on the method of tangent hyperbolas on banach spaces abstract as an application of the technique employed by the author in a series of papers 15 18 some results are established concerning convergence of the method of tangent hyperbolas for solving nonlinear equations in banach spaces as well as existence and uniqueness of solution the results are compared with those obtained by doring 5

new bounds for the first inequality of ostrowski gruss type and applications abstract some new bounds for the first inequality of ostrowski gruss type are derived these new bounds can be much better than some recently obtained bounds applications in numerical integration are also given

the convergence of the perturbed newton method and its application for ill conditioned problems abstract iterative methods such as newton s behave poorly when solving ill conditioned problems they become slow first order and decrease their accuracy in this paper we analyze deeply and widely the convergence of a modified newton method which we call perturbed newton in order to overcome the usual disadvantages newton s one presents the basic point of this method is the dependence of a parameter affording a degree of freedom that introduces regularization choices for that parameter are proposed the theoretical analysis will be illustrated through examples

equivalence test and confidence interval for the difference in proportions for the paired sample design summary this paper considers a model for the di erence of two proportions in a paired or matched design of clinical trials case control studies and also sensitivity comparison studies of two laboratory tests this model includes a parameter indicating both interpatient variability of response probabilities and their correlation under the proposed model we derive a one sided test for equivalence based upon the e cient score equivalence is defined here as not more than 100 per cent inferior mcnemar s test for significance is shown to be a special case of the proposed test further a score based confidence interval for the di erence of two proportions is derived one of the features of these methods is applicability to the 2 2 table with o diagonal zero cells all the mcnemar type tests and confidence intervals published so far cannot apply to such data a monte carlo simulation study shows that the proposed test has empirical significance levels closer to the nominal a level than the other tests recently proposed and further that the proposed confidence interval has better empirical coverage probability than those of the four published methods 1998 john wiley sons ltd

avoiding the computation of the second fre chet derivative in the convex acceleration of newton s method we introduce a new two step method to approximate a solution of a nonlinear operator equation in a banach space an existence uniqueness theorem and error estimates are provided for this iteration using newton kantorovich type assumptions and a technique based on a new system of recurrence relations for a special choice of the parameter involved we use our method is of fourth order

a kantorovich type convergence analysis for the gauss newton method we present a semilocal kantorovich type convergence analysis for the gauss newton method which reduces to the wellknown newton kantorovich theorem for the newton method in a natural way additionnally a classification of the nonlinear regression problem into adequate and not adequate models is obtained

a note on halley s method we introduce the degree of logarithmic convexity which provides a measure of the convexity of a function at each point making use of this concept we obtain a new theorem of global convergence for halley s method

analysis of clustered matched pair data for a non inferiority study design hypothesis testing of matched pair data has been adapted for equivalence one and two sided study designs however a statistical problem arises when more than one measurement is recorded for each study subject ignoring the correlation between the repeated measurements per subject may underestimate the standard error of the parameter estimate a method of testing for non inferiority or equivalence of clustered matched pair data is not yet available this paper proposes a wald type test statistic for a non inferiority hypothesis and applies it to the comparison of two diagnostic procedures

disturbance attenuation and h sub infinity control via measurement feedback in nonlinear systems a solution to the problem of disturbance attenuation via measurement feedback with internal stability is presented for an affine nonlinear system it is shown that the concept of disturbance attenuation in the sense of truncated l sub 2 norms can be given an interpretation in terms of the response to periodic inputs in the sense of rms amplitude even in the nonlinear setup in the case of state feedback a family of controllers is also provided the proofs of all these results are simple and provide deeper insight even in the analysis of the corresponding linear control problem

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

historical developments in convergence analysis for newton s and newton like methods historical developments in convergence theory as well as error estimates for newton s method and newton like methods for nonlinear equations are described mainly for differentiable equations in banach spaces

on the semilocal convergence of newton kantorovich method under center lipschitz conditions in this work we study newton s method for solving nonlinear equations with operators defined between two banach spaces together with the classical kantorovich theory we consider a center lipschitz condition for the frechet derivative of the involved operator this fact allow us to obtain a majorizing sequence for the sequence defined in banach spaces and to give conditions for the convergence in this way we obtain a generalization of kantorovich s theorem that improves the values of the universal constant that appears in it as well as the radius where the solution is located and where it is unique finally we illustrate the main theoretical result by means of some examples

comments on non inferiority tests for clustered matched pair data by j nam and d kwon statistics in medicine 2009 28 1668 1679

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

on the comparison of correlated proportions for clustered data mcnemar s test is often used to compare two proportions estimated from paired observations we propose a method extending this to the case where the observations are sampled in clusters the proposed method is simple to implement and makes no assumptions about the correlation structure we conducted a monte carlo simulation study to compare the size and power of the proposed method with a test developed earlier by eliasziw and donner in the presence of intracluster correlation the size of mcnemar s test can greatly exceed the nominal level the size of eliasziw and donner s test is also inflated for some correlation patterns the proposed method on the other hand is close to the nominal size for a variety of correlation patterns although it is slightly less powerful than eliasziw and donner s procedure the proposed method is a good alternative to eliasziw and donner s test when in practice little is known about the correlation pattern of the data

least change secant update methods for undetermined systems least change secant updates for nonsquare matrices have been addressed recently in 6 here the use of these updates in iterative procedures for the numerical solution of underde retrained systems is considered the model method is the normal flow algorithm used in homotopy or continuation methods for determining points on an implicitly defined curve a kantorovich type local convergence analysis is given which supports the use of least change secant updates in this al gorithm this analysis also provides a kantorovich type local convergence analysis for least change secant update methods in the usual case of an equal number of equations and unknowns this in turn gives a local convergence analysis for augmented jacobian algorithms which use least change secant updates in conclusion the results of some numerical experiments are given

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

instantaneous impulse response of nonlinear system mathematical models for nonlinear systems which admit both standard integrable and impulsive inputs are developed it is shown that instantaneous response of a nonlinear system to an impulse is not in general single valued whereas a particular response can be selected through the construction of an auxiliary system the complete integrability of the auxiliary system is proven to guarantee uniqueness of the impulse response

on diagonally structured problems in unconstrained optimization using an inexact super halley method we consider solving the unconstrained minimization problem using an iterative method derived from the third order super halley method each iteration of the super halley method requires the solution of two linear systems of equations we show a practical implementation using an iterative method to solve the linear systems this paper introduces an array of arrays jagged data structure for storing the second and third derivative of a multivariate function and suitable termination criteria for the inner iterative method to achieve a cubic rate of convergence using a jagged compressed diagonal storage of the hessian matrices and for the tensor numerical results show that storing the diagonals are more efficient than the row or column oriented approach when we use an iterative method for solving the linear systems of equations

notes on a paper considering nonlinear equations in abstract of the paper a rafiq a note on a family of methods for solving nonlinear equations appl math comput 195 2008 819 821 we can find the following sentences we cite ujevic et al introduced a family of methods for solving nonlinear equations however the main algorithm 1 put forward by ujevic et al p 7 is wrong this is the main aim of this note we also point out some major bugs in the results of ujevic et al the end of the citation here it is shown that all of the mentioned assertions are not true in other words the algorithm 1 is correct up to an obvious misprint which is not mentioned in the above paper and there are no major bugs in the paper by ujevic et al in fact these observations which will be given in this note show that the main aim of the paper by rafiq is wrong

solution of underdetermined nonlinear equations by stationary iteration methods nonlinear stationary fixed point iterations inr n are considered the perron ostrowski theorem 23 guarantees convergence if the iteration functiong possesses an isolated fixed pointu in this paper a sufficient condition for convergence is given ifg possesses a manifold of fixed points as an application convergence of a nonlinear extension of the method of kaczmarz is proved this method is applicable to underdetermined equations it is appropriate for the numerical treatment of large and possibly ill conditioned problems with a sparse nonsquare jacobian matrix a practical example of this type nonlinear image reconstruction in ultrasound tomography is included

non inferiority test and confidence interval for the difference in correlated proportions in diagnostic procedures based on multiple raters the efficacy of diagnostic procedures is generally evaluated on the basis of the results from multiple raters however there are few adequate methods of performing non inferiority tests with confidence intervals to compare the accuracies sensitivities or specificities when multiple raters are considered we propose new statistical methods for comparing the accuracies of two diagnostic procedures in a non inferiority trial on the basis of the results from multiple independent raters who are also independent of the study centers we consider a study design in which each patient is subjected to two diagnostic procedures and all images are read by all raters by assuming a multinomial distribution for matched pair categorical data arising from the study design we derive a score based full menu that is a non inferiority test confidence interval and sample size formula for inference of the difference in correlated proportions between the two diagnostic procedures we conduct monte carlo simulation studies to examine the validity of the proposed methods which showed that the proposed test has a size closer to the nominal significance level than a wald type test and that the proposed confidence interval has better empirical coverage probability than a wald type confidence interval we illustrate the proposed methods with data from a study of diagnostic procedures for the diagnosis of oesophageal carcinoma infiltrating the tracheobronchial tree copyright 2011 john wiley sons ltd

nonlinear stochastic programming with a case study in continuous switching the optimal solution as well as the objective of stochastic programming problems vary with the underlying probability measure this paper addresses stability with respect to the underlying probability measure and stability of the objective the techniques presented are employed to make problems numerically tractable which are formulated by involving numerous scenarios or even by involving a continuous probability measure the results justify clustering techniques which significantly reduce computation times while guaranteeing a desired approximation quality the second part of the paper highlights newton s method to solve the reduced stochastic recourse problems the techniques presented exploit the particular structure of the recourse function of the stochastic optimization problem the tools are finally demonstrated on a benchmark problem which is taken from electrical power flows

newton like methods for solving underdetermined nonlinear equations with nondifferentiable terms abstract in this paper we consider newton like methods for solving underdetermined systems of nonlinear equations with nondifferentiable terms after presenting local convergence analysis for the methods we prove a semilocal convergence theorem as well as uniqueness of solution in a generalized sense another semilocal convergence theorem for the newton chord method is also established finally a numerical example is given

a convergence theorem for newton like methods in banach spaces on the basis of the results obtained in a series of papers 25 28 a convergence theorem for newton s method in banach spaces is given which improves the theorems of kantorovich 4 lancaster 8 and ostrowski 10 the error bounds obtained improve the recent results of potra 17

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

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

a unified derivation of several error bounds for newton s process abstract this paper gives a unified derivation of the upper and lower bounds of the errors in newton s process obtained by dennis 4 tapia 27 gragg tapia 6 and recently by potra ptak 20 and miel 15 with the use of different techniques our argument is based upon the kantorovich theorem and the kantorovich recurrence relations from the proof it is concluded that the recent upper and lower bounds of miel are finer than those of potra ptak but do not improve the basic error bounds which are directly obtained from the kantorovich theorem further it is shown that the error bounds obtained with the use of the recurrence relations are the same as those obtained by the majorant principle finally results on the newton method in partially ordered space are surveyed a method is also described for estimating the componentwise errors for computed solutions

newton homotopy analysis method for nonlinear equations abstract in this paper we present an efficient numerical algorithm for solving nonlinear algebraic equations based on newton raphson method and homotopy analysis method also we compare homotopy analysis method with adomian s decomposition method and homotopy perturbation method some numerical illustrations are given to show the efficiency of algorithm

a note on the halley method in banach spaces abstract we provide a convergence theorem under the usual ostrowski kantorovich conditions for an equivalent form of the halley method in banach spaces the best possible upper error bound for the method is obtained some applications to the solution of quadratic operator equations are provided

on robust h sub spl infin control for nonlinear discrete and sampled data systems the linearization technique for h sub spl infin control for nonlinear discrete and sampled data systems is developed in a unified framework under minimal assumptions it is shown directly without using any riccati or hamilton jacobi isaacs equation that h sub spl infin control for a linearized system is also h sub spl infin control for a nonlinear original one

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

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

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

convergence analysis of some algorithms for solving nonsmooth equations this paper presents convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally lipschitzian functions for the directional derivative based and the generalized jacobian based newton methods both the iterates and the corresponding function values are locally superlinearly convergent globally a limiting point of the iterate sequence generated by the damped directional derivative based newton method is a zero of the system if and only if the iterate sequence converges to this point and the stepsize eventually becomes one provided that the system is strongly bd regular and semismooth at this point in this case the convergence is superlinear a general attraction theorem is presented which can be applied to two algorithms proposed by han pang and rangaraj a hybrid method which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent is also presented

improvement of some ostrowski gruss type inequalities abstract in this paper we improve and further generalize some ostrowski gruss type inequalities involving bounded once and twice differentiable mappings

a family of methods for solving nonlinear equations we present a family of methods for solving nonlinear equations some well known classical methods and their modifications belong to our family for example newton potra ptak chebyshev halley and ostrowski s methods convergence analysis shows that our family contains methods of convergence order from 2 to 4 all our fourth order methods are optimal in terms of the kung and traub conjecture several examples are presented and compared

error bounds for newton s iterates derived from the kantorovich theorem in this paper it is shown that the upper and lower bounds of the errors in the newton iterates recently obtained by potra ptak 11 and miel 7 with the use of nondiscrete induction and majorizing sequence respectively follow immediately from the kantorovich theorem and the kantorovich recurrence relations it is also shown that the upper and lower bounds of miel are finer than those of potra ptak

a globally convergent successive approximation method for severely nonsmooth equations this paper presents a globally convergent successive approximation method for solving f x 0 where f is a continuous function at each step of the method f is approximated by a smooth function f k with pa f k f pa rightarrow 0 as k rightarrow infty the direction f k x k 1 f x k is then used in a line search on a sum of squares objective the approximate function f k can be constructed for nonsmooth equations arising from variational inequalities maximal monotone operator problems nonlinear complementarity problems and nonsmooth partial differential equations numerical examples are given to illustrate the method

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

on some third order iterative methods for solving nonlinear equations using the iteration formulas of second order mamta v kanwar v k kukreja sukhjit singh on a class of quadratically convergent iteration formulae appl math comput 2004 in press for solving single variable nonlinear equations two classes of third order multipoint methods without using second derivative are derived the main advantage of these classes is that they do not fail if the derivative of the function is either zero or very small in the vicinity of the required root further a new family of secant like method with guaranteed super linear convergence is obtained by discrete modifications and their comparison with respect to the existing classical methods is given

application of the mcnemar test to non independent matched pair data mcnemar s one degree of freedom chi square test for the equality of proportions appears frequently in the analysis of pairs of matched binary outcome data y1i y2i an assumption underlying this test is that the responses from pair to pair are mutually independent in certain applications however the pairs may represent repeated measurements on the same experimental unit and hence this assumption is violated in this paper we suggest an adjustment to the mcnemar test to account for the repeated measures clustering effect and we report on a monte carlo simulation that evaluates the effectiveness of this approach

a note on a family of methods for solving nonlinear equations abstract ujevic et al nenad ujevic goran erceg ivan lekic a family of methods for solving nonlinear equations appl math comput 2007 doi 10 1016 j amc 2007 03 007 introduced a family of methods for solving nonlinear equations for certain choices of parameters firstly they showed that the classical newton s method is a member of this family and then they introduced a particular method however the main algorithm 1 put forward by ujevic et al p 7 is wrong this is the main aim of this note we also point out some major bugs in the results of ujevic et al

solution of nonlinear equations by modified adomian decomposition method in this paper we modify the standard adomian method for solution of the nonlinear equation f x 0 four examples are presented and compared using standard and modified adomian methods

on the sample size for one sided equivalence of sensitivities based upon mcnemar s test in this paper we formalize the problem of testing the one sided equivalence in the sensitivities of two medical diagnostic tests under a matched pair study design we derive conditional and unconditional sample size formulae which are decreasing functions of the probability of being diagnosed by both tests we calculate upper boundary and midpoint sample sizes results of a monte carlo simulation study that compares the proposed sample size formulae with that of lachenbruch suggest that our midpoint conditional sample size is the best choice to obtain the desired power for the type of equivalence studies discussed in the paper

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

convergence of newton like methods for singular operator equations using outer inverses we present a semilocal kantorovich type analysis for newton like methods for singular operator equations using outer inverses we establish sharp generalizations of the kantorovich theory and the mysovskii theory for operator equations when the derivative is not necessarily invertible the results reduce in the case of an invertible derivative to well known theorems of kantorovich and mysovskii with no additional assumptions unlike earlier theorems which impose strong conditions the strategy of the analysis is based on banach type lemmas and perturbation bounds for outer inverses which show that the set of outer inverses to a given bounded linear operator admits selections that behave like bounded linear inverses in contrast to inner inverses or generalized inverses which do not depend continuously on perturbations of the operator we give two examples to illustrate our results and compare them with earlier results and another numerical example to relate our results to computational issues

the construction of rational iterative methods for solving nonlinear equations abstract in this paper based on the idea of constructing iterative methods by sharma j r sharma a family of third order methods to solve nonlinear equations appl math comput 184 2007 210 215 jiang and han dongdong jiang danfu han some one parameter families of third order methods for solving nonlinear equations appl math comput 195 2008 392 396 we propose some rational iterative methods by linearizing the quadratic curve equation by analysis of asymptotic errors we prove the new families are all cubically convergent as an example we present a new concrete iterative family with better convergent performance than classic third order methods through the adjustment of parameter n automatically

non inferiority tests for clustered matched pair data non inferiority tests for matched pair data where pairs are mutually independent may not be appropriate when pairs are clustered the tests may require an adjustment to account for the correlation within a cluster we consider the adjusted score and wald type tests and a modification of obuchowski s method for non inferiority and compare them with the non inferiority test based on a method of moments estimate in terms of type 1 error rate and power by simulations for a small cluster size under various correlation structures in general the score test adjusted by an inflation factor and the modified obuchowski s method perform as good as the test based on moments estimate in the accuracy of type 1 error rates the latter does not provide reasonably close type 1 error rates to the nominal level when the number of clusters is 25 or smaller and a positive response rate for the standard procedure is 20 per cent or lower the adjusted score test the method based on moments estimate and the modified test are comparable in power the adjusted wald type test is too anti conservative and we should caution use of the test since number of clusters is strongly related to the accuracy of empirical type 1 error rate and power it is very important to have a sufficiently large number of clusters in designing a clustered matched pair study for non inferiority

approximation of a nondifferentiable nonlinear problem related to mhd equilibria we present a simple method based on a variant of the implicit function theorem which leads to the existence of a part of a nontrivial solution branch of the nonlinear eigenvalue problem u u in u 1 on where is a two dimensional domain with boundary the advantage of this method is that we can apply it for analysing the approximation of the above problem by a finite element method the error analysis of the discrete problem appears immediately we give also an iteration scheme which allows to solve the approximate problem

h control of linear time varying systems a state space approach in this paper the standard problem of h infty control theory for finite dimensional linear time varying continuous time plants is considered the problem is given a real number gamma 0 f

numerical stability of the halley iteration for the solution of a system of nonlinear equations let f rq rq and x a simple root in rq of the system of nonlinear equations f x 0 abstract pade approximants apa and abstract rational approximants ara for the operator f have been introduced in 2 and 3 the adjective abstract refers to the use of abstract polynomials 5 for the construction of the rational operators the apa and ara have been used for the solution of a system of nonlinear equations in 4 of particular interest was the following third order iterative procedure a2 xi xi 2 il iai f fi ai2 with f the 1st frechet derivative of f in xi ai f fi the newton correction where fi f xi fi the 2nd frechet derivative of f in xi where fi a2 is the bilinear operator fi evaluated in ai ai and componentwise multiplication and division in rq for q i this technique is known as the halley iteration 6 p 91 in this paper the numerical stability 7 of the halley iteration for the case q i is investigated and illustrated by a numerical example 1 numerical stability of iterations we consider the numerical solution of the equation 1 f x 0 with f rq rq x f x abstract analytic in 0 5 assume that 1 has a simple root x we briefly repeat the definition of condition number given by wozniakowski 7 the condition number should measure the sensitivity of the solution output with respect to changes in the data input we assume that f depends parametrically on a vector d e rp called data vector f x f x d and instead of the exact value f x d we only have the computed value fl f x d in t digit floating point binary arithmetic at best we can expect that fl f x d is the exact value of a slightly perturbed operator at slightly perturbed data 2 fl f x d i af f x ax d ad where i is the q x q unit matrix and 1axii 1 let us now suppose that f x d 0 is solved by an iterative procedure 4 xi f where f can use several fj thejth frechet derivative of f at xi if j i or 2 a single or double prime is used instead of the superscript j if xi is the sequence of successive approximations of x we can at best expect xi to be the representation of a computed value for x llxi x 1 00 is numerically stable if limi c iiii 00 00 lim 1i4 1i 1 we will now prove numerical stability of this iteration under assumptions similar to the assumptions for the newton iteration we will also assume that the divisions in 4 are such that 5 0 iiaiiij kpk 1 2f ji ka fi f7 ai j 0p this content downloaded from 157 55 39 161 on mon 23 may 2016 05 03 48 utc all use subject to http about jstor org terms 174 annie a m cuyt condition 5 takes care of the fact that the denominator of the correction term in 4 does not become too small in comparison with o iai li kpk the assumption of 5 is a natural generalization of the following relations a forq 1 lim a 1l i oco ar f7 fl a72 5a and so 3l e n vi l a i d case 1 k 0 1 0 with d e r in a convergent process 4 lim iix xill 0 and thus lim ai o i e 3m e n v vi m ai 0 p 00 and so vi m a72 o ilaillp also 2 lim ai 0 i e i ooii a ai 2 3n e n dvi n 0 p 5b a 2 a2 and so vi max n m i

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

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

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

nonlinear h control of nonsmooth time varying systems with application to friction mechanical manipulators nonlinear h controller synthesis is developed for nonsmooth time varying systems via measurement feedback a local h controller is derived by means of a certain perturbation of the differential riccati equations appearing in solving the h control problem for the linearized system when these unperturbed equations have bounded positive semidefinite solutions stabilizability and detectability properties of the control system are thus ensured by the existence of the proper solutions of the unperturbed differential riccati equations and hence the proposed synthesis procedure obviates an extra formidable in the nonlinear case work on verification of these properties theoretical results are applied to a position tracking control problem and particularly a regulation problem for mechanical systems with friction performance issues of the nonlinear h tracking controller are illustrated in a simulation study made for a two degrees of freedom robot manipulator

nonlinear h sub spl infin control of time varying systems a unified distribution based formalism for continuous and sampled data measurement feedback design nonlinear h sub spl infin controller synthesis is presented for time varying systems via sampled data measurement feedback the h sub spl infin control problem for a nonlinear system with a sampled data measurement output is shown to be equivalent to the h sub spl infin control problem for a certain nonlinear system with a time continuous measurement output special mathematical tools are developed to straightforwardly derive a local solution of the sampled data measurement feedback smf h sub spl infin control problem from that of the time continuous measurement feedback cmf h sub spl infin control problem

estimates of majorizing sequences in the newton kantorovich method a further improvement let f b x0 r x y be an operator with x and y banach spaces and f be holder continuous with exponent the convergence of the sequence of newton kantorovich approximations xn xn 1 f xn 1 1f xn 1 n n is a classical tool to solve the equation f x 0 the convergence of xn is often reduced to the study of the majorizing sequence rn defined by r0 0 r1 a rn 1 rn bk rn rn 1 1 1 1 bkrn n n with a b k parameters related to f and f in the paper f cianciaruso e de pascale estimates of majorizing sequences in the newton kantorovich method submitted for publication we proved that if a bk 1 1 1 1 1 then the following estimates for rn hold rn bk 1 1 1 1 1 1 1 n n n in the present paper we give a stronger at least asymptotically estimates on rn under a weaker condition on the techniques employed in the paper are similar to the ones used in f cianciaruso e de pascale estimates of majorizing sequences in the newton kantorovich method submitted for publication finally we make a comparison with previous results

a method for solving nonlinear equations a 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

a new modified secant like method for solving nonlinear equations in this paper we present a new secant like method for solving nonlinear equations analysis of the convergence shows that the asymptotic convergence order of this method is 1 3 some numerical results are given to demonstrate its efficiency

improving newton raphson method for nonlinear equations by modified adomian decomposition method in this paper we present some efficient numerical algorithms for solving nonlinear equations based on newton raphson method the modified adomian decomposition method is applied to construct the numerical algorithms some numerical illustrations are given to show the efficiency of algorithms

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

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

a method for finding sharp error bounds for newton s method under the kantorovic assumptions this paper gives a method for finding sharpa posteriori error bounds for newton s method under the assumptions of kantorovich s theorem on the basis of this method new error bounds are derived and comparison is made among the known bounds of dennis 2 doring 4 gragg tapia 5 kantorovich 6 7 kornstaedt 9 lancaster 10 miel 11 13 moret 14 ostrowski 17 18 potra 19 and potra ptak 20

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

newton s method s basins of attraction revisited in this paper we revisit the chaotic number of iterations needed by newton s method to converge to a root here we consider a simple modified newton method depending on a parameter it is demonstrated using polynomiography that even in the simple algorithm the presence and the position of the convergent regions i e regions where the method converges nicely to a root can be complicatedly a function of the parameter

assessing screening tests extensions of mcnemar s test we address the problem of comparing a new screening test to a currently available screening test in the absence of a gold standard when both tests are given to each participant in a clinical trial the usual analytical approach is to apply mcnemar s test for equality of the off diagonal probabilities with rejection of the null hypothesis implying that the tests differ for assessing equivalence however we consider a compound null hypothesis that the new test gives either fewer or more positive results than the standard if both parts of this hypothesis are rejected we assert equivalence in the rate of positive responses we propose an extension of mcnemar s test for this situation a companion step is to construct a confidence interval for the ratio of the marginal probabilities and assert equivalence if the interval is sufficiently small it is also important that the tests agree a large proportion of the time this can be verified with a complementary two tailed binomial test another situation arises when there is a gold standard for disease diagnosis and we wish to compare the sensitivity and specificity of two screening tests we show that a 2 degrees of freedom chi square test based on two mcnemar like tables is an appropriate test

establishing equivalence of two treatments and sample size requirements in matched pairs design statistical methods for testing the null hypothesis of a nonzero difference between two treatments and the sample size determination for matched pairs studies are investigated a wald type test proposed by lu and bean 1995 statistics in medicine 14 1831 1839 is anticonservative i e its false positive error rate is greater than specified score method and normal deviate based on a restricted maximum likelihood estimation are presented these two test statistics are shown to be algebraically equal their actual type i error probabilities are satisfactorily close to a nominal level numerical examinations demonstrate that the sample size formulas using these alternative methods are reasonable while that by lu and bean is not the efficiency of matching in equivalence studies is positively related to intercorrelation or the kappa coefficient of agreements we recommend the score or ml methods to establish equivalence of two treatments for individually matched samples

convergence of newton s method for singular smooth and nonsmooth equations using adaptive outer inverses we present a local convergence analysis of generalized newton methods for singular smooth and nonsmooth operator equations using adaptive constructs of outer inverses we prove that for a solution x of f x 0 there exists a ball s s x r r 0 such that for any starting point x 0 in s the method converges to a solution bar x in s of gamma f x 0 where gamma is a bounded linear operator that depends on the frechet derivative of f at x 0 or on a generalized jacobian of f at x 0 point bar x may be different from x when x is not an isolated solution moreover we prove that the convergence is quadratic if the operator is smooth and superlinear if the operator is locally lipschitz these results are sharp in the sense that they reduce in the case of an invertible derivative or generalized derivative to earlier theorems with no additional assumptions the results are illustrated by a system of smooth equations and a system of nonsmooth equations each of which is equivalent to a nonlinear complementarity problem

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

the prediction approach to robust variance estimation in two stage cluster sampling abstract robust statistics are derived for estimating the error variance when a finite population total or a ratio of totals is estimated from a two stage cluster sample in which the first stage sampling fraction is small the analysis is based on prediction superpopulation models that allow for very general correlation structure within clusters results are obtained for when all cluster sizes are known and for when sizes are known only for sample clusters ratio regression and mean of ratios statistics are in the class of population total estimators considered

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

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

tests for equivalence or non inferiority for paired binary data assessment of therapeutic equivalence or non inferiority between two medical diagnostic procedures often involves comparisons of the response rates between paired binary endpoints the commonly used and accepted approach to assessing equivalence is by comparing the asymptotic confidence interval on the difference of two response rates with some clinical meaningful equivalence limits this paper investigates two asymptotic test statistics a wald type sample based test statistic and a restricted maximum likelihood estimation rmle based test statistic to assess equivalence or non inferiority based on paired binary endpoints the sample size and power functions of the two tests are derived the actual type i error and power of the two tests are computed by enumerating the exact probabilities in the rejection region the results show that the rmle based test controls type i error better than the sample based test to establish an equivalence between two treatments with a symmetric equivalence limit of 0 15 a minimal sample size of 120 is needed the rmle based test without the continuity correction performs well at the boundary point 0 a numerical example illustrates the proposed procedures

concerning the terra incognita between convergence regions of two newton methods abstract the majorizing principle is used to show local and semilocal convergence of newton methods to a locally unique solution of a nonlinear operator in a banach space when the frechet derivative of the operator involved satisfies a center holder and a holder continuity condition then we investigate an unknown area terra incognita between the convergence regions of newton s method and the corresponding modified newton s method our approach compares favorably with other corresponding ones in this direction

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

quasi newton methods for solving underdetermined nonlinear simultaneous equations abstract we analyze iterative processes of type xk 1 xk xk ek f xk for solving f x 0 f r n r m m n parameters ek are updated at each iteraction using least change secant update procedures we prove local linear and superlinear convergence results we introduce two new superlinearly convergent methods of this type and one linearly convergent quasi newton generalization of cimmino s parallel algorithm for solving linear systems some numerical experiments are presented

a novel cubically convergent iterative method for computing complex roots of nonlinear equations abstract a fast and simple iterative method with cubic convergent is proposed for the determination of the real and complex roots of any function f x 0 the idea is based upon passing a defined function g x tangent to f x at an arbitrary starting point choosing g x in the form of x k or k x where k is obtained for the best correlation with the function f x gives an added freedom which in contrast to all existing methods accelerates the convergence also this new method can find complex roots just by a real initial guess this is in contrast to many other methods like the famous newton method that needs complex initial guesses for finding complex roots the proposed method is compared to some new and famous methods like newton method and a modern solver that is fsolve command in matlab the results show the effectiveness and robustness of this new method as compared to other methods