequences and we extend the applicability of ntm using the same information before chen and yamamoto 1989 13 deuflhard 2004 16 kantorovich and akilov 1982 19 miel 1979 20 miel 1980 21 and rheinboldt 1968 30 applications special cases and examples are also provided in this study to justify the theoretical results of our new approach

convergence and numerical analysis of a family of two step steffensen s methods we provide sufficient conditions for the semilocal convergence of a family of two step steffensen s iterative methods on a banach space the main advantage of this family is that it does not need to evaluate neither any frechet derivative nor any bilinear operator but having a high speed of convergence some numerical experiments are also presented

on newton s method for solving equations containing frechet differentiable operators of order at least two we provide sufficient conditions for the semilocal convergence of newton s method to a locally unique solution of a nonlinear operator equation containing operators that are frechet differentiable of order at least two in a banach space setting numerical examples are also provided to show that our results apply to solve nonlinear equations in cases earlier ones cannot j m gutierrez a new semilocal convergence theorem for newton s method j comput appl math 79 1997 131 145 z huang a note of kantorovich theorem for newton iteration j comput appl math 47 1993 211 217 f a potra sharp error bounds for a class of newton like methods libertas mathematica 5 1985 71 84

a steffensen s type method in banach spaces with applications on boundary value problems in this paper a modified steffensen s type iterative scheme for the numerical solution of a system of nonlinear equations is studied two convergence theorems are presented the numerical solution of boundary value problems by the multiple shooting method using the proposed iterative scheme is analyzed

general 2 2 system of nonlinear integral equations and its approximate solution abstract in this note we consider a general 2 2 system of nonlinear volterra type integral equations the modified newton method modified nm is used to reduce the nonlinear problems into 2 2 linear system of algebraic integral equations of volterra type the latter equation is solved by discretization method nystrom method with gauss legendre quadrature is applied for the kernel integrals and newton forwarded interpolation formula is used for finding values of unknown functions at the selected node points existence and uniqueness solution of the problems are proved and accuracy of the quadrature formula together with convergence of the proposed method are obtained finally numerical examples are provided to show the validity and efficiency of the method presented numerical results reveal that the proposed methods is efficient and accurate comparisons with other methods for the same problem are also presented

procedures for calculating reversible one dimensional cellular automata we describe two algorithms for calculating reversible one dimensional cellular automata of neighborhood size 2 we explain how this kind of automaton represents all the other cases using two basic properties of reversible automata such as uniform multiplicity of ancestors and welch indices these algorithms only require matrix products and transitive closures of binary relations to classify all the possible reversible automata of neighborhood size 2 we expose the features advantages and differences with other well known methods finally we present results for reversible automata from three to six states and neighborhood size 2

semilocal convergence of secant like methods for differentiable and nondifferentiable operator equations abstract from well known secant like methods we observe that we can construct a new family of secant like methods that includes the secant method and kurchatov s method we analyse the local orders of convergence and the efficiencies of the methods of the family and study the semilocal convergence for differentiable and nondifferentiable operators finally we apply our results to conservative problems

the secant method and divided differences holder continuous we apply the secant method to solve non linear operator equations in banach spaces a semilocal convergence result is obtained where the first order divided difference of the non linear operator is holder continuous for that we use a technique based on a new system of recurrence relations to obtain domains of existence and uniqueness of the solution and give an explicit expression for the a priori error bounds moreover we apply our results to the numerical solution of a non linear boundary value problem of second order

schubert polynomials for the classical groups the task of a theory of schubert polynomials is to produce explicit representatives for schubert classes in the cohomology ring of a flag variety and to do so in a manner that is as natural as possible from a combinatorial point of view to explain more fully let us review a special case the schubert calculus for grassmannians where one asks for the number of linear spaces of given dimension satisfying certain geometric conditions a typical problem is to find the number of lines meeting four given lines in general position in 3 space answer below for each of the four given lines the set of lines meeting it is a schubert variety in the grassmannian and we want the number of intersection points of these four subvarieties in the modem solution of this problem the schubert varieties induce canonical elements of the cohomology ring of the grassmannian called schubert classes the product of these schubert classes is the class of a point times the number of intersection points counted with appropriate multiplicities this reformulation of the problem though one of the great achievements of algebraic geometry is only part of a solution it remains to give a concrete model for the cohomology ring that makes explicit computation with schubert classes possible as it happens the cohomology rings of grassmannians can be identified with quotients of a polynomial ring so that schubert classes correspond to schur functions intersection numbers such as we are considering then turn out to be littlewood richardson coefficients for example the answer to our four lines 4 problem is the coefficient of the schur function s 2 2 in the product s 1 x or 2 for an extended treatment and history of the subject see 10 11 18 the identification of schur functions as schubert polynomials for grassmannians is a consequence of a more general and now highly developed theory of schubert polynomials for the flag varieties of the special linear groups sl n c the starting point for this more general theory is a construction of

three step iterative methods with eighth order convergence for solving nonlinear equations a family of eighth order iterative methods for the solution of nonlinear equations is presented the new family of eighth order methods is based on king s fourth order methods and the family of sixth order iteration methods developed by chun et al per iteration the new methods require three evaluations of the function and one evaluation of its first derivative therefore this family of methods has the efficiency index which equals 1 682 kung and traub conjectured that a multipoint iteration without memory based on n evaluations could achieve optimal convergence order 2 n 1 thus we provide a new example which agrees with the conjecture of kung traub for n 4 numerical comparisons are made to show the performance of the presented methods

on the local convergence of fast two step newton like methods for solving nonlinear equations a local convergence analysis is presented for a fast two step newton like method tsnlm for solving nonlinear equations in a banach space setting the tsnlm unifies earlier methods such as newton s secant newton like chebyshev secant chebyshev newton steffensen stirling s and other single or multistep methods numerical examples and a comparative study of these methods validating our theoretical results are also given in the concluding section of this paper

a closed formula for the inverse of a reversible cellular automaton with 2r 1 cyclic rule abstract reversibility of cellular automata ca has been an extensively studied problem from both a theoretical and a practical point of view it is known when a 2 r 1 cyclic cellular automaton with periodic boundary conditions p b c is reversible see siap et al 2013 but as far as we know no explicit expression is given for its inverse cellular automaton apart from the case r 1 see encinas and del rey 2007 in this paper we give a closed formula for the inverse rule of a reversible 2 r 1 cyclic cellular automaton with p b c over the finite field f 2 for any value of the neighbourhood radius r it turns out that the inverse of a reversible 2 r 1 cyclic ca with p b c is again a cyclic ca with p b c but with a different neighbourhood radius and this radius depends on certain numbers which need to be computed by a new algorithm we introduce finally we apply our results to the case r 1 which is the eca with wolfram rule number 150 to introduce an alternative and improved expression for the inverse transition dipolynomial formulated in encinas and del rey 2007 we also illustrate these results by giving explicit computations for the inverse transition dipolynomial of a reversible cellular automaton with penta cyclic rule

derivative free iterative methods for nonlinear systems in this work we introduce a new operator of divided differences that preserves the convergence order when it is used for approximating the jacobian matrix in iterative method for solving nonlinear systems we obtain derivative free iterative methods with lower computational cost than the corresponding ones with different operators of divided differences we also study the global convergence of these methods by analyzing their dynamical behavior

a weak kantorovich existence theorem for the solution of nonlinear equations the kantorovich theorem is a fundamental tool in nonlinear analysis for proving the existence and uniqueness of solutions of nonlinear equations arising in various fields this theorem was weakened recently by argyros who used a combination of lipschitz and center lipschitz conditions in place of the lipschitz conditions of the kantorovich theorem in the present paper we prove a weak kantorovich type theorem that gives the same conclusions as the previous two results under weaker conditions illustrative examples are provided in the paper

extending the applicability of the local and semilocal convergence of newton s method we present a local as well a semilocal convergence analysis for newton s method in a banach space setting using the same lipschitz constants as in earlier studies we extend the applicability of newton s method as follows local case a larger radius is given as well as more precise error estimates on the distances involved semilocal case the convergence domain is extended the error estimates are tighter and the information on the location of the solution is at least as precise as before numerical examples further justify the theoretical results

an improvement of the euler chebyshev iterative method abstract we present a new method for the computation of the solutions of nonlinear equations when it is necessary to use high precision we improve the euler chebyshev iterative method which is a generalization of an improvement of newton s method a symbolic computation allows us to find the best coefficients respect to the local order of convergence the adaptation of the strategy presented here gives an additional iteration function with an additional evaluation of the function it provides a lower cost if we use adaptive multi precision arithmetics the numerical results computed using this system with a floating point representing a maximum of 210 decimal digits support this theory

some variants of ostrowski s method with seventh order convergence in this paper we present a class of new variants of ostrowski s method with order of convergence seven per iteration the new methods require three evaluations of the function and one evaluation of its first derivative and therefore this class of methods has the efficiency index equal to 1 627 numerical tests verifying the theory are given and multistep iterations based on the present methods are developed

a class of quasi newton generalized steffensen methods on banach spaces we consider a class of generalized steffensen iterations procedure for solving nonlinear equations on banach spaces without any derivative we establish the convergence under the kantarovich ostrowski s conditions the majorizing sequence will be a newton s type sequence thus the convergence can have better properties finally a numerical comparation with the classical methods is presented

higher order derivative free iterative methods with and without memory for systems of nonlinear equations abstract a derivative free family of iterations without memory consisting of three steps for solving nonlinear systems of equations is brought forward then the main aim of the paper is furnished by proposing several novel schemes with memory possessing higher rates of convergence analytical discussions are reported and the theoretical efficiency of the methods is studied application of the schemes in solving partial differential equations is finally provided to support the theoretical discussions