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

measuring and testing dependence by correlation of distances distance correlation is a new measure of dependence between random vectors distance covariance and distance correlation are analogous to product moment covariance and correlation but unlike the classical definition of correlation distance correlation is zero only if the random vectors are independent the empirical distance dependence measures are based on certain euclidean distances between sample elements rather than sample moments yet have a compact representation analogous to the classical covariance and correlation asymptotic properties and applications in testing independence are discussed implementation of the test and monte carlo results are also presented

dimension reduction for the conditional mean in regressions with categorical predictors consider the regression of a response y on a vector of quantitative predictors x and a categorical predictor w in this article we describe a first method for reducing the dimension of x without loss of information on the conditional mean mathrm e y x w and without requiring a prespecified parametric model the method which allows for but does not require parametric versions of the subpopulation mean functions mathrm e y x w w includes a procedure for inference about the dimension of x after reduction this work integrates previous studies on dimension reduction for the conditional mean mathrm e y x in the absence of categorical predictors and dimension reduction for the full conditional distribution of y x w the methodology we describe may be particularly useful for constructing low dimensional summary plots to aid in model building at the outset of an analysis our proposals provide an often parsimonious alternative to the standard technique of modeling with interaction terms to adapt a mean function for different subpopulations determined by the levels of w examples illustrating this and other aspects of the development are presented

new modifications of hansen patrick s family with optimal fourth and eighth orders of convergence in this paper we present two new second derivative free classes of iterative methods based on hansen patrick s family hansen and patrick 1977 4 for solving nonlinear equations numerically in terms of computational cost both families require only three and four functional evaluations to achieve optimal fourth and eighth orders of convergence respectively moreover the local convergence analysis of the proposed methods is also given using hypotheses only on the first derivative and lipschitz constants furthermore the proposed schemes can also determine the complex zeros without having to start from a complex initial guess as would be necessary with other methods numerical examples and comparisons with some existing methods are included to confirm the theoretical results and high computational efficiency

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

martingale difference correlation and its use in high dimensional variable screening in this article we propose a new metric the so called martingale difference correlation to measure the departure of conditional mean independence between a scalar response variable v and a vector predictor variable u our metric is a natural extension of distance correlation proposed by szekely rizzo and bahirov which is used to measure the dependence between v and u the martingale difference correlation and its empirical counterpart inherit a number of desirable features of distance correlation and sample distance correlation such as algebraic simplicity and elegant theoretical properties we further use martingale difference correlation as a marginal utility to do high dimensional variable screening to screen out variables that do not contribute to conditional mean of the response given the covariates further extension to conditional quantile screening is also described in detail and sure screening properties are rigorously justified both simulation results and real data illustrations demonstr

some novel and optimal families of king s method with eighth and sixteenth order of convergence in this study our principle aim is to provide some novel eighth and sixteenth order families of king s method for solving nonlinear equations which should be superior than the existing schemes of same order the relevant optimal orders of the proposed families satisfy the classical kung traub conjecture which was made in 1974 the derivations of the proposed schemes are based on the weight function and rational approximation approaches respectively in addition convergence properties of the proposed families are fully investigated along with one lemma and two main theorems describing their order of convergence we consider a concrete variety of real life problems e g the trajectory of an electron in the air gap between two parallel plates chemical engineering problem van der waal s equation which explains the behavior of a real gas by introducing in the ideal gas equations and fractional conversion in a chemical reactor in order to check the validity applicability and effectiveness of our proposed methods further it is found from the numerical results that our proposed methods perform better than the existing ones of the same order when the accuracy is checked in the multi precision digits

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

an interpolated controller for stabilization of a plant with variable operating conditions this paper considers stabilization of a linear time invariant single input single output plant with variable operating conditions it is assumed that the plant is described by a linear interpolation of proper stable coprime factorizations of the transfer functions of two representative models which are defined at two representative operating points a linear interpolation of stabilizing controllers for the representative models is applied to the interpolated plant the necessary and sufficient condition for the interpolated plant to be stabilizable by the interpolated controller is presented using the nevanlinna pick theory it is also shown that simultaneous stabilizability of the two representative models is necessary for such stabilization an example is given which illustrates that the class of stabilizable plants via interpolated controllers is larger than that of plants stabilizable by fixed controllers

gain scheduling via linear fractional transformations abstract a linear finite dimensional plant with rational state space parameter dependence is controlled using a parameter dependent controller the parameters are known to take on values in a unit ball and are known in real time the goal of control is to stabilize the parameter dependent closed loop system and provide disturbance error attenuation as measured in induced l 2 norms the approach taken uses the optimally scaled small gain theorem and solves the control synthesis problem by reformulating the existence conditions into a finite dimensional convex optimization

sure independence screening in generalized linear models with np dimensionality ultrahigh dimensional variable selection plays an increasingly important role in contemporary scientific discoveries and statistical research among others fan and lv 2008 propose an independent screening framework by ranking the marginal correlations they showed that the correlation ranking procedure possesses a sure independence screening property within the context of the linear model with gaussian covariates and responses in this paper we propose a more general version of the independent learning with ranking the maximum marginal likelihood estimates or the maximum marginal likelihood itself in generalized linear models we show that the proposed methods with fan and lv 2008 as a very special case also possess the sure screening property with vanishing false selection rate the conditions under which that the independence learning possesses a sure screening is surprisingly simple this justifies the applicability of such a simple method in a wide spectrum we quantify explicitly the extent to which the dimensionality can be reduced by independence screening which depends on the interactions of the covariance matrix of covariates and true parameters simulation studies are used to illustrate the utility of the proposed approaches in addition we supported in part by grant nsf grants dms 0714554 and dms 0704337 the bulk of the work was conducted when rui song was a postdoctoral research fellow at princeton university the authors would like to thank the associate editor and two referees for their constructive comments that improve the presentation and the results of the paper ams 2000 subject classifications primary 68q32 62j12 secondary 62e99 60f10

an approach to simultaneous system design part 11 nonswitching gain and dynamic feedback compensation by algebraic geometric methods this paper studies structured uncertainty problems in feedback system design considers a compact parameterization of the space of linear dynamical systems and introduces base points and critical points as two algebraic geometric objects that have significance in sensitivity and robustness studies respectively using the nevanlinna pick interpolation theory the author obtains a necessary and sufficient condition for simultaneous stabilization of a structured one parameter family of plants a recent result due to kharitonov on the simultaneous stability of a parameterized family of polynomials leads to a sufficiency condition for simultaneous stabilization of a structured multiparameter family of plants furthermore the author considers simultaneous pole placement of an r tuple of plants as a means to arbitrarily tune the natural frequencies of a multimode linear dynamical system the concept of nondegenerate and twisted r tuples of plants is introduced as the pole placement problem is studie

martingale difference divergence matrix and its application to dimension reduction for stationary multivariate time series in this article we introduce a new methodology to perform dimension reduction for a stationary multivariate time series our method is motivated by the consideration of optimal prediction and focu

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

dimension reduction for conditional mean in regression in many situations regression analysis is mostly concerned with inferring about the conditional mean of the response given the predictors and less concerned with the other aspects of the conditional distribution in this paper we develop dimension reduction methods that incorporate this consideration we introduce the notion of the central mean subspace cms a natural inferential object for dimension reduction when the mean function is of interest we study properties of the cms and develop methods to estimate it these methods include a new class of estimators which requires fewer conditions than phd and which displays a clear advantage when one of the conditions for phd is violated cms also reveals a transparent distinction among the existing methods for dimension reduction ols phd sir and save we apply the new methods to a data set involving recumbent cows

conditional quantile screening in ultrahigh dimensional heterogeneous data summary to accommodate the heterogeneity that is often present in ultrahigh dimensional data we propose a conditional quantile screening method which enables us to select features that contribute to the conditional quantile of the response given the covariates the method can naturally handle censored data by incorporating a weighting scheme through redistribution of the mass to the right moreover it is invariant to monotone transformation of the response and requires substantially weaker conditions than do alternative methods we establish sure independent screening properties for both the complete and the censored response cases we also conduct simulations to evaluate the finite sample performance of the proposed method and compare it with existing approaches

quantile regression for analyzing heterogeneity in ultra high dimension ultra high dimensional data often display heterogeneity due to either heteroscedastic variance or other forms of non location scale covariate effects to accommodate heterogeneity we advocate a more general interpretation of sparsity which assumes that only a small number of covariates influence the conditional distribution of the response variable given all candidate covariates however the sets of relevant covariates may differ when we consider different segments of the conditional distribution in this framework we investigate the methodology and theory of nonconvex penalized quantile regression in ultra high dimension the proposed approach has two distinctive features 1 it enables us to explore the entire conditional distribution of the response variable given the ultra high dimensional covariates and provides a more realistic picture of the sparsity pattern 2 it requires substantially weaker conditions compared with alternative methods in the literature thus it greatly alleviates the

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

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

feature screening via distance correlation learning this article is concerned with screening features in ultrahigh dimensional data analysis which has become increasingly important in diverse scientific fields we develop a sure independence screening procedure based on the distance correlation dc sis the dc sis can be implemented as easily as the sure independence screening sis procedure based on the pearson correlation proposed by fan and lv however the dc sis can significantly improve the sis fan and lv established the sure screening property for the sis based on linear models but the sure screening property is valid for the dc sis under more general settings including linear models furthermore the implementation of the dc sis does not require model specification e g linear model or generalized linear model for responses or predictors this is a very appealing property in ultrahigh dimensional data analysis moreover the dc sis can be used directly to screen grouped predictor variables and multivariate response variables we establish

quantile adaptive model free variable screening for high dimensional heterogeneous data we introduce a quantile adaptive framework for nonlinear variable screening with high dimensional heterogeneous data this framework has two distinctive features 1 it allows the set of active variables to vary across quantiles thus making it more flexible to accommodate heterogeneity 2 it is model free and avoids the difficult task of specifying the form of a statistical model in a high dimensional space our nonlinear independence screening procedure employs spline approximations to model the marginal effects at a quantile level of interest under appropriate conditions on the quantile functions without requiring the existence of any moments the new procedure is shown to enjoy the sure screening property in ultra high dimensions furthermore the quantile adaptive framework can naturally handle censored data arising in survival analysis we prove that the sure screening property remains valid when the response variable is subject to random right censoring numerical studies confirm the fine performance of the proposed method for various semiparametric models and its effectiveness to extract quantile specific information from heteroscedastic data

a new highly efficient and optimal family of eighth order methods for solving nonlinear equations the principle aim of this manuscript is to present a new highly efficient and optimal eighth order family of iterative methods to solve nonlinear equations in the case of simple roots the derivation of this scheme is based on weight function and rational approximation approaches the proposed family requires only four functional evaluations viz f xn f xn f yn and f zn per iteration therefore the proposed family is optimal in the sense of kung traub hypotheses in addition we given a theorem which describing the order of convergence of the proposed family moreover we present a local convergence analysis using hypotheses only on the first order derivative since in our preceding theorem we used hypotheses on higher order derivatives that do not appear in these methods in this way we expand the applicability of these methods even further furthermore a variety of nonlinear equations is considered for the numerical experiments it is observed from the numerical experiments that our proposed methods perform better than the existing optimal methods of same order when the accuracy is checked in multi precision digits

a new optimal eighth order family of iterative methods for the solution of nonlinear equations in this paper a new optimal family of methods for solving nonlinear equations is presented it is proved by analysis of convergence that each member of the family which requires three evaluations of the function and one evaluation of its first derivative per iteration is of optimal order eight numerical comparisons are made with several other existing eighth order methods by using the basins of attraction and through numerical computations to demonstrate the efficiency and the performance of the presented methods

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

testing for marginal linear effects in quantile regression this paper develops a new marginal testing procedure to detect the presence of significant predictors associated with the conditional quantiles of a scalar response the idea is to fit the marginal quantile regression on each predictor one at a time and then base the test on the t statistics associated with the most predictive predictors a resampling method is devised to calibrate this test statistic which has non regular limiting behavior due to the selection of the most predictive variables asymptotic validity of the procedure is established in a general quantile regression setting in which the marginal quantile regression models can be misspecified even though a fixed dimension is assumed to derive the asymptotic results the proposed test is applicable and computationally feasible for large dimensional predictors the method is more flexible than existing marginal screening test methods based on mean regression and has the added advantage of being robust against outliers in the response the approach is illustrated using an application to an hiv drug resistance dataset

iterative methods for solving nonlinear equations with finitely many roots in an interval in this paper we consider a nonlinear equation f x 0 having finitely many roots in a bounded interval based on the so called numerical integration method b i yun a non iterative method for solving non linear equations appl math comput 198 2008 691 699 without any initial guess we propose iterative methods to obtain all the roots of the nonlinear equation in the result an algorithm to find all of the simple roots and multiple ones as well as the extrema of f x is developed moreover criteria for distinguishing zeros and extrema are included in the algorithm availability of the proposed method is demonstrated by some numerical examples

generating function method for constructing new iterations abstract in this paper we propose a generating function method for constructing new two and three point iterations with p 3 p 8 order of convergence this approach allows us to derive a new family of the optimal order iterative methods that include well known methods as special cases the necessary and sufficient conditions for pth order convergence of the proposed iterations are given in terms of parameters n and n we also propose some generating functions for n and n we give the extension of a class of optimal fourth order jarratt s type iterations with a 2 3 we develop a unified representation of all optimal eighth order methods several numerical results are given to demonstrate the efficiency and the performance of the presented methods and compare them with some other existing 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

comparison of several families of optimal eighth order methods several families of optimal eighth order methods to find simple roots are compared to the best known eighth order method due to wang and liu 2010 we have tried to improve their performance by choosing the free parameters of each family using two different criteria

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

composite coefficient of determination and its application in ultrahigh dimensional variable screening in this article we propose to measure the dependence between two random variables through a composite coefficient of determination ccd of a set of nonparametric regressions these regressions ta

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

non euclidian metrics and the robust stabilization of systems with parameter uncertainty this paper considers from a complex function theoretic point of view certain kinds of robust synthesis problems in particular we use a certain kind of metric on the disk the hyperbolic metric which allows us to reduce the problem of robust stabilization of systems with many types of real and complex parameter variations to an easily solvable problem in non euclidian geometry it is shown that several apparently different problems can be treated in a unified general framework a new result on the gain margin problem for multivariable plants is also given finally we apply our methods to systems with real zero or pole variations

independent screening for single index hazard rate models with ultrahigh dimensional features summary in data sets with many more features than observations independent screening based on all univariate regression models leads to a computationally convenient variable selection method recent efforts have shown that in the case of generalized linear models independent screening may suffice to capture all relevant features with high probability even in ultrahigh dimension it is unclear whether this formal sure screening property is attainable when the response is a right censored survival time we propose a computationally very efficient independent screening method for survival data which can be viewed as the natural survival equivalent of correlation screening we state conditions under which the method admits the sure screening property within a class of single index hazard rate models with ultrahigh dimensional features and describe the generally detrimental effect of censoring on performance an iterative variant of the method is also described which combines screening with penalized regression to handle more complex feature covariance structures the methodology is evaluated through simulation studies and through application to a real gene expression data set

model free feature screening for ultrahigh dimensional data with the recent explosion of scientific data of unprecedented size and complexity feature ranking and screening are playing an increasingly important role in many scientific studies in this article we propose a novel feature screening procedure under a unified model framework which covers a wide variety of commonly used parametric and semiparametric models the new method does not require imposing a specific model structure on regression functions and thus is particularly appealing to ultrahigh dimensional regressions where there are a huge number of candidate predictors but little information about the actual model forms we demonstrate that with the number of predictors growing at an exponential rate of the sample size the proposed procedure possesses consistency in ranking which is both useful in its own right and can lead to consistency in selection the new procedure is computationally efficient and simple and exhibits a competent empirical performance in our intensive simulations and real