third and fifth order newton gauss methods for solving nonlinear equations with n variables based on the mean value theorem of multivariable vectors function f x two new iterative schemes with third order and fifth order convergence are constructed respectively by using gauss quadrature formula for solving systems of nonlinear equations their error equations and asymptotic numerical convergence constants are obtained the two suggested methods are compared with the related methods for solving systems of nonlinear equations and boundary value problems of nonlinear odes in the numerical examples to demonstrate the efficiency and practicality

root finding by high order iterative methods based on quadratures we discuss a recursive family of iterative methods for the numerical approximation of roots of nonlinear functions in one variable these methods are based on newton cotes closed quadrature rules we prove that when a quadrature rule with n 1 nodes is used the resulting iterative method has convergence order at least n 2 starting with the case n 0 which corresponds to the newton s method

products of involutions in o v abstract let v be a finite dimensional vector space over a commutative field of characteristic distinct from 2 let v carry a symmetric nondegenerate bilinear form the special orthogonal group is o v o v det 1 main result an element o v is a product of two involutions in o v ifand only if dim v 2 mod 4 or an orthogonal decomposition of v into orthogonally indecomposable modules contains a module of odd dimension

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

accelerated convergence in newton s method newton s method is based on a linear approximation of the function whose roots are to be determined taken at the current point and the resulting algorithm is known to converge quadratically in a procedure to increase the rate of convergence the author modifies the target function in such a way that newton s method applied to the modified function will yield a faster rate of convergence

linear systems and normality this paper is concerned with responses of linear systems to non gaussian random excitation and with the measurement of the departure of the responses from gaussian behavior first we show the classical rosenblatt result and its nonapplicability to the most popular practical systems described by differential equations of first and second order then using a simple measure of departure from normality the asymmetry and excess coefficients and performing numerical calculations we give quantitative information about the effect of system parameters and the radius of correlation of the excitation process on the distance from normality

a note on the local convergence of iterative methods based on adomian decomposition method and 3 node quadrature rule abstract darvishi and barati m t darvishi a barati super cubic iterative methods to solve systems of nonlinear equations appl math comput 2006 10 1016 j amc 2006 11 022 derived a super cubic method from the adomian decomposition method to solve systems of nonlinear equations the authors showed that the method is third order convergent using classical taylor expansion but the numerical experiments conducted by them showed that the method exhibits super cubic convergence in the present work using ostrowski s technique based on point of attraction we show that their method is in fact fourth order convergent we also prove the local convergence of another fourth order method from 3 node quadrature rule using point of attraction

a versatile technique for the optimal approximation of random processes by functional quantization this paper presents the mathematical foundation of a novel and versatile technique for the approximation of random processes using the functional quantization concept in this approach random processes are represented by optimal quantizers that is finite collections of deterministic functions and corresponding probability masses that are carefully constructed to attain certain optimal properties although the computational cost of obtaining such optimal quantizers is not negligible their use in applications is simple and relatively inexpensive the paper has two objectives first we present an accessible overview of functional quantization theory second we introduce a novel methodology to compute optimal quantizers which is based on the classical lloyd s method and monte carlo simulation for validation purposes the proposed methodology is tested using two fundamental gaussian random processes brownian motion and fractional brownian motion for which optimal quantizers are known the results obtained compare very well with previous results reported in the literature then to demonstrate the unique versatility of the methodology the application to a non gaussian process is presented

stochastic reduced order models for inverse problems under uncertainty abstract this work presents a novel methodology for solving inverse problems under uncertainty using stochastic reduced order models sroms given statistical information about an observed state variable in a system unknown parameters are estimated probabilistically through the solution of a model constrained stochastic optimization problem the point of departure and crux of the proposed framework is the representation of a random quantity using a srom a low dimensional discrete approximation to a continuous random element that permits efficient and non intrusive stochastic computations characterizing the uncertainties with sroms transforms the stochastic optimization problem into a deterministic one the non intrusive nature of sroms facilitates efficient gradient computations for random vector unknowns and relies entirely on calls to existing deterministic solvers furthermore the method is naturally extended to handle multiple sources of uncertainty in cases where state variable data system parameters and boundary conditions are all considered random the new and widely applicable srom framework is formulated for a general stochastic optimization problem in terms of an abstract objective function and constraining model for demonstration purposes however we study its performance in the specific case of inverse identification of random material parameters in elastodynamics we demonstrate the ability to efficiently recover random shear moduli given material displacement statistics as input data we also show that the approach remains effective for the case where the loading in the problem is random as well

efficient high order methods based on golden ratio for nonlinear systems abstract we derive new iterative methods with order of convergence four or higher for solving nonlinear systems by composing iteratively golden ratio methods with a modified newton s method we use different efficiency indices in order to compare the new methods with other ones and present several numerical tests which confirm the theoretical results

on modified newton methods with cubic convergence abstract in this paper we present two new modified newton methods for solving a non linear equation permitting f x 0 in some points these new methods are showed to be cubically convergent from the practical application we show that these new methods are vast superior and possible in global convergence

third order and fourth order iterative methods for finding multiple and distinct zeros of non linear equations in this paper we point out and analyse six two step iterative methods for finding multiple as well as distinct zeros of non linear equations we prove that the methods for multiple zeros have third order convergence where as the methods for distinct zeros have fourth order convergence the methods calculate the multiple as well as distinct zeros with high accuracy the numerical tests show their better performance in case of algebraic as well as non algebraic equations

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

kepler equation and accelerated newton method we study the efficiency of the accelerated newton method garlach siam rev 36 1994 272 276 for several orders of convergence versus danby s method for the resolution of kepler s equation we find that the cited method of order three is competitive with danby s method and the classical newton s method we also generalize the accelerated newton method for the resolution of system of algebraic equations obtaining a formula of order three and a proof of its convergence its application to several examples shows that its efficiency is greater than newton s method

an iterative method for solving nonlinear equations an iterative method for finding a solution of the equation f x 0 is presented the method is based on some specially derived quadrature rules it is shown that the method can give better results than the newton method

reversible quaternionic hyperbolic isometries abstract let g be a group an element g g is called reversible if it is conjugate to g 1 within g and called strongly reversible if it is conjugate to g 1 by an order two element of g let h h n be the n dimensional quaternionic hyperbolic space let psp n 1 be the isometry group of h h n in this paper we classify reversible and strongly reversible elements in sp n and sp n 1 also we prove that all the elements of psp n 1 are strongly reversible

pseudocomposition a technique to design predictor corrector methods for systems of nonlinear equations abstract a new technique for designing iterative methods for solving nonlinear systems is presented this procedure called pseudocomposition uses a known method as a predictor and the gaussian quadrature as a corrector the order of convergence of the resulting scheme depends among other factors on the order of the last two steps of the predictor we also introduce a new iterative algorithm of order six and apply the mentioned technique to generate a new method of order ten finally some numerical test are shown

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

bireflectionality of the weak orthogonal and the weak symplectic groups let v be a finite dimensional vector space q a quadratic form and fa f the bilinear form associated with q we assume v is regular then the orthogonal group o v is bireflectional i e every isometry in o v is a product of two involutory isometries in o v this has been shown in 8 if the field of scalars k has characteristic distinct from 2 and in 4 and 5 if char k 2 the latter papers also establish the bireflectionality for the symplectic group sp v again under the assumption that v f is regular and char k 2 for char k 2 the symplectic group is not bireflectional see 31 we shall extend the results just mentioned we shall drop the assumptions that v is finite dimensional and that v f is regular i e the vector space v may be infinite dimensional and the radical of v may be distinct from zero we shall use the notation and the concepts in 2 for every 7c e hom v v we define f n ue v m u and b n vz u zj e v the spaces f r and b n are called fix and path of z respectively the groups o v n e o v rad vc f z and dim b n co and sp v 71 e sp v rad v c f n and dim b n co are called the weak orthogonal and the weak symplectic group respectively

a method for solving stochastic equations by reduced order models and local approximations a method is proposed for solving equations with random entries referred to as stochastic equations ses the method is based on two recent developments the first approximates the response surface giving the solution of a stochastic equation as a function of its random parameters by a finite set of hyperplanes tangent to it at expansion points selected by geometrical arguments the second approximates the vector of random parameters in the definition of a stochastic equation by a simple random vector referred to as stochastic reduced order model srom and uses it to construct a srom for the solution of this equation the proposed method is a direct extension of these two methods it uses sroms to select expansion points rather than selecting these points by geometrical considerations and represents the solution by linear and or higher order local approximations the implementation and the performance of the method are illustrated by numerical examples involving random eigenvalue problems and stochastic algebraic differential equations the method is conceptually simple non intrusive efficient relative to classical monte carlo simulation accurate and guaranteed to converge to the exact solution

third order methods from quadrature formulae for solving systems of nonlinear equations we extend to p dimensional problems a modification of the newton method based on quadrature formulas of order at least one which produces iterative methods with order of convergence three a general error analysis providing the higher order of convergence is given these new methods may be more efficient then other third order methods as they do not require the use of the second order frechet derivative

a uniparametric chebyshev type method free from second derivatives in this paper we present a family of new chebyshev type methods free from second derivatives for solving non linear equations analysis of convergence shows that these new methods are cubically convergent as particular cases we introduce two efficient methods in this family of the new methods the practical utility of which is demonstrated by numerical examples

a unified model for solving a system of nonlinear equations in this paper we suggest and analyze a unified model of two step iterative method for solving a system of nonlinear equations we prove that the unified model achieves fourth order convergence several numerical examples are given to illustrate the efficiency and the performance of the new iterative methods from the comparison with some known methods it is observed that the present methods are of practical utility

on products of two involutions in the orthogonal group of a vector space let v be a finite dimentional vector space over a commutative field of characteristic distinct from 2 let v carry a symmetric nondegenerate bilinear form results a let where o v and are involutions there exists an orthogonal decomposition of v into orthogonally indecomposable modules which are simultaneously invariant under and b let o v one can find involutions o v such that and b b b holds if and only if an orthogonal decomposition of v into orthogonally indecomposable modules does not contain a term whose minimum polynomial is x 1 where is even

third order convergence theorem by using majorizing function for a modified newton method in banach space abstract following the ideas of frontini and sormani we present a modified newton method in banach space which is used to solve the nonlinear operator equation we establish the newton kantorovich convergence theorem for the modified newton method with third order convergence in banach space by using majorizing function we also get the error estimate finally two examples are provided to show the application of our theorem

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

an efficient fifth order method for solving systems of nonlinear equations in this paper we present a three step iterative method of convergence order five for solving systems of nonlinear equations the methodology is based on the two step homeier s method with cubic convergence homeier 2004 computational efficiency in its general form is discussed and a comparison between the efficiency of proposed technique and existing ones is made the performance is tested through numerical examples moreover theoretical results concerning order of convergence and computational efficiency are verified in the examples it is shown that the present method has an edge over existing methods particularly when applied to large systems of equations

accelerated methods of order 2p for systems of nonlinear equations we present a new iterative method of order of convergence 5 for solving nonlinear systems by composing the midpoint method with newton s method and using an approximation for the jacobian matrix in order to reduce the required number of functional evaluations per iteration in addition we compare the efficiency index of these methods with that of newton s method and present several numerical tests which confirm the theoretical results and allow us to compare these variants with newton s method

reversible complex hyperbolic isometries let pu n 1 denote the isometry group of the n dimensional complex hyperbolic space hn an isometry g is called reversible if g is conjugate to g 1 in pu n 1 if g can be expressed as a product of two involutions it is called strongly reversible we classify reversible and strongly reversible elements in pu n 1 we also investigate reversibility and strong reversibility in su n 1

derivative free iterative methods for nonlinear systems in this work we introduce a new operator of divided differences that preserves the convergence order when it is used for approximating the jacobian matrix in iterative method for solving nonlinear systems we obtain derivative free iterative methods with lower computational cost than the corresponding ones with different operators of divided differences we also study the global convergence of these methods by analyzing their dynamical behavior

linear random vibration by stochastic reduced order models a practical method is developed for calculating statistics of the states of linear dynamic systems with deterministic properties subjected to non gaussian noise and systems with uncertain properties subjected to gaussian and non gaussian noise these classes of problems are relevant as most systems have uncertain properties physical noise is rarely gaussian and the classical theory of linear random vibration applies to deterministic systems and can only deliver the first two moments of a system state if the noise is non gaussian the method 1 is based on approximate representations of all or some of the random elements in the definition of linear random vibration problems by stochastic reduced order models sroms that is simple random elements having a finite number of outcomes of unequal probabilities 2 can be used to calculate statistics of a system state beyond its first two moments and 3 establishes bounds on the discrepancy between exact and srom based solutions of linear random vibration problems the implementation of the method has required to integrate existing and new numerical algorithms examples are presented to illustrate the application of the proposed method and assess its accuracy copyright 2009 john wiley sons ltd

on iterative methods for nonlinear equations several iterative methods have been suggested and analyzed for solving nonlinear equation f x 0 all these methods can be classified as one step and two step methods in this paper we have suggested and analyzed several two step methods for solving nonlinear equations we have also shown that these two step methods are quadratically convergent several numerical examples are given to illustrate this comparison our results can be viewed as important refinement and improvement of the previously known results

on a family of second derivative free variants of chebyshev s method in this paper we present a family of new second derivative free variants of chebyshev s method in which f x 0 in some points is permitted analysis of convergence shows that the new methods are cubically convergent analysis of efficiency in term of function evaluations shows that the new methods have the great practical utility which is demonstrated by numerical examples