on an efficient family of derivative free three point methods for solving nonlinear equations abstract new three step derivative free families of three point methods for solving nonlinear equations are presented first a new family without memory of optimal order eight consuming four function evaluations per iteration is proposed by using two weight functions the improvement of the convergence rate of this basic family even up to 50 is obtained without any additional function evaluation using a self accelerating parameter this varying parameter is calculated in each iterative step employing only information from the current and the previous iteration defining in this way a family with memory the self accelerating parameter is calculated applying newton s interpolating polynomials of degree scaling from 1 to 4 the corresponding r orders of convergence are increased from 8 to 10 11 6 4 2 11 66 and 12 providing very high computational efficiency of the proposed methods with memory another convenient fact is that these methods do not use derivatives numerical examples and comparison with the existing three point methods are included to confirm theoretical results

covariance regularization by thresholding this paper considers regularizing a covariance matrix of p variables estimated from n observations by hard thresholding we show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense the variables are gaussian or sub gaussian and log p n 0 and obtain explicit rates the results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity we discuss an intuitive resampling scheme for threshold selection and prove a general cross validation result that justifies this approach we also compare thresholding to other covariance estimators in simulations and on an example from climate data 1 introduction estimation of covariance matrices is important in a number of areas of statistical analysis including dimension reduction by principal component analysis pca classification by linear or quadratic discriminant analysis lda and qda establishing independence and conditional independence relations in the context of graphical models and setting confidence intervals on linear functions of the means of the components in recent years many application areas where these tools are used have been dealing with very high dimensional datasets and sample sizes can be very small relative to dimension examples include genetic data brain imaging spectroscopic imaging climate data and many others it is well known by now that the empirical covariance matrix for samples of size n from a p variate gaussian distribution np p is not a good estimator of the population covariance if p is large many results in random matrix theory illustrate this from the classical mary law 29 to the more recent work of johnstone and his students on the theory of the largest eigenvalues 12 23 30 and associated eigenvectors 24 however with the exception of a method for estimating the covariance spectrum 11 these probabilistic results do not offer alternatives to the sample covariance matrix alternative estimators for large covariance matrices have therefore attracted a lot of attention recently two broad classes of covariance estimators have emerged those that rely on a natural ordering among variables and assume that variables

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

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

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

construction of optimal order nonlinear solvers using inverse interpolation abstract there is a vast literature on finding simple roots of nonlinear equations by iterative methods these methods can be classified by order by the information used or by efficiency there are very few optimal methods that is methods of order 2 m requiring m 1 function evaluations per iteration here we give a general way to construct such methods by using inverse interpolation and any optimal two point method the presented optimal multipoint methods are tested on numerical examples and compared to existing methods of the same order of convergence

matrix positivity preservers in fixed dimension i a classical theorem of i j schoenberg characterizes functions that preserve positivity when applied entrywise to positive semidefinite matrices of arbitrary size obtaining similar characterizations in fixed dimension is intricate in this note we provide a solution to this problem in the polynomial case as consequences we derive tight linear matrix inequalities for hadamard powers of positive semidefinite matrices and a sharp asymptotic bound for the matrix cube problem involving hadamard powers

hub discovery in partial correlation graphs one of the most important problems in large scale inference problems is the identification of variables that are highly dependent on several other variables when dependence is measured by partial correlations these variables identify those rows of the partial correlation matrix that have several entries with large magnitudes i e hubs in the associated partial correlation graph this paper develops theory and algorithms for discovering such hubs from a few observations of these variables we introduce a hub screening framework in which the user specifies both a minimum partial correlation rho and a minimum degree delta to screen the vertices the choice of rho and delta can be guided by our mathematical expressions for the phase transition correlation threshold rho c governing the average number of discoveries they can also be guided by our asymptotic expressions for familywise discovery rates under the assumption of large number p of variables fixed number n of multivariate samples and weak dependence under the null hypothesis that the dispersion covariance matrix is sparse these limiting expressions can be used to enforce familywise error constraints and to rank the discoveries in order of increasing statistical significance for n ll p the computational complexity of the proposed partial correlation screening method is low and is therefore highly scalable thus it can be applied to significantly larger problems than previous approaches the theory is applied to discovering hubs in a high dimensional gene microarray dataset

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

a non iterative method for solving non linear equations abstract in this paper using a hyperbolic tangent function tanh x 0 we develop a non iterative method to estimate a root of an equation f x 0 the problem of finding root is transformed to evaluating an integral and thus we need not take account of choosing initial guess the larger the value of the better the approximation to the root alternatively we employ the signum function sgn x instead of the hyperbolic tangent function which results in an exact formula for the root availability of the present method is shown by some numerical examples for which the traditional newton s method is not appropriate

positive definite hankel matrix completions and hamburger moment completions abstract in this paper we give solutions to hamburger moment problems with missing entries the problem of completing partial positive sequences is considered the main result is a characterization of positive definite completable patterns namely patterns that guarantee the existence of hamburger moment completion of a partial positive definite sequence moreover several patterns which are not positive definite completable are given

brief paper decentralized coordination of autonomous swarms using parallel gibbs sampling in this paper we present analysis of a discrete time decentralized stochastic coordination algorithm for a group of mobile nodes called an autonomous swarm on a finite spatial lattice all nodes take their moves by sampling in parallel their locally perceived gibbs distributions corresponding to a pairwise nearest neighbor potential the algorithm has no explicit requirements on the connectedness of the underlying information graph which varies with the swarm configuration it is established that with an appropriate annealing schedule the algorithm results in swarm configurations converging to the global minimizers of a modified potential energy function the extent of discrepancy between the modified and original potential energy functions is determined by the maximum node travel between time steps and when such distance is small the ultimate swarm configurations are close to the global minimizers of the original potential energy simulation results are further presented to illustrate the capability of the sampling algorithm in approximate global optimization for swarms

gibbs sampler based coordination of autonomous swarms in this paper a novel gibbs sampler based algorithm is proposed for coordination of autonomous swarms the swarm is modeled as a markov random field mrf on a graph with a time varying neighborhood system determined by local interaction links the gibbs potential is designed to reflect global objectives and constraints it is established that with primarily local sensing communications the swarm configuration converges to the global minimizer s of the potential function the impact of the gibbs potential on the convergence speed is investigated finally a hybrid algorithm is developed to improve the efficiency of the stochastic scheme by integrating the gibbs sampler based method with the deterministic gradient flow method simulation results are presented to illustrate the proposed approach and verify the analyses

self organization in systems of self propelled particles we investigate a discrete model consisting of self propelled particles that obey simple interaction rules we show that this model can self organize and exhibit coherent localized solutions in one and in two dimensions in one dimension the self organized solution is a localized flock of finite extent in which the density abruptly drops to zero at the edges in two dimensions we focus on the vortex solution in which the particles rotate around a common center and show that this solution can be obtained from random initial conditions even in the absence of a confining boundary furthermore we develop a continuum version of our discrete model and demonstrate that the agreement between the discrete and the continuum model is excellent

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

a new eighth order iterative method for solving nonlinear equations abstract in this paper we present an improvement of the fourth order newton type method for solving a nonlinear equation the new newton type method is shown to converge of the order eight per iteration the new method requires three evaluations of the function and one evaluation of its first derivative and therefore the new method has the efficiency index of 8 4 which is better than the well known newton type methods of lower order we shall examine the effectiveness of the new eighth order newton type method by approximating the simple root of a given nonlinear equation numerical comparisons are made with several other existing methods to show the performance of the presented method

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

monotonicity for entrywise functions of matrices abstract we characterize real functions f on an interval for which the entrywise matrix function a ij f a ij is positive monotone and convex respectively in the positive semidefiniteness order fractional power functions are exemplified and related weak majorizations are shown

on a general class of optimal order multipoint methods for solving nonlinear equations abstract we develop a class of n point iterative methods with optimal 2 n order of convergence for solving nonlinear equations newton s second order and ostrowski s fourth order methods are special cases corresponding to n 1 and n 2 eighth and sixteenth order methods that correspond to n 3 and n 4 of the class are special cases of the eighth and sixteenth order methods proposed by sharma et al 25 the methodology is based on employing the previously obtained n 1 step scheme and modifying the n th step by using rational hermite interpolation unlike that of existing higher order techniques the proposed technique is attractive since it leads to a simple implementation local convergence analysis is provided to show that the iterations are locally well defined and convergent theoretical results are verified through numerical experimentations the performance is also compared with already established methods in literature it is observed that new algorithms are more accurate than existing counterparts and very effective in high precision computations

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

retaining positive definiteness in thresholded matrices abstract positive definite p d matrices arise naturally in many areas within mathematics and also feature extensively in scientific applications in modern high dimensional applications a common approach to finding sparse positive definite matrices is to threshold their small off diagonal elements this thresholding sometimes referred to as hard thresholding sets small elements to zero thresholding has the attractive property that the resulting matrices are sparse and are thus easier to interpret and work with in many applications it is often required and thus implicitly assumed that thresholded matrices retain positive definiteness in this paper we formally investigate the algebraic properties of p d matrices which are thresholded we demonstrate that for positive definiteness to be preserved the pattern of elements to be set to zero has to necessarily correspond to a graph which is a union of complete components this result rigorously demonstrates that except in special cases positive definiteness can be easily lost we then proceed to demonstrate that the class of diagonally dominant matrices is not maximal in terms of retaining positive definiteness when thresholded consequently we derive characterizations of matrices which retain positive definiteness when thresholded with respect to important classes of graphs in particular we demonstrate that retaining positive definiteness upon thresholding is governed by complex algebraic conditions

complete characterization of hadamard powers preserving loewner positivity monotonicity and convexity abstract entrywise powers of symmetric matrices preserving positivity monotonicity or convexity with respect to the loewner ordering arise in various applications and have received much attention recently in the literature following fitzgerald and horn 1977 8 it is well known that there exists a critical exponent beyond which all entrywise powers preserve positive definiteness similar phenomena have also recently been shown by hiai 2009 to occur for monotonicity and convexity in this paper we complete the characterization of all the entrywise powers below and above the critical exponents that are positive monotone or convex on the cone of positive semidefinite matrices we then extend the original problem by fully classifying the positive monotone or convex powers in a more general setting where additional rank constraints are imposed on the matrices we also classify the entrywise powers that are super sub additive with respect to the loewner ordering finally we extend all the previous characterizations to matrices with negative entries our analysis consequently allows us to answer a question raised by bhatia and elsner 2007 regarding the smallest dimension for which even extensions of the power functions do not preserve loewner positivity

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

remarks on on a general class of multipoint root finding methods of high computational efficiency an oversight on the number of function evaluations of the n point root finding methods proposed in siam j numer anal 47 2010 pp 4402 4414 is discussed to correct this flaw modified methods of optimal computational efficiency are presented by applying higher order hermite interpolatory polynomials

efficient polynomial root refiners a survey and new record efficiency estimates a typical iterative polynomial root finder begins with a relatively slow process of computing a crude but sufficiently close initial approximation to a root and then rapidly refines it the policy of using the same iterative process at both stages of computing an initial approximation and refining it however is neither necessary nor most effective the efficiency of an iteration at the former stage resists formal study and is usually decided empirically whereas formal study of the efficiency at the latter stage of refinement is not hard and is the subject of the current paper we define this local efficiency as log 1 0qd log 1 0 q 1 d q is the convergence order and d is the number of function evaluations per iteration it is inversely proportional to the number of flops involved assuming that about 2n flops are needed per evaluation of a polynomial of a degree n at a single point we extend the definition to cover the recent matrix methods for polynomial root finding as well as some methods that combine n approximations to all n roots to refine them simultaneously for the approximation of a single root of a polynomial of degree n the maximum local efficiency achieved so far is log 1 02 0 301 but we show its growth to infinity for simultaneous approximation of all n roots as n grows to infinity

a fourth order iterative method for solving nonlinear equations the present paper illustrates an iterative numerical method to solve nonlinear equations of the form f x 0 especially those containing the partial and non partial involvement of transcendental terms comparative analysis shows that the present method is faster than newton raphson method hybrid iteration method new hybrid iteration method and others cost is also found to be minimum than these methods the beauty in our method can be seen because of the optimization in important effecting factors i e lesser number of iteration steps lesser number of functional evaluations and lesser value of absolute error in final as well as in individual step as compared to the other methods this work also demonstrates the higher order convergence of the present method as compared to others without going to the computation of second derivative

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 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

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

eighth order family of iterative methods for nonlinear equations and their basins of attraction abstract we present a new family of eighth order methods for solving nonlinear equations the order of convergence of the considered methods is proved and corresponding asymptotic error constants are expressed in terms of four parameters numerical examples obtained using mathematica with high precision arithmetic demonstrate convergence and efficacy of our family of methods for some combinations of parameter values the new eighth order methods produce very good results on tested examples compared to the results produced by some of the eighth order methods existing in the related literature an exploration of the relevant dynamics of the proposed methods is presented along with illustrative basins of attraction for various polynomials

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

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

equilibria and steering laws for planar formations abstract this paper presents a lie group setting for the problem of control of formations as a natural outcome of the analysis of a planar two vehicle formation control law the vehicle trajectories are described using the planar frenet serret equations of motion which capture the evolution of both the vehicle position and orientation for unit speed motion subject to curvature steering control the set of all possible relative equilibria for arbitrary g invariant curvature controls is described where g se 2 is a symmetry group for the control law and a global convergence result for the two vehicle control law is proved an n vehicle generalization of the two vehicle control law is also presented and the corresponding relative equilibria for the n vehicle problem are characterized work is on going to discover stability and convergence results for the n vehicle problem

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

interpolatory multipoint methods with memory for solving nonlinear equations abstract a general way to construct multipoint methods for solving nonlinear equations by using inverse interpolation is presented the proposed methods belong to the class of multipoint methods with memory in particular a new two point method with memory with the order 5 17 2 4 562 is derived computational efficiency of the presented methods is analyzed and their comparison with existing methods with and without memory is performed on numerical examples it is shown that a special choice of initial approximations provides a considerably great accuracy of root approximations obtained by the proposed interpolatory iterative 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

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

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 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

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

an optimal steffensen type family for solving nonlinear equations abstract in this paper a general family of steffensen type methods with optimal order of convergence for solving nonlinear equations is constructed by using newton s iteration for the direct newtonian interpolation it satisfies the conjecture proposed by kung and traub h t kung j f traub optimal order of one point and multipoint iteration j assoc comput math 21 1974 634 651 that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 2 m 1 its error equations and asymptotic convergence constants are obtained finally it is compared with the related methods for solving nonlinear equations in the numerical examples

stability analysis of swarms in this note we specify an individual based continuous time model for swarm aggregation in n dimensional space and study its stability properties we show that the individuals autonomous agents or biological creatures will form a cohesive swarm in a finite time moreover we obtain an explicit bound on the swarm size which depends only on the parameters of the swarm model

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 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

three point methods with and without memory for solving nonlinear equations abstract a new family of three point derivative free methods for solving nonlinear equations is presented it is proved that the order of convergence of the basic family without memory is eight requiring four function evaluations which means that this family is optimal in the sense of the kung traub conjecture further accelerations of convergence speed are attained by suitable variation of a free parameter in each iterative step this self accelerating parameter is calculated using information from the current and previous iteration so that the presented methods may be regarded as the methods with memory the self correcting parameter is calculated applying the secant type method in three different ways and newton s interpolatory polynomial of the second degree the corresponding r order of convergence is increased from 8 to 4 1 5 2 8 472 9 10 and 11 the increase of convergence order is attained without any additional function calculations providing a very high computational efficiency of the proposed methods with memory another advantage is a convenient fact that these methods do not use derivatives numerical examples and the comparison with existing three point methods are included to confirm theoretical results and high computational efficiency

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

solving systems of nonlinear equations when the nonlinearity is expensive construction of multi step iterative method for solving system of nonlinear equations is considered when the nonlinearity is expensive the proposed method is divided into a base method and multi step part the convergence order of the base method is five and each step of multi step part adds additive factor of five in the convergence order of the base method the general formula of convergence order is 5 m 2 where m 3 is the step number for a single instance of the iterative method we only compute two jacobian and inversion of one jacobian is required the direct inversion of jacobian is avoided by computing lu factors the computed lu factors are used in the multi step part for solving five systems of linear equations that make the method computational efficient the distinctive feature of the underlying multi step iterative method is the single call to the computationally expensive nonlinear function and thus offers an increment of additive factor of five in the convergence order per single call the numerical simulations reveal that our proposed iterative method clearly shows better performance where the computational cost of the involved nonlinear function is higher than the computational cost for solving five lower and upper triangular systems

on generalized biparametric multipoint root finding methods with memory a general family of biparametric n point methods with memory for solving nonlinear equations is proposed using an original accelerating procedure with two parameters this family is based on derivative free classes of n point methods without memory of interpolatory type and steffensen like method with two free parameters the convergence rate of the presented family is considerably increased by self accelerating parameters which are calculated in each iteration using information from the current and previous iteration and newton s interpolating polynomials with divided differences the improvement of convergence order is achieved without any additional function evaluations so that the proposed family has a high computational efficiency numerical examples are included to confirm theoretical results and demonstrate convergence behavior of the proposed methods

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

large scale correlation screening this article addresses the problem of screening for variables with high correlations in high dimensional data in which there can be many fewer samples than variables we focus on threshold based correlation screening methods for three related applications screening for variables with large correlations within a single treatment autocorrelation screening screening for variables with large cross correlations over two treatments cross correlation screening and screening for variables that have persistently large autocorrelations over two treatments persistent correlation screening the novelty of correlation screening is that it identifies a smaller number of variables that are highly correlated with others compared with identifying a number of correlation parameters correlation screening suffers from a phase transition phenomenon as the correlation threshold decreases the number of discoveries increases abruptly we obtain asymptotic expressions for the mean number of discoveries and the phase t

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

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

general approach to constructing optimal multipoint families of iterative methods using hermite s rational interpolation abstract we discuss accelerating convergence of multipoint iterative methods for solving scalar equations using particular type rational interpolant both derivative free and newton type methods are investigated simultaneously as a conclusion a theorem of konig s type for multipoint iterations is stated a new optimal multipoint family of methods based on rational interpolation is constructed the iteration uses n function evaluations per cycle and o j operations in j th step of a single iteration to obtain 2 n 1 order of convergence several equivalent forms of the obtained iterates and development techniques are presented

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

a note on the priority of optimal multipoint methods for solving nonlinear equations the goal of this note is to clarify some questions of the priority and chronology of three point optimal methods which could provide a better insight into the topic we present two methods that appeared before the method of bi et al w bi q wu h ren a new family of eight order iterative methods for solving nonlinear equations appl math comput 214 2009 236 245 which is often cited in the literature as the successor of the kung traub method from 1974

an inequality for tensor product of positive operators and its applications abstract we present an inequality for tensor product of positive operators on hilbert spaces by considering the tensor products of operators as words on certain alphabets i e a set of letters as applications of the operator inequality and by a multilinear approach we show some matrix inequalities concerning induced operators and generalized matrix functions including determinants and permanents as special cases

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

low complexity root finding iteration functions with no derivatives of any order of convergence in this paper a procedure to design steffensen type methods of different orders for solving nonlinear equations is suggested by using a particular divided difference of first order we can transform many iterative methods into derivative free iterative schemes holding the order of convergence of the departure original method numerical examples and the study of the dynamics are made to show the performance of the presented schemes and to compare them with another ones

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

stability of multiagent systems with time dependent communication links we study a simple but compelling model of network of agents interacting via time dependent communication links the model finds application in a variety of fields including synchronization swarming and distributed decision making in the model each agent updates his current state based upon the current information received from neighboring agents necessary and or sufficient conditions for the convergence of the individual agents states to a common value are presented thereby extending recent results reported in the literature the stability analysis is based upon a blend of graph theoretic and system theoretic tools with the notion of convexity playing a central role the analysis is integrated within a formal framework of set valued lyapunov theory which may be of independent interest among others it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence even for the simple models discussed in the present paper

a class of optimal eighth order derivative free methods for solving the danchick gauss problem abstract a derivative free optimal eighth order family of iterative methods for solving nonlinear equations is constructed using weight functions approach with divided first order differences its performance along with several other derivative free methods is studied on the specific problem of danchick s reformulation of gauss method of preliminary orbit determination numerical experiments show that such derivative free high order methods offer significant advantages over both the classical and danchick s newton approach