approximating grammar probabilities solution of a conjecture it is proved that the production probabilities of a probabilistic context free grammar may be obtained as the limit of the estimates inferred from an increasing sequence of randomly drawn samples from the language generated by the grammar

a posteriori error estimates for the finite element solution of a singularly perturbed linear ordinary differential equation an error analysis is presented which establishes local error estimates for the finite element solution of a boundary value problem for a singularly perturbed ordinary differential equation herewith a posteriors error estimates are developed which suggest an adaptive refinement of the mesh the efficiency of this method is demonstrated by numerical examples

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

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

a posteriori error analysis for numerical approximations of friedrichs systems the global error of numerical approximations for symmetric positive systems in the sense of friedrichs is decomposed into a locally created part and a propagating component residual based two sided local a posteriori error bounds are derived for the locally created part of the global error these suggest taking the l 2 norm as well as weaker dual norms of the computable residual as local error indicators the dual graph norm of the residual vec r h is further bounded from above and below in terms of the l 2 norm of h vec r h where h is the local mesh size the theoretical results are illustrated by a series of numerical experiments

adams like techniques for zero finder methods two families of zero finding iterative methods for nonlinear equations are presented we derive them solving an initial value problem using adams like multistep techniques namely adams methods have been used to solve the problem that consists in a differential equation in what appears the inverse function of the one which zero will be computed and the condition given by the value attained by it at the initial approximation performing this procedure several methods of different local orders of convergence have been obtained

probabilistic hyperedge replacement grammars abstract in a probabilistic graph grammar each production has a probability attached to it this induces a probability assigned to each derivation tree and to each derived graph conditions for this probability function to be a probabilistic measure are discussed the statistical properties of the generated language are investigated we show how to compute the average size of an inductive function and the probability of an inductive graph predicate a relationship can be established between production probabilities and some statistical information of the generated language this way probabilities can be assigned to productions so that the generated graphs satisfy some statistical conditions

full linear multistep methods as root finders root finders based on full linear multistep methods lmms use previous function values derivatives and root estimates to iteratively find a root of a nonlinear function as ode solvers full lmms are typically not zero stable however used as root finders the interpolation points are convergent so that such stability issues are circumvented a general analysis is provided based on inverse polynomial interpolation which is used to prove a fundamental barrier on the convergence rate of any lmm based method we show using numerical examples that full lmm based methods perform excellently finally we also provide a robust implementation based on brents method that is guaranteed to converge

a weighted variant family of newton s method with accelerated third order convergence abstract an improvement to the iteration of newton s method that generalizes previous known improvements of it are presented a symbolic computation that allows us to find the best coefficients with regard to the local order of convergence is given furthermore a set of functions have been tested and all the methods considered present third theoretical and computational order of convergence

construction of newton like iteration methods for solving nonlinear equations in this paper we present a simple and yet powerful and easily applicable scheme in constructing the newton like iteration formulae for the computation of the solutions of nonlinear equations the new scheme is based on the homotopy analysis method applied to equations in general form equivalent to the nonlinear equations it provides a tool to develop new newton like iteration methods or to improve the existing iteration methods which contains the well known newton iteration formula in logic those all improve the newton method the orders of convergence and corresponding error equations of the obtained iteration formulae are derived analytically or with the help of maple some numerical tests are given to support the theory developed in this paper

a family of fifth order iterations composed of newton and third order methods abstract in this paper we present a family of fifth order methods formed by the composition of newton and third order methods for solving non linear equations per iteration the new methods require two evaluations of the function one of its first derivative and one of its second derivative analysis of efficiency shows that the new methods can compete with newton s method and the classical third order methods numerical results show that the methods have definite practical utility

a family of composite fourth order iterative methods for solving nonlinear equations in this paper we present a family of new fourth order iterative methods for solving nonlinear equations per iteration the methods consisting of the family require only two evaluations of the function and one evaluation of its derivative several numerical examples are given to illustrate the efficiency and performance of some of the presented methods

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

an axiomatic definition of context free rewriting and its application to nlc graph grammars abstract an abstract notion of context free grammar is introduced it deals with abstract objects that can be words trees graphs or other combinatorial objects it is applied to nlc graph grammars introduced by rozenberg and janssens the monadic second order theory of a context free nlc set of graphs is decidable

monadic second order evaluations on tree decomposable graphs abstract every graph generated by a hyperedge replacement graph grammar can be represented by a tree namely the derivation tree of the derivation sequence that produced it certain functions on graphs can be computed recursively on the derivation trees of these graphs by using monadic second order logic and semiring homomorphisms we describe in a single formalism a large class of such functions polynomial and even linear algorithms can be constructed for some of these functions we unify similar results obtained by takamizawa 1982 bern 1987 arnborg 1991 and habel 1989

an asymptotic finite element method for improvement of solutions of boundary layer problems a scheme that uses singular perturbation theory to improve the performance of existing finite element methods is presented the proposed scheme improves the error bounds of the standard galerkin finite element scheme by a factor of o en 1 where e is the small parameter andn is the order of the asymptotic approximation numerical results for linear second order o d e s are given and are compared with several other schemes

global uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems higher order elements abstract in this paper we develop a general higher order finite element method for solving singularly perturbed elliptic linear and quasilinear problems in two space dimensions we prove that a quasioptimal global uniform convergence rate of 0 n x m 1 in m 1 n x n v m 1 in m 1 n v in l 2 norm is obtained for a reaction diffusion model by using the m th order m 2 tensor product element thus answering some open problems posed by roos in h g roos layer adapted grids for singular perturbation problems z angew math mech 78 5 1998 291 309 and h g ross m stynes and l tobiska numerical methods for singularly perturbed differential equations springer verlag berlin 1996 278 here n x and n v are the number of partitions in the x and y directions respectively numerical results are provided supporting our theoretical analysis

a new iterative method for solving nonlinear equations abstract in this paper we present a new efficient iterative method for solving nonlinear equations improving newton raphson method this method is based on a modification of the proposal of abbasbandy on improving the order of accuracy of newton raphson method s abbasbandy improving newton raphson method for nonlinear equations by modified adomian decomposition method appl math comput 145 2003 887 893 the convergence properties of the method and some other methods are discussed some numerical illustrations are given to show that the proposed method behaves equal or better performance compared to the other methods of same kind

an improvement of the euler chebyshev iterative method abstract we present a new method for the computation of the solutions of nonlinear equations when it is necessary to use high precision we improve the euler chebyshev iterative method which is a generalization of an improvement of newton s method a symbolic computation allows us to find the best coefficients respect to the local order of convergence the adaptation of the strategy presented here gives an additional iteration function with an additional evaluation of the function it provides a lower cost if we use adaptive multi precision arithmetics the numerical results computed using this system with a floating point representing a maximum of 210 decimal digits support this theory

zero finder methods derived from obreshkov s techniques in this paper two families of zero finding iterative methods for solving nonlinear equations f x 0 are presented the key idea to derive them is to solve an initial value problem applying obreshkov like techniques more explicitly obreshkov s methods have been used to numerically solve an initial value problem that involves the inverse of the function f that defines the equation carrying out this procedure several methods with different orders of local convergence have been obtained an analysis of the efficiency of these methods is given finally we introduce the concept of extrapolated computational order of convergence with the aim of numerically test the given methods a procedure for the implementation of an iterative method with an adaptive multi precision arithmetic is also presented