a modified newton method for rootfinding with cubic convergence we consider a modification of the newton method for finding a zero of a univariate function the case of multiple roots is not treated it is proven that the modification converges cubically per iteration it requires one evaluation of the function and two evaluations of its derivative thus the modification is suitable if the calculation of the derivative has a similar or lower cost than that of the function itself classes of such functions are sketched and a numerical example is given

the monotone iterative technique for a parabolic boundary value problem with discontinuous nonlinearity let be a bounded lebesgue measurable domain of r n with a lipschitz boundary the following initial boundary value problem is considered u t lu f u h in q bu o on u 0 where q o t o t and 0 0 l is a uniformly elliptic operator b a boundary operator defined by bu u v bu in which v denotes the outer normal derivative on and f represents a discontinuous non linearity

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

choosing the most stable members of kou s family of iterative methods abstract in this manuscript we analyze the dynamical anomalies of a parametric family of iterative schemes designed by kou et al it is known that its order of convergence is three for any arbitrary value of the parameter but it has order four and it is optimal in the sense of kung traub s conjecture when a specific value is selected among all the elements of this family one can choose this fourth order element or any of the infinite members of third order of convergence if only the speed of convergence is considered however the stability of the methods plays an important role in their reliability when they are applied on different problems this is the reason why we analyze in this paper the dynamical behavior on quadratic polynomials of the mentioned family the study of fixed points and their stability joint with the critical points and their associated parameter planes show the richness of the class and allow us to find members of it with excellent numerical properties as well as other ones with very unstable behavior some test functions are analyzed for confirming the theoretical results

third order iterative methods free from second derivatives for nonlinear equation in this paper we suggest and analyze a new two step iterative method for solving nonlinear equations which is called the modified householder method without second derivatives for nonlinear equation we also prove that the modified method has cubic convergence several examples are given to illustrate the efficiency and the performance of the new method new method can be considered as an alternative to the present cubic convergent methods for solving nonlinear equations

korevaar schoen s directional energy and ambrosio s regular lagrangian flows we develop korevaar schoen s theory of directional energies for metric valued sobolev maps in the case of textsf rcd source spaces to do so we crucially rely on ambrosio s concept of regular lagrangian flow our review of korevaar schoen s spaces brings new even in the smooth category insights on some aspects of the theory in particular concerning the notion of differential of a map along a vector field and about the parallelogram identity for textsf cat 0 targets to achieve these one of the ingredients we use is a new even in the euclidean setting stability result for regular lagrangian flows

the improvements of chebyshev halley methods with fifth order convergence abstract in this paper we present a family of new fifth order methods which improves the classical chebyshev halley methods with cubic convergence the new methods add one evaluation of the function at the point iterated by chebyshev halley methods analysis of efficiency shows that the new methods can be of practical interest which is also corroborated by numerical tests

differential of metric valued sobolev maps abstract we introduce a notion of differential of a sobolev map between metric spaces the differential is given in the framework of tangent and cotangent modules of metric measure spaces developed by the first author we prove that our notion is consistent with kirchheim s metric differential when the source is a euclidean space and with the abstract differential provided by the first author when the target is r we also show compatibility with the concept of co local weak differential introduced by convent and van schaftingen

modified newton s method with third order convergence and multiple roots in recent papers appl math comput 140 2003 419 426 appl math lett 13 2000 87 93 a new modification of the newton s method mnm which produces iterative methods with order of convergence three have been proposed here we study the order of convergence of such methods when we have multiple roots we prove that the order of convergence of the mnm go down to one but when the multiplicity p is known it may be rised up to two by using two different types of correction when p is unknown we show that the two most efficient methods in the family of the mnm converge faster than the classical newton s method

on newton type methods with cubic convergence recently there has been some progress on newton type methods with cubic convergence that do not require the computation of second derivatives weerakoon and fernando appl math lett 13 2000 87 derived the newton method and a cubically convergent variant by rectangular and trapezoidal approximations to newton s theorem while frontini and sormani j comput appl math 156 2003 345 140 2003 419 derived further cubically convergent variants by using different approximations to newton s theorem homeier j comput appl math 157 2003 227 169 2004 161 independently derived one of the latter variants and extended it to the multivariate case here we show that one can modify the werrakoon fernando approach by using newton s theorem for the inverse function and derive a new class of cubically convergent newton type methods

means based modifications of newton s method for solving nonlinear equations in this paper we consider a family of six sets of means based modifications of newton s method for solving nonlinear equations each set is a parametric class of methods some well known methods belong to our family for example the arithmetic mean newton s method s weerakoon t g i fernando a variant of newton s method with accelerated third order convergence appl math lett 13 2000 87 93 the harmonic mean newton s method a y ozban some new variants of newton s method appl math lett 17 2004 677 682 the geometric mean newton s method t lukic n m ralevic geometric mean newton s method for simple and multiple roots appl math lett 21 2008 30 36 the power mean newton s method x zhou a class of newton s methods with third order convergence appl math lett 20 2007 1026 1030 the convergence analysis shows third order of our family comparison of the family members shows that there are no big differences between them twelve numerical examples were tested and two characteristic ones are presented

a uniqueness result for the decomposition of vector fields in mathbb r d rd given a vector field rho 1 mathbf b in l 1 mathrm loc mathbb r times mathbb r d mathbb r d 1 such that mathrm div t x rho 1 mathbf b is a measure we consider the problem of uniqueness of the representation eta of rho 1 mathbf b mathcal l d 1 as a superposition of characteristics gamma t gamma t gamma rightarrow mathbb r d dot gamma t mathbf b t gamma t we give conditions in terms of a local structure of the representation eta on suitable sets in order to prove that there is a partition of mathbb r d 1 into disjoint trajectories wp mathfrak a mathfrak a in mathfrak a such that the pde begin aligned mathrm div t x big u rho 1 mathbf b big in mathcal m mathbb r d 1 quad u in l infty mathbb r times mathbb r d end aligned can be disintegrated into a family of odes along wp mathfrak a with measure r h s the decomposition wp mathfrak a is essentially unique we finally show that mathbf b in l 1 t mathrm bv x mathrm loc satisfies this local structural assumption and this yields in particular the renormalization property for nearly incompressible mathrm bv vector fields

existence and uniqueness of maximal regular flows for non smooth vector fields in this paper we provide a complete analogy between the cauchy lipschitz and the diperna lions theories for ode s by developing a local version of the diperna lions theory more precisely we prove the existence and uniqueness of a maximal regular flow for the diperna lions theory using only local regularity and summability assumptions on the vector field in analogy with the classical theory which uses only local regularity assumptions we also study the behaviour of the ode trajectories before the maximal existence time unlike the cauchy lipschitz theory this behaviour crucially depends on the nature of the bounds imposed on the spatial divergence of the vector field in particular a global assumption on the divergence is needed to obtain a proper blow up of the trajectories

some variant of newton s method with third order convergence we present a new modification of the newton s method which produces iterative methods with order of convergence three a general error analysis providing the higher order of convergence is given and the best efficiency in term of function evaluations of two of this new methods is provided

calculus and heat flow in metric measure spaces and applications to spaces with ricci bounds from below this paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces x mathsf d mathfrak m our main results are a general study of the relations between the hopf lax semigroup and hamilton jacobi equation in metric spaces x d the equivalence of the heat flow in l 2 x mathfrak m generated by a suitable dirichlet energy and the wasserstein gradient flow of the relative entropy functional mathrm ent mathfrak m in the space of probability measures open image in new window the proof of density in energy of lipschitz functions in the sobolev space w 1 2 x mathsf d mathfrak m a fine and very general analysis of the differentiability properties of a large class of kantorovich potentials in connection with the optimal transport problem is the fourth achievement of the paper our results apply in particular to spaces satisfying ricci curvature bounds in the sense of lott and villani ann math 169 903 991 2009 and sturm acta math 196 65 131 2006 and acta math 196 133 177 2006 and require neither the doubling property nor the validity of the local poincare inequality

metric measure spaces with riemannian ricci curvature bounded from below in this paper we introduce a synthetic notion of riemannian ricci bounds from below for metric measure spaces x d m which is stable under measured gromov hausdorff convergence and rules out finsler geometries it can be given in terms of an enforcement of the lott sturm and villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow besides stability it enjoys the same tensorization global to local and local to global properties in these spaces which we call rcd k spaces we prove that the heat flow which can be equivalently characterized either as the flow associated to the dirichlet form or as the wasserstein gradient flow of the entropy satisfies wasserstein contraction estimates and several regularity properties in particular bakry emery estimates and the l lip feller regularization we also prove that the distance induced by the dirichlet form coincides with d that the local energy measure has density given by the square of cheeger s relaxed slope and as a consequence that the underlying brownian motion has continuous paths all these results are obtained independently of poincare and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact as the infinite dimensional ones

iterative methods with fourth order convergence for nonlinear equations in this paper we suggest and analyze a new three step iterative method for solving nonlinear equations involving only first derive of the function using a new decomposition technique which is due to noor m aslam noor numerical analysis and optimization lecture notes comsats institute of information technology islamabad pakistan 2006 m aslam noor new iterative schemes for nonlinear equations appl math comput in press and noor and noor m aslam noor k inayat noor some iterative schemes for nonlinear equations appl math comput in press we show that this new iterative method has fourth order convergence several numerical examples are given to illustrate the efficiency and performance of the new method

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

on starshaped rearrangement and applications a radial symmetrization technique is investigated and new properties are proven the method transforms functions u into new functions u with starshaped level sets and is therefore called starshaped rearrangement this rearrangement is in general not equimeasurable a circumstance with some surprising consequences the method is then applied to certain variational and free boundary problems and yields new results on the geometrical properties of solutions to these problems in particular the lipschitz continuity of free boundaries can now be easily obtained in a new fashion introduction in 26 g szego introduced the concept of radial symmetrization this is a nonequimeasurable rearrangement which transforms functions u q r q c r2 with starshaped level sets qc x e flu x c into functions u with symmetric starshaped level sets later 2 3 15 16 22 his approach was generalized to transform functions with level sets containing the origin into starshaped and not necessarily symmetric level sets and to higher dimensions we call a set d c rn starshaped with respect to xo if x e d implies tx 1 t xo e d for every t e 0 1 we call a set d starshaped if it is starshaped with respect to the origin therefore a more appropriate name for this rearrangement might be starshaped rearrangement in particular since the notion of radial symmetrization can be easily confused with circular or spherical symmetrization or spherically symmetric rearrangement one of the standard results in the theory of rearrangements is that the capacity of a condenser decreases under rearrangement i e j vu x lpdx 1 vu x ipdx for p 1 where u 1 on ql u 0 on 9q and u o c q c q e 0 0 j ivu p x ipdx for p 1 provided q c rn is a bounded starshaped domain containing ue 0 and u 0 1 u e c0 1 q u 1 in u 0 and u 0 on iq another convenient feature of other rearrangements is their equimeasurability a rearrangement u u is called equimeasurable if the n dimensional lebesgue received by the editors march 26 1984 and in revised form september 24 1985 1980 mathematics subject classification primary 35j20 35r35 35b05 26d20 lthis research was supported by the deutsche forschungsgemeinschaft dfg and by e reid 1986 american mathematical society 0002 9947 86 1 00 25 per page

lipschitz continuity of harmonic maps between alexandrov spaces in 1997 jost calc var pde 5 1 19 1997 and lin collection of papers on geometry analysis and mathematical physics world sci publ river edge 1997 independently proved that every energy minimizing harmonic map from an alexandrov space with curvature bounded from below to an alexandrov space with non positive curvature is locally holder continuous lin 1997 proposed an open problem can the holder continuity be improved to lipschitz continuity j jost also asked a similar problem about lipschitz regularity of harmonic maps between singular spaces see page 38 in jost in jost kendall mosco rockner sturm eds new directions in dirichlet forms international press boston 1998 the main theorem of this paper gives a complete resolution to it

on a characterization of some newton like methods of r order at least three in this paper we extend to banach spaces the result given by gander in amer math monthly 92 1985 131 to obtain a characterization of newton like iterative process with r order of convergence at least three to do this we consider the construction and the study of the semilocal convergence of a multiparametric family of iterative processes in banach spaces for solving the nonlinear equation f x 0

a steffensen like method and its higher order variants for solving nonlinear equations we suggest a second order parametric steffensen like method which is derivative free and only uses two evaluations of the function in one step we also suggest a variant of the steffensen like method which is still derivative free and uses four evaluations of the function to achieve cubic convergence moreover a fast steffensen like method with super quadratic convergence and a fast variant of the steffensen like method with super cubic convergence are proposed by using a parameter estimation the error equations and asymptotic convergence constants are obtained for the discussed methods the numerical results and the basins of attraction support the proposed methods

a simply constructed third order modifications of newton s method in this paper we present a simple and easily applicable approach to construct some third order modifications of newton s method for solving nonlinear equations it is shown by way of illustration that existing third order methods can be employed to construct new third order iterative methods the proposed approach is applied to the classical chebyshev halley methods to derive their second derivative free variants numerical examples are given to support that the methods thus obtained can compete with known third order methods

the cubic semilocal convergence on two variants of newton s method in this paper we discuss two variants of newton s method without using any second derivative for solving nonlinear equations by using the majorant function and confirming the majorant sequences we obtain the cubic semilocal convergence and the error estimation in the kantorovich type theorems the numerical examples are presented to support the usefulness and significance

on an elliptic operator with discontinuous nonlinearity abstract a family of elliptic operators with discontinuous nonlinearity affected by the presence of a positive parameter is considered a critical value c of and the zero set ec of the corresponding critical solution u c play a special role in the treatment of these operators sharp estimates for c as a functional of boundary functions are given and the structure of ec is investigated an asymptotic formula for the solution u and its free boundary is presented as

a geometric construction of iterative functions of order three to solve nonlinear equations in this paper we consider a geometric construction of iteration functions of order three to develop cubically convergent iterative methods for solving nonlinear equations this construction can be applied to any iteration function of order two to develop an iteration function of order three some examples are given of deriving several third order iteration methods and several numerical results follow to illustrate the performance of the derived methods

a modified newton method with cubic convergence the multivariate case recently a modification of the newton method for finding a zero of a univariate function with local cubic convergence has been introduced here we extend this modification to the multi dimensional case i e we introduce a modified newton method for vector functions that converges locally cubically without the need to compute higher derivatives the case of multiple roots is not treated per iteration the method requires one evaluation of the function vector and solving two linear systems with the jacobian as coefficient matrix where the jacobian has to be evaluated twice since the additional computational effort is nearly that of an additional newton step the proposed method is useful especially in difficult cases where the number of iterations can be reduced by a factor of two in comparison to the newton method this much better convergence is indeed possible as shown by a numerical example also the modified newton method can be advantageous in cases where the evaluation of the function is more expensive than solving a linear system with the jacobian as coefficient matrix an example for this is given where numerical quadrature is involved finally we discuss shortly possible extensions of the method to make it globally convergent