new eighth order iterative methods for solving nonlinear equations in this paper three new families of eighth order iterative methods for solving simple roots of nonlinear equations are developed by using weight function methods per iteration these iterative methods require three evaluations of the function and one evaluation of the first derivative this implies that the efficiency index of the developed methods is 1 682 which is optimal according to kung and traub s conjecture 7 for four function evaluations per iteration notice that bi et al s method in 2 and 3 are special cases of the developed families of methods in this study several new examples of eighth order methods with efficiency index 1 682 are provided after the development of each family of methods numerical comparisons are made with several other existing methods to show the performance of the presented methods

directional newton methods in n variables directional newton methods for functions f of n variables are shown to converge under standard assumptions to a solution of f x 0 the rate of convergence is quadratic for near gradient directions and directions along components of the gradient of f with maximal modulus these methods are applied to solving systems of equations without reversion of the jacobian matrix

tsallis conditional mutual information in investigating long range correlation in symbol sequences abstract the presence of long range correlation lrc as opposed to markov chain of a finite order is of primary interest in the study of symbol sequences such as dna among others methods using information theory have been developed for markov chain order estimation in this work we consider the tsallis entropy and define the tsallis conditional mutual information tcmi for markov chain order estimation which is more suitable to identify large orders suggesting effectively lrc the tcmi of order m for a tsallis parameter q is computed for increasing m and a significance test is performed for each m until tcmi is found non significant randomization and parametric significance tests for tcmi are developed for the latter the null distribution of tcmi is approximated with a gamma distribution deriving analytic expressions for its parameters we assess the accuracy of order estimation with the two tests for tcmi and compare them to the respective tests using the shannon entropy extended simulations on markov chains of different orders and structures of the transition probability matrix show that shannon and tsallis tests with parameter q 2 have similar performance for small orders when the problem becomes more demanding for higher markov chain orders and lrcs tsallis tests for q 2 have better approximation to the correct order with the sequence length n finally the tsallis tests are favorably compared to shannon tests to real dna sequences

sixth order derivative free family of iterative methods in this study we develop a four parameter family of sixth order convergent iterative methods for solving nonlinear scalar equations methods of the family require evaluation of four functions per iteration these methods are totally free of derivatives convergence analysis shows that the family is sixth order convergent which is also verified through the numerical work though the methods are independent of derivatives computational results demonstrate that family of methods are efficient and demonstrate equal or better performance as compared with other six order methods and the classical newton method

predictor corrector halley method for nonlinear equations abstract in this paper we suggest and analyze a new two step predictor corrector type iterative methods for solving nonlinear equations of the type f x 0 by using the technique of updating the solution this method can be viewed as a predictor corrector iterative halley s method we also consider the convergence analysis of the proposed method to illustrate the efficiency of this new method we give several examples and a comparison with the method of grau and diaz barrero m grau j l diaz barrero an improvement to ostrowski root finding method appl math comput 173 2006 450 456 is given this method can be considered as a refinement and improvement of the halley method

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

entropy estimates of small data sets estimating entropies from limited data series is known to be a non trivial task naive estimations are plagued with both systematic bias and statistical errors here we present a new balanced estimator for entropy functionals shannon renyi and tsallis specially devised to provide a compromise between low bias and small statistical errors for short data series this new estimator outperforms other currently available ones when the data sets are small and the probabilities of the possible outputs of the random variable are not close to zero otherwise other well known estimators remain a better choice the potential range of applicability of this estimator is quite broad specially for biological and digital data series

entropy and local uncertainty of data from sensory neurons we present an empirical comparison between neural interspike interval sequences obtained from two different kinds of sensory receptors both differ in their internal structure as well as in the strength of correlations and the degree of predictability found in the respective spike trains as a further tool in this context we suggest the local uncertainty assigning a well defined predictability to individual spikes the local uncertainty is demonstrated to reveal significant patterns within the interspike interval sequences even when its overall structure is almost random our approach is based on the concept of symbolic dynamics and information theory

new iterative methods for solving nonlinear equation by using homotopy perturbation method in this paper we suggest and analyze a new class of iterative methods for solving nonlinear equations by using the homotopy perturbation method convergence of their method is also considered here we also discuss the efficiency index and computational order of convergence of new methods several numerical examples are given to illustrate the efficiency and performance of these new methods these new iterative methods may be viewed as an extension and generalization of the existing methods for solving nonlinear equations

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

a multi parameter family of three step eighth order iterative methods locating a simple root a multi parameter family of three step eighth order iterative methods free from second derivatives are proposed in this paper to find a simple root of nonlinear algebraic equations convergence analysis as well as numerical experiments confirms the eighth order convergence and asymptotic error constants

a family of higher order iterations free from second derivative for nonlinear equations in r abstract the aim of this article is to develop a family of higher order iterations free from second derivative for solving nonlinear equations in r their theoretical computational and dynamical aspects are fully investigated and theorems are established to provide their order of convergence and asymptotic error constant it is observed that the family includes sixth order methods and for a particular case its eighth order can be achieved in this family methods use three functions and one first derivative evaluations the family of methods can be shown to be optimal by using kung traub conjecture a number of numerical examples are worked out to demonstrate the applicability of these methods the improved results are obtained in comparison to some of the existing robust methods on the considered test examples local convergence analysis and dynamical study of the proposed family of methods are also carried out

a hummel seebeck family of iterative methods with improved convergence and efficiency several improvements to a family of iterative methods deduced from a hummel seebeck theorem are presented a symbolic computation that allows to find the best coefficients respect to the local order of convergence is also given the theoretical and computational order of convergence for all methods is increased furthermore the efficiency of these methods applied to the functions tested is improved adapting the strategy presented here a new iteration function with a new evaluation of the function is obtained and using adaptive multi precision arithmetic a smaller cost is got the numerical results computed carrying out this procedure with a floating point system representing 1000 decimal digits support this theory

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 stable family with high order of convergence for solving nonlinear equations recently li et al 2014 have published a new family of iterative methods without memory with order of convergence five or six which are not optimal in the sense of kung and traub s conjecture therefore we attempt to modify this suggested family in such a way that it becomes optimal to this end we consider the same two first steps of the mentioned family and furthermore we introduce a better approximation for f z in the third step based on interpolation idea as opposed to the taylor s series used in the work of li et al theoretical dynamical and numerical aspects of the new family are described and investigated in details

on a general transformation of multipoint root solvers optimal multipoint methods for solving nonlinear equations of arbitrary order of convergence are investigated a low cost transformation that converts newton preconditioned methods into a derivative free variant is presented this transforming procedure preserves both algorithm body structure and order of convergence of the original scheme another useful application of the proposed transformation is the acceleration of convergence order of non optimal methods

sixteenth order method for nonlinear equations modification of newton s method with higher order convergence is presented the modification of newton s method is based on king s fourth order method the new method requires three step per iteration analysis of convergence demonstrates that the order of convergence is 16 some numerical examples illustrate that the algorithm is more efficient and performs better than classical newton s method and other methods

some iterative methods free from second derivatives for nonlinear equations in a recent paper noor m aslam noor new classes of iterative methods for nonlinear equations appl math comput 2007 doi 10 1016 j amc 2007 suggested and analyzed a generalized one parameter halley method for solving nonlinear equations using in this paper we modified this method which has fourth order convergence as special cases we obtain a family of third order iterative methods for appropriate and suitable choice of the parameter we have compared this modified noor method with some other iterative methods which shows that this new iterative method is robust and efficient one several examples are given to illustrate the efficiency and the performance of this new method

two new families of sixth order methods for solving non linear equations in this paper we developed two new families of sixth order methods for solving simple roots of non linear equations per iteration these methods require two evaluations of the function and two evaluations of the first order derivatives which implies that the efficiency indexes of our methods are 1 565 these methods have more advantages than newton s method and other methods with the same convergence order as shown in the illustration examples finally using the developing methodology described in this paper two new families of improvements of jarratt method with sixth order convergence are derived in a straightforward manner notice that kou s method in jisheng kou yitian li an improvement of the jarratt method appl math comput 189 2007 1816 1821 and wang s method in xiuhua wang jisheng kou yitian li a variant of jarratt method with sixth order convergence appl math comput 204 2008 14 19 are the special cases of the new improvements

a new family of optimal eighth order methods with dynamics for nonlinear equations we propose a simple yet efficient family of three point iterative methods for solving nonlinear equations each method of the family requires four evaluations namely three functions and one derivative per full iteration and possesses eighth order of convergence thus the family is optimal in the sense of kung traub conjecture and has the efficiency index 1 682 which is better than that of newton s and many other higher order methods various numerical examples are considered to check the performance and to verify the theoretical results computational results including the elapsed cpu time confirm the efficient and robust character of proposed technique moreover the presented basins of attraction also confirm better performance of the methods as compared to other established methods in literature

a new two step biparametric family of sixth order iterative methods free from second derivatives for solving nonlinear algebraic equations in this paper a new two step biparametric family of sixth order iterative methods free from second derivatives are proposed in this paper to find a simple root of a nonlinear algebraic equation their efficiency indices are all found to be 6 1 4 convergence analysis together with numerical experiments confirms sixth order convergence and asymptotic error constants

guessing probability distributions from small samples we propose a new method for the calculation of the statistical properties e g the entropy of unknown generators of symbolic sequences the probability distributionp k of the elementsk of a population can be approximated by the frequenciesf k of a sample provided the sample is long enough so that each elementk occurs many times our method yields an approximation if this precondition does not hold for a givenf k we recalculate the zipf ordered probability distribution by optimization of the parameters of a guessed distribution we demonstrate that our method yields reliable results

prediction and entropy of nonlinear dynamical systems and symbolic sequences with lro abstract following shannon we introduce higher order entropies and derive dynamic entropies the nth order dynamic entropy conditional entropy is a measure of the uncertainty of the next state which follows after the observation of n foregoing states the asymptotic behaviour of the dynamic entropies at large n is studied for several nonlinear model systems and for symbolic sequences with long range order lro for example we investigate 1d maps texts dna strings and time series it is shown that the existence of long correlations improves the possibility of predictions characteristic scaling laws for the higher order shannon entropies and the conditional entropies are derived and a new interpolation formula is tested finally instead of the dynamic entropies which yield mean values of the uncertainty predictability we investigate the local values of the uncertainty predictability

reply to comments on exact evaluation of the maliuzhinets half plane function for original paper see i s koh et al ibid vol 55 no 2 p 483 5 2007 for comment paper see s nadarajah ibid vol 55 no 11 p 3366 2007 this article is a reply to comment on exact evaluation of the maliuzhinets half plane function

a family of optimal three point methods for solving nonlinear equations using two parametric functions abstract using an interactive approach which combines symbolic computation and taylor s series a wide family of three point iterative methods for solving nonlinear equations is constructed these methods use two suitable parametric functions at the second and third step and reach the eighth order of convergence consuming only four function evaluations per iteration this means that the proposed family supports the kung traub hypothesis 1974 on the upper bound 2 m of the order of multipoint methods based on m 1 function evaluations providing very high computational efficiency different methods are obtained by taking specific parametric functions the presented numerical examples demonstrate exceptional convergence speed with only few function evaluations

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

some sixth order zero finding variants of chebyshev halley methods a zero finding technique in which the order of convergence is improved and nonlinear equations are solved more efficiently than they are solved by traditional iterative methods is derived composing a modified chebyshev halley method with a variant of this method that just introduces one evaluation of the function the iterative methods presented are obtained by carrying out this procedure the output numerical results show that the new methods compete in both order and efficiency with the modified chebyshev halley methods

new family of seventh order methods for nonlinear equations abstract a family of seventh order iterative methods for the solution of nonlinear equations is presented the new methods are based on king s fourth order methods and without using the second derivatives per iteration the new methods require three evaluations of the function and one evaluation of its first derivative therefore this family of methods has efficiency index equal to 1 627 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

modified householder iterative method for nonlinear equations abstract in this paper we suggest and analyze a new two step predictor corrector type iterative method for solving nonlinear equations of the type f x 0 this new method includes the two step newton method as a special case we show that this new two step method is a sixth order convergent method several examples are given to illustrate the efficiency of this new method and its comparison with other sixth order methods this method can be considered as a significant improvement of the newton method and its variant forms

highly efficient family of iterative methods for solving nonlinear models abstract in this study we present a new highly efficient sixth order family of iterative methods for solving nonlinear equations along with convergence properties further we extend this family to the multidimensional case preserving the same order of convergence in order to illustrate the applicability of our methods we choose some real world problems namely kinematic syntheses boundary value bratu s 2d fisher s and hammerstein integral problems in the case of nonlinear systems see farhane et al 2017 1 kaur and arora 2017 2 for more applications in addition numerical comparisons are made to show the performance of our iterative techniques with the existing ones moreover we find that our techniques perform better as compared to the existing ones of same order in terms of residual error and the errors between two consecutive iterations finally the stability analysis of our methods also support this to great extent

a family of modified ostrowski methods with accelerated sixth order convergence based on ostrowski fourth order multipoint method we derive a one parameter family of sixth order methods for solving equations each member of the family requires three evaluations of the given function and one evaluation of its derivative per iteration numerical examples are presented and performance is compared with ostrowski method

directional secant method for nonlinear equations we present a directional secant method a secant variant of the directional newton method for solving a single nonlinear equation in several variables under suitable assumptions we prove the convergence and the quadratic convergence speed of this new method numerical examples show that the directional secant method is feasible and efficient and has better numerical behaviour than the directional newton method

an acceleration of newton s method super halley method from a study of the convexity we give an acceleration for newton s method and obtain a new third order method then we use this method for solving non linear equations in banach spaces establishing conditions on convergence existence and uniqueness of solution as well as error estimates

eighth order methods with high efficiency index for solving nonlinear equations in this paper we derive a new family of eighth order methods for solving simple roots of nonlinear equations by using weight function methods per iteration these methods require three evaluations of the function and one evaluation of its first derivative which implies that the efficiency indexes are 1 682 numerical comparisons are made to show the performance of the derived methods as shown in the illustration examples

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

a sixth order method for nonlinear equations abstract in this paper a sixth order method is developed by extending a third order method of weerakoon and fernando s weerakoon t g i fernando a variant of newton s method with accelerated third order convergence appl math lett 13 2000 87 93 for finding the real roots of nonlinear equations in r starting with a suitably chosen x 0 the method generates a sequence of iterates converging to the root the convergence analysis is provided to establish its sixth order of convergence in terms of computational cost it requires evaluations of only two functions and two first derivatives per iteration this implies that efficiency index of our method is 1 565 our method is comparable with the methods of neta b neta a sixth order family of methods for nonlinear equations intern j computer math 7 1979 157 161 and that of kou and li jisheng kou yitian li an improvement of jarratt method appl math comput 189 2007 1816 1821 it does not require the evaluation of the second order derivative of the given function as required in the family of chebyshev halley type methods jisheng kou xiuhua wang sixth order variants of chebyshev halley methods for solving non linear equations appl math comput 190 2007 1839 1843 jisheng kou on chebyshev halley methods with sixth order convergence for solving non linear equations appl math comput 190 2007 126 131 the efficacy of the method is tested on a number of numerical examples it is observed that our method takes less number of iterations than newton s method and the method of weerakoon and fernando on comparison with the other sixth order methods it behaves either similarly or better for the examples considered

a penta parametric family of fifteenth order multipoint methods for nonlinear equations abstract a penta parametric family of four step multipoint iterative methods of order fifteen for nonlinear algebraic equations are developed and their convergence properties are established the efficiency indices are all found to be 15 1 5 1 71877 better than 14 1 5 1 69522 of a family of fourteenth order methods suggested by neta 9 numerical examples are demonstrated to verify the developed theory

on computational efficiency for multi precision zero finding methods an adapted definition related to multi precision arithmetic zero finding methods is considered and several improvements to iterative methods to compute nonlinear equation solutions which increase the local order of convergence are revisited furthermore a higher efficiency when we use a definition related to adaptive multi precision arithmetic is also given numerical results computed with a floating point system representing 200 and 1000 decimal digits support this theory

iterative methods improving newton s method by the decomposition method in this paper we present a sequence of iterative methods improving newton s method for solving nonlinear equations the adomian decomposition method is applied to an equivalent coupled system to construct the sequence of the methods whose order of convergence increases as it progresses the orders of convergence are derived analytically and then rederived by applying symbolic computation of maple some numerical illustrations are given

a biparametric family of optimally convergent sixteenth order multipoint methods with their fourth step weighting function as a sum of a rational and a generic two variable function a biparametric family of four step multipoint iterative methods of order sixteen to numerically solve nonlinear equations are developed and their convergence properties are investigated the efficiency indices of these methods are all found to be 16 1 5 1 741101 being optimally consistent with the conjecture of kung traub numerical examples as well as comparison with existing methods developed by kung traub and neta are demonstrated to confirm the developed theory in this paper

a family of optimal sixteenth order multipoint methods with a linear fraction plus a trivariate polynomial as the fourth step weighting function a new family of four step optimal multipoint iterative methods of order sixteen for solving nonlinear equations are developed along with their convergence properties numerical experiments with comparison to some existing methods are demonstrated to support the underlying theory

some sixth order variants of ostrowski root finding methods abstract in this paper we present some sixth order class of modified ostrowski s methods for solving nonlinear equations per iteration each class member requires three function and one first derivative evaluations and is shown to be at least sixth order convergent several numerical examples are given to illustrate the performance of some of the presented methods

a variant of jarratt method with sixth order convergence in this paper we present a variant of jarratt method for solving non linear equations per iteration the new method adds the evaluation of the function at another point in the procedure iterated by jarratt method as a consequence the local order of convergence is improved from four for jarratt method to six for the new method a new multistep iteration scheme based on the new method is developed and numerical tests verifying the theory are also given

a one parameter square root family of two step root finders a one parameter family of iterative methods for solving nonlinear equations is constructed all the methods of the proposed family are cubically convergent for a simple root except one particular method which attains the fourth order without the increase of computational cost these methods belong to the class of two step methods and require three function evaluations per iteration the square root structure of the family provides finding a complex zero of real functions in some cases a comparison analysis shows that the presented family generates the methods which are comparable or even superior than the existing two step iterative methods of the third order

exact evaluation of the maliuzhinets half plane function the maliuzhinets half plane function psipi z has been used for many applications but its closed form has not been known in this communication we exactly evaluate a known integral representation of psipi z to formulate an exact closed form of the function which is expressed in terms of the dilogarithm function the exact formulation is numerically verified by comparing it with the results obtained by numerical integration of a integral representation of general maliuzhinets function a closed form of psipi 2 z is also obtained since it can be expressed in terms of psipi z

fifth order iterative methods for solving nonlinear equations abstract in this paper we suggest and analyze a new two step iterative method for solving nonlinear equation f x 0 by rewriting the given nonlinear equation as a coupled system of equations and using the taylor series it is shown that this new iterative method is of fifth order several examples are given to illustrate its performance and efficiency comparison with other methods is also given this method can be viewed as an improvement of the halley s method

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 note on some quadrature based three step iterative methods for non linear equations in a recent paper n a mir t zaman some quadrature based three step iterative methods for non linear equations appl math comput 193 2007 366 373 some new three step iterative methods for non linear equations have been proposed in this note we show that the algorithm 2 2 and algorithm 2 3 given by the authors have twelfth order and ninth order convergence respectively not seventh order one as claimed in their work

the jarratt method in banach space setting abstract in this note we extend the jarratt method of order four into banach spaces we also establish a kantorovich type convergence theorem see the references and give an explicit expression for the error bound of the method jarratt 1966 1969

new modifications of potra ptak s method with optimal fourth and eighth orders of convergence in this paper we present two new iterative methods for solving nonlinear equations by using suitable taylor and divided difference approximations both methods are obtained by modifying potra ptak s method trying to get optimal order we prove that the new methods reach orders of convergence four and eight with three and four functional evaluations respectively so kung and traub s conjecture kung and traub 1974 2 that establishes for an iterative method based on n evaluations an optimal order p 2 n 1 is fulfilled getting the highest efficiency indices for orders p 4 and p 8 which are 1 587 and 1 682 we also perform different numerical tests that confirm the theoretical results and allow us to compare these methods with potra ptak s method from which they have been derived and with other recently published eighth order methods

some higher order modifications of newton s method for solving nonlinear equations in this paper we consider constructing some higher order modifications of newton s method for solving nonlinear equations which increase the order of convergence of existing iterative methods by one or two or three units this construction can be applied to any iteration formula and per iteration the resulting methods add only one additional function evaluation to increase the order some illustrative examples are provided and several numerical results are given to show the performance of the presented methods

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

some higher order iteration functions for solving nonlinear models abstract in this paper we present a new efficient sixth order family of jarratt type methods for solving scalar equations then we extend this family to the multidimensional case preserving the same order of convergence we also discuss the theoretical convergence properties of the proposed scheme in the case of scalar as well as multidimensional case the derivation of these schemes are based on weight function approach and free disposable parameters we also demonstrate the applicability of them on total six number of problems first five are real life problems namely continuous stirred tank reactor cstr chemical engineering the trajectory of an electron in the air gap between two parallel plates hammerstein integration and boundary value problems last one is the standard academic test problem in addition numerical comparisons are made to show the performance of the proposed iterative techniques with the existing techniques of the same order in the scalar as well as multi dimensional case finally on the basis of numerical results we conclude that our techniques perform better in terms of residual error error between the two consecutive iterations asymptotic error constant term and approximated root as compared to the existing ones of same order in scalar as well as multidimensional case

a family of modified super halley methods with fourth order convergence in this paper we present a family of modified super halley methods for solving non linear equations analysis of convergence shows that the methods have fourth order convergence the superiority of the new methods is that they require no additional evaluations of the function the first derivative or second derivative as compared with the classical third order methods although their order is improved numerical results show that the new methods can be efficient

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

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

the improvements of modified newton s method abstract in this paper we improve some third order modifications of newton s method and obtain many new methods for solving non linear equations the new methods have the order of convergence five or six per iteration these methods require two evaluations of the function and two evaluations of its first derivative and therefore the efficiency of the new methods may also be improved these methods can compete with newton s method as we show in some examples

a family of three point methods of optimal order for solving nonlinear equations a family of three point iterative methods for solving nonlinear equations is constructed using a suitable parametric function and two arbitrary real parameters it is proved that these methods have the convergence order eight requiring only four function evaluations per iteration in this way it is demonstrated that the proposed class of methods supports the kung traub hypothesis 1974 3 on the upper bound 2 n of the order of multipoint methods based on n 1 function evaluations consequently this class of root solvers possesses very high computational efficiency numerical examples are included to demonstrate exceptional convergence speed with only few function evaluations

improvements of the efficiency of some three step iterative like newton methods an improvement of the local order of convergence is presented to increase the efficiency of the iterative method with an appropriate number of evaluations of the function and its derivative the third and fourth order of known two step like newton methods have been improved and the efficiency has also been increased

modified householder iterative method free from second derivatives for nonlinear equations abstract in this paper we suggest and analyze a new two step predictor corrector type iterative method free from second derivatives for solving nonlinear equations of the type f x 0 this new method includes the two step newton method as a special case we prove that the new iterative method is of fourth order several examples are given to illustrate the efficiency of this new method and its comparison with other iterative methods this method can be considered as a significant improvement of the newton method and its variant forms

comments on exact evaluation of the maliuzhinets half plane function this paper presents the comments on exact evaluation of the maliuzhinets half plane function

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

modifications of higher order convergence for solving nonlinear equations abstract in liang fang guoping he some modifications of newton s method with higher order convergence for solving nonlinear equations j comput appl math 228 2009 296 303 the authors pointed out that the iteration constructed in y m ham c b chun and s g lee some higher order modifications of newton s method for solving nonlinear equations j comput appl math 222 2008 477 486 failed when p 2 they gave some counterexamples and obtained a modified result however they did not show the essential reason which leads to the incorrect result in this paper we shall show that reason and present more general results than the above mentioned papers

a convergence improvement factor and higher order methods for solving nonlinear equations abstract a convergence improvement factor is introduced for solving nonlinear equations so as to increase the efficiency of iterative methods many well known iterative methods are covered

statistical linguistic characterization of variability in observed and synthetic daily precipitation series abstract this paper deals with an application of zipf law in climatology this analysis allows the extraction of information not available by standard methods in particular rainfall temporal aggregation patterns associated with different climates are characterized by means of exponents derived from the resulting scaling laws the analogy with linguistic analysis is obtained using a particular coding of precipitation as a discrete variable with four states corresponding to four standard precipitation thresholds each weekly symbolic sequence of observed precipitation is considered as a word and each local station defines a language characterized by the observed words in a period representative of the climatology to characterize these precipitation languages we obtained characteristic exponents derived from the zipf law for a set of representative stations of the main koppen s climates and subclimates we found different scaling behaviors for different subclimates given by a single exponent in the range 0 6 humid tropical climates to 1 4 polar climates some humid middle latitude subclimates exhibit a crossover with two different characteristic exponents corresponding to high and low frequency aggregation patterns no explanation for this behavior is provided as an application of the proposed methodology the same analysis was applied to stochastic time series obtained from standard weather generators we found that the language defined by a first order markov process agrees with the observed language of the observed series

a variant of super halley method with accelerated fourth order convergence in this paper we present a new variant of super halley method for solving non linear equations analysis of convergence shows that the method has the order of convergence four per iteration the new method requires the same evaluations of the function the first derivative and second derivative as its classical predecessor super halley method numerical results show that the method has definite practical utility

an improvement of the jarratt method in this paper we present a variant of jarratt method for solving non linear equations per iteration the new method adds the evaluation of the function at another point in the procedure iterated by jarratt method as a consequence the local order of convergence is improved from four for jarratt method to six for the new method the new multistep iteration scheme based on the new method is developed and numerical tests verifying the theory are also given

a class of three point root solvers of optimal order of convergence the construction of a class of three point methods for solving nonlinear equations of the eighth order is presented these methods are developed by combining fourth order methods from the class of optimal two point methods and a modified newton s method in the third step obtained by a suitable approximation of the first derivative based on interpolation by a nonlinear fraction it is proved that the new three step methods reach the eighth order of convergence using only four function evaluations which supports the kung traub conjecture on the optimal order of convergence numerical examples for the selected special cases of two step methods are given to demonstrate very fast convergence and a high computational efficiency of the proposed multipoint methods some computational aspects and the comparison with existing methods are also included

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

a class of two step steffensen type methods with fourth order convergence abstract based on steffensen s method we derive a one parameter class of fourth order methods for solving nonlinear equations in the proposed methods an interpolating polynomial is used to get a better approximation to the derivative of the given function each member of the class requires three evaluations of the given function per iteration therefore this class of methods has efficiency index which equals 1 587 kung and traub conjectured an iteration using n evaluations of f or its derivatives without memory is of convergence order at most 2 n 1 the new class of fourth order methods agrees with the conjecture of kung traub for the case n 3 numerical comparisons are made to show the performance of the presented methods

a fifth order iterative method for solving nonlinear equations in this paper we present a new fifth order method for solving nonlinear equations per iteration the new method requires two function and two first derivative evaluations it is shown that the new method is fifth order convergent several numerical examples are given to illustrate the performance of the presented method

some improvements of jarratt s method with sixth order convergence abstract in this paper we present a one parameter family of variants of jarratt s fourth order method for solving nonlinear equations it is shown that the order of convergence of each family member is improved from four to six even though it adds one evaluation of the function at the point iterated by jarratt s method per iteration several numerical examples are given to illustrate the performance of the presented methods

some modification of newton s method by the method of undetermined coefficients in this paper we construct some modifications of newton s method for solving nonlinear equations which is based on the method of undetermined coefficients it is shown by way of illustration that the method of undetermined coefficients is a promising tool for developing new methods and reveals its wide applicability by obtaining some existing methods as special cases two new sixth order methods are developed numerical examples are given to support that the methods thus obtained can compete with other iterative methods

some quadrature based three step iterative methods for non linear equations in this paper we present three step quadrature based iterative methods for solving non linear equations the convergence analysis of the methods is discussed it is established that the new methods have convergence order six seven and eight respectively numerical tests show that the new methods are comparable with the well known existing methods and in many cases give better results our results can be considered as an improvement and refinement of the previously known results in the literature

three step iterative methods with optimal eighth order convergence in this paper based on ostrowski s method a new family of eighth order methods for solving nonlinear equations is derived in terms of computational cost each iteration of these methods requires three evaluations of the function and one evaluation of its first derivative so that their efficiency indices are 1 682 which is optimal according to kung and traub s conjecture numerical comparisons are made to show the performance of the new family

an infinite family of one step iterators for solving nonlinear equations to increase the order of convergence and a new algorithm of global convergence in this paper we present an infinite family of one step iterative formulas for solving nonlinear equations present method one from now on pmi that can be expressed as x n 1 f m x n with email protected m 1 we will prove that the corresponding iteration formula of the family x n 1 f m 0 x n has an order of convergence m 0 1 the increment of the velocity of convergence of the sequence of the iterator family x n 1 f m 1 x n with respect to the previous one x n 1 f m x n is attained at the expense of one derivative evaluation more besides we introduce a new algorithm present method two from now on pmii that plays the role of seeker for an initial value to guarantee the local convergence of the pmi both of them can be combined as an algorithm of global convergence included the case of singular roots that does not depend on the chosen initial value and that allows to find all the roots in a feasible interval in a general and complete way these are in my opinion the main results of this work