local convergence of newton s method on the heisenberg group in the present paper we study newton s method on the heisenberg group for solving the equation f x 0 where f is a mapping from heisenberg group to its lie algebra under certain generalized lipschitz condition we obtain the convergence radius of newton s method and the estimation of the uniqueness ball of the zero point of f some applications to special cases including kantorovich s condition and condition are provided the determination of an approximate zero point of an analytic mapping is also presented concrete examples are given to illustrate applications of our results

majorizing functions and convergence of the gauss newton method for convex composite optimization we introduce a notion of quasi regularity for points with respect to the inclusion f x in c where f is a nonlinear frechet differentiable function from mathbb r v to mathbb r m when c is the set of minimum points of a convex real valued function h on mathbb r m and f satisfies the l average lipschitz condition of wang we use the majorizing function technique to establish the semilocal linear quadratic convergence of sequences generated by the gauss newton method with quasi regular initial points for the convex composite function h circ f results are new even when the initial point is regular and f is lipschitz

extending the applicability of gauss newton method for convex composite optimization on riemannian manifolds we present a semi local convergence analysis of the gauss newton method for solving convex composite optimization problems in riemannian manifolds using the notion of quasi regularity for an initial point using a combination the l average lipszhitz condition and the center l 0 average lipschitz condition we introduce majorizing sequences for the gauss newton method that are more precise than in earlier studies consequently our semi local convergence analysis for the gauss newton method has the following advantages under the same computational cost weaker sufficient convergence conditions more precise estimates on the distances involved and an at least as precise information on the location of the solution

convergence of newton s method and inverse function theorem in banach space under the hypothesis that the derivative satisfies some kind of weak lipschitz condition a proper condition which makes newton s method converge and an exact estimate for the radius of the ball of the inverse function theorem are given in a banach space also the relevant results on premises of kantorovich and smale types are improved in this paper

convergence analysis of the secant type methods in this paper under the hypothesis that derivative satisfies some kinds of weak lipschitz condition the radius estimates of the convergence balls of secant type methods for operator equations are established in banach space some applications and numerical experiments are also given

local convergence analysis of newton s method for solving strongly regular generalized equations abstract in this paper we consider newton s method for solving a generalized equation of the form f x f x 0 where f y is continuously differentiable x and y are banach spaces x is open and f x y has a nonempty closed graph we show that under strong regularity of the equation the method is locally convergent to a solution with superlinear quadratic rate our analysis which is based on general majorant condition enables us to obtain a convergence result under the lipschitz smale s and nesterov nemirovskii s self concordant conditions

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

on newton s method under holder continuous derivative a mysovskii type theorem for newton s method under k p holder continuous derivative is considered in this paper for the application studied by others the new condition is weaker than ones in the literature also we prove that the optimal convergent order is p 1 for 0 p 1

extended newton methods for multiobjective optimization majorizing function technique and convergence analysis we consider the extended newton method for approaching a pareto optimum of a multiobjective optimization problem establish quadratic convergence criteria and estimate a radius of convergence ball

newton method under weak lipschitz continuous derivative in banach spaces in the present paper a relax kantorovich type condition to guarantee the convergence of newton method with order 2 in banach space is obtained which takes the well known kantorovich and smale conditions as its special cases the relations between the new condition and the old ones are also studied

point estimation of simultaneous methods for solving polynomial equations a survey ii the construction of computationally verifiable initial conditions which provide both the guaranteed and fast convergence of the numerical root finding algorithm is one of the most important problems in solving nonlinear equations smale s point estimation theory from 1981 was a great advance in this topic it treats convergence conditions and the domain of convergence in solving an equation f z 0 using only the information of f at the initial point z 0 the study of a general problem of the construction of initial conditions of practical interest providing guaranteed convergence is very difficult even in the case of algebraic polynomials in the light of smale s point estimation theory an efficient approach based on some results concerning localization of polynomial zeros and convergent sequences is applied in this paper to iterative methods for the simultaneous determination of simple zeros of polynomials we state new improved initial conditions which provide the guaranteed convergence of frequently used simultaneous methods for solving algebraic equations ehrlich aberth s method ehrlich aberth s method with newton s correction borsch supan s method with weierstrass correction and halley like or wang zheng method the introduced concept offers not only a clear insight into the convergence analysis of sequences generated by the considered methods but also explicitly gives their order of convergence the stated initial conditions are of significant practical importance since they are computationally verifiable they depend only on the coefficients of a given polynomial its degree n and initial approximations to polynomial zeros

improved local convergence of newton s method under weak majorant condition we provide a local convergence analysis for newton s method under a weak majorant condition in a banach space setting our results provide under the same information a larger radius of convergence and tighter error estimates on the distances involved than before 14 special cases and numerical examples are also provided in this study

improving upon standard estimators in discrete exponential families with applications to poisson and negative binomial cases assume that xi xp are independent random observations having discrete exponential densities p1 0 ti x rj 1 i 1 p respectively a general technique of improving upon the uniform minimum variance unbiased estimator umvue of o1 op is developed under possibly weighted squared error loss functions it is shown that improved estimators can be constructed by solving a difference inequality typical difference inequalities of a fairly general type are presented and solved when specialized to poisson and negative binomial cases broad classes of estimators are given that dominate the umvue these results unify many known results in this rapidly diverging field and some of them are new especially those related to negative binomial distributions improved estimators are also obtained for the problems in which some of the observations are from poisson families and some from negative binomial families for sum of squared errors loss estimators which dominate the umvue in the discrete exponential families are also given explicitly

kantorovich s theorem on newton s method for solving generalized equations under the majorant condition in this paper we consider a version of the kantorovich s theorem for solving the generalized equation f x t x 0 where f is a frechet derivative function and t is a set valued and maximal monotone acting between hilbert spaces we show that this method is quadratically convergent to a solution of f x t x 0 we have used the idea of majorant function which relaxes the lipschitz continuity of the derivative f it allows us to obtain the optimal convergence radius uniqueness of solution and also to solving generalized equations under smale s condition

upper estimate of martingale dimension for self similar fractals we study upper estimates of the martingale dimension dm of diffusion processes associated with strong local dirichlet forms by applying a general strategy to self similar dirichlet forms on self similar fractals we prove that dm 1 for natural diffusions on post critically finite self similar sets and that dm is dominated by the spectral dimension for the brownian motion on sierpinski carpets

on independent statistical decision problems and products of diffusions letzt be a null recurrent diffusion on p with generatorg 1 r r for smooth positiver this note constructs an independent recurrent diffusionz ton 1 such that zt z t is transient in p 1 this resolves negatively an old question in simultaneous estimation is there an admissible but not bayes estimator x of the mean of a multivariate normal distribution for quadratic loss with the property for every admissible x wherex is normal and independent ofx remains admissible in the combined problem obtained by summing the component losses

convergence of gauss newton s method and uniqueness of the solution abstract in this paper we study the convergence of gauss newton s method for nonlinear least squares problems under the hypothesis that derivative satisfies some kinds of weak lipschitz condition we obtain the exact estimates of the radii of convergence ball of gauss newton s method and the uniqueness ball of the solution new results can be used to determinate approximation zero of gauss newton s method

two conditions concerning newton s method in this note we compare two conditions concerning newton s method and find out that even under some weaker condition one has the convergence of the newton s method and a better error bound than the relative strong condition

a complete class theorem for statistical problems with finite sample spaces this paper contains a complete class theorem theorem 3 2 which applies to most statistical estimation problems having a finite sample space this theorem also applies to many other statistical problems with finite sample spaces the description of this complete class involves a stepwise algorithm at each step of the process it is necessary to construct the bayes procedures in a suitably modified version of the original problem the complete class is a minimal complete class if the loss function is strictly convex some examples are given to illustrate the application of this complete class theorem among these is a new result concerning the estimation of the parameters of a multinomial distribution under a normalized quadratic loss function see example 4 5

existence of eigenforms on fractals with three vertices in this paper i prove that on every fractal structure with three vertices for suitable weights the renormalization operator has an eigenform

the guaranteed convergence of laguerre like method abstract the construction of initial conditions that provide a guaranteed convergence of zero finding methods has attracted a great deal of attention for many years in this paper we consider convergent properties of the laguerre like method of the fourth order for the simultaneous approximation of polynomial zeros using a procedure based on smale s point estimation theory and some recent results concerned with localization of complex polynomial zeros we state initial conditions which enable both the guaranteed and fast convergence of this method these conditions are computationally verifiable since they depend only on initial approximations polynomial coefficients and polynomial degree which is of practical importance

approximate zeros of quadratically convergent algorithms smale s condition for a point to be an approximate zero of a func tion for newton s method is extended to the general quadratically convergent iterative algorithm it is shown in which way the bound in the condition is affected by the characteristics of the algorithm this puts the original condition of smale for newton s method in a more general perspective the results are also discussed in the light of numerical evidence

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

convergence criterion of newton s method for singular systems with constant rank derivatives abstract the present paper is concerned with the convergence problem of newton s method to solve singular systems of equations with constant rank derivatives under the hypothesis that the derivatives satisfy a type of weak lipschitz condition a convergence criterion based on the information around the initial point is established for newton s method for singular systems of equations with constant rank derivatives applications to two special and important cases the classical lipschitz condition and the smale s assumption are provided the latter in particular extends and improves the corresponding result due to dedieu and kim in j p dedieu m kim newton s method for analytic systems of equations with constant rank derivatives j complexity 18 2002 187 209

convergence of the variants of the chebyshev halley iteration family under the holder condition of the first derivative the present paper is concerned with the convergence problem of the variants of the chebyshev halley iteration family with parameters for solving nonlinear operator equations in banach spaces under the assumption that the first derivative of the operator satisfies the holder condition of order p a convergence criterion of order 1 p for the iteration family is established an application to a nonlinear hammerstein integral equation of the second kind is provided

extended convergence results for the newton kantorovich iteration we present new semilocal and local convergence results for the newton kantorovich method these new results extend the applicability of the newton kantorovich method on approximate zeros by improving the convergence domain and ratio given in earlier studies by argyros 2003 cianciaruso 2007 smale 1986 and wang 1999 these advantages are also obtained under the same computational cost numerical examples where the old sufficient convergence criteria are not satisfied but the new convergence criteria are satisfied are also presented in this study

local convergence results of gauss newton s like method in weak conditions abstract in this paper we study the convergence of gauss newton s like method for nonlinear least squares problems under the hypothesis that derivative satisfies some kind of weak lipschitz condition we obtain the sharp estimates of the radii of convergence ball of gauss newton s like method and the uniqueness ball of the solution

approximate solutions for abstract inequality systems we consider conic inequality systems of the type f x ge k 0 with approximate solution x 0 associated to a parameter tau where f is a twice frechet differentiable function between hilbert spaces x and y and ge k is the partial order in y defined by a nonempty convex not necessarily closed cone k subseteq y we prove that under the suitable conditions the system f x ge k 0 is solvable and the ratio of the distance from x 0 to the solution set s over the distance from f x 0 to the cone k has an upper bound given explicitly in terms of tau and x 0 we show that the upper bound is sharp applications to analytic function inequality equality systems on euclidean spaces are given and the corresponding results of dedieu siam j optim 11 2000 pp 411 425 are extended and significantly improved

convergence of the family of the deformed euler halley iterations under the holder condition of the second derivative the convergence problem of the family of the deformed euler halley iterations with parameters for solving nonlinear operator equations in banach spaces is studied under the assumption that the second derivative of the operator satisfies the holder condition a convergence criterion of the order 2 p of the iteration family is established an application to a nonlinear hammerstein integral equation of the second kind is provided

newton s method for solving generalized equations kantorovich s and smale s approaches abstract in this paper we study newton type methods for solving generalized equations involving set valued maps in banach spaces kantorovich type theorems both local and global versions are proved as well as the quadratic convergence of the newton sequence we also extend smale s classical theory to generalized equations these results are new and can be considered as an extension of many known ones in the literature for classical nonlinear equations our approach is based on tools from variational analysis the metric regularity concept plays an important role in our analysis

proper analytic free maps this paper concerns analytic free maps these maps are free analogs of classical analytic functions in several complex variables and are defined in terms of non commuting variables amongst which there are no relations they are free variables analytic free maps include vector valued polynomials in free non commuting variables and form a canonical class of mappings from one non commutative domain d in say g variables to another non commutative domain d in g variables as a natural extension of the usual notion an analytic free map is proper if it maps the boundary of d into the boundary of d assuming that both domains contain 0 we show that if f d d is a proper analytic free map and f 0 0 then f is one to one moreover if also g g then f is invertible and f 1 is also an analytic free map these conclusions on the map f are the strongest possible without additional assumptions on the domains d and d

an improved convergence theorem for a class of secant like methods an improved semilocal convergence theorem for a class of secant like methods in banach spaces is provided for solving a nonlinear operator equation it is assumed that the first order divided difference of the nonlinear operator is holder continuous our convergence conditions strategically proposed are weaker than some existing ones and are easily satisfied finally two simple examples are provided to show that our results apply while earlier ones fail

a convergence theorem for the newton like methods under some kind of weak lipschitz conditions we provide sufficient conditions for the convergence of the newton like methods in the assumption that the derivative satisfies some kind of weak lipschitz conditions consequently some important convergence theorems follow from our main result in this paper

expanding the applicability of newton s method using smale s α theory we present a tighter convergence analysis than earlier studies such as in cianciaruso 2007 guo 2007 shen and li 2010 smale 1986 1987 wang and zhao 1995 wang 1999 wang and han 1990 of newton s method using smale s a theory by introducing the notion of the center c 0 condition in particular in the semilocal convergence case we show that if the center c 0 condition is smaller than the c condition then the new majorizing sequence is tighter than the old majorizing sequence the new convergence criteria are weaker than the older convergence criteria furthermore in the local convergence case we obtain a larger radius of convergence and tighter error estimates on the distances involved these improvements are obtained under the same computational cost numerical examples and applications are also provided in this study to show that the older results cannot apply but the new results apply to solve equations

a new fourth order family of simultaneous methods for finding polynomial zeros using a suitable fixed point relation with a complex parameter a new one parameter family of simultaneous methods of the fourth order for finding complex zeros of a polynomial is derived in ordinary complex arithmetic convergence analysis of the presented family is performed under computationally verifiable initial conditions which depend only on polynomial coefficients and initial approximations further improvements of the proposed family of methods are discussed and a modification for finding multiple zeros is presented some numerical results for various values of the parameter are given

the convergence of a family of parallelzero finding methods abstract one of the crucial problems in solving polynomial equations is to state some kind of quantitative conditions that provide both safe and fast convergence in the spirit of smale s point estimation theory introduced in 1981 we construct in this paper computationally verifiable initial conditions which guarantee the convergence of the recently proposed one parameter family of methods of the fourth order for the simultaneous finding polynomial zeros these conditions are computationally verifiable since they depend only on initial approximations polynomial coefficients and polynomial degree which is of significant practical importance

extended newton methods for conic inequalities approximate solutions and the extended smale α theory abstract using the convex process theory and the majorizing technique we study the convergence issues of the iterative sequences generated by the extended newton method for solving the conic inequality system f c 0 defined by a cone c and a frechet differentiable function f satisfying the extended weak condition convergence criterion of the extended newton method is presented as an application we establish when f is analytic an extended theory similar to smale s theory for nonlinear analytic equations

denjoy wolff theorems hilbert metric nonexpansive maps and reproduction decimation operators let k be a closed cone with nonempty interior in a banach space x suppose that f intk intk is order preserving and homogeneous of degree one let q k 0 be a continuous homogeneous of degree one map such that q x 0 for all x k 0 let t x f x q f x we give conditions on the cone k and the map f which imply that there is a convex subset of k which contains the omega limit set x t for every x intk we show that these conditions are satisfied by reproduction decimation operators we also prove that x t k for a class of operator valued means

energy measures and indices of dirichlet forms with applications to derivatives on some fractals we introduce the concept of index for regular dirichlet forms by means of energy measures and discuss its properties in particular it is proved that the index of strong local regular dirichlet forms is identical with the martingale dimension of the associated diffusion processes as an application a class of self similar fractals is taken up as an underlying space we prove that first order derivatives can be defined for functions in the domain of the dirichlet forms and their total energies are represented as the square integrals of the derivatives

convergence ball of a modified secant method with convergence order 1 839 abstract a local convergence theorem on a modified secant method with convergence order 1 839 for solving nonlinear equations is established under the hypotheses that the second order and third order derivative of the function involved are bounded an estimate of the radius of the convergence ball of this method is obtained moreover an error estimate is provided which matches the convergence order of the method