comparative study of methods of various orders for finding repeated roots of nonlinear equations abstract in this paper we are considering 20 families of methods for finding repeated roots of a nonlinear equation the methods are of order up to 8 we use the idea of basin of attraction to compare the methods we found that 4 methods performed best based on 3 quantitative criteria

higher order stable generalized finite element method for the elliptic eigenvalue and source problems with an interface in 1d abstract we study the generalized finite element methods gfems for the second order elliptic eigenvalue problem with an interface in 1d the linear stable generalized finite element methods sgfem were recently developed for the elliptic source problem with interfaces we first generalize sgfem to arbitrary order elements and establish the optimal error convergence of the approximate solutions for the elliptic source problem with an interface we then apply the abstract theory of spectral approximation of compact operators to establish the error estimation for the eigenvalue problem with an interface the error estimations on eigenpairs strongly depend on the estimation of the discrete solution operator for the source problem we verify our theoretical findings in various numerical examples including both source and eigenvalue problems

on optimal fourth order iterative methods free from second derivative and their dynamics in this paper new fourth order optimal root finding methods for solving nonlinear equations are proposed the classical jarratt s family of fourth order methods are obtained as special cases we then present results which describe the conjugacy classes and dynamics of the presented optimal method for complex polynomials of degree two and three the basins of attraction of existing optimal methods and our method are presented and compared to illustrate their performance

a class of two point sixth order multiple zero finders of modified double newton type and their dynamics under the assumption of the known multiplicity of zeros of nonlinear equations a class of two point sextic order multiple zero finders and their dynamics are investigated in this paper by means of extensive analysis of modified double newton type of methods with the introduction of a bivariate weight function dependent on function to function and derivative to derivative ratios higher order convergence is obtained additional investigation is carried out for extraneous fixed points of the iterative maps associated with the proposed methods along with a comparison with typically selected cases through a variety of test equations numerical experiments strongly support the theory developed in this paper in addition relevant dynamics of the proposed methods is successfully explored for various polynomials with a number of illustrative basins of attraction

dynamics of a family of chebyshev halley type methods in this paper the dynamics of the chebyshev halley family is studied on quadratic polynomials a singular set that we call cat set appears in the parameter space associated to the family this set has interesting similarities with the mandelbrot set the parameter space has allowed us to find different elements of the family which have bad convergence properties since periodic orbits and attractive strange fixed points appear in the dynamical plane of the corresponding method

a new family of optimal eighth order methods with dynamics for nonlinear equations we propose a simple yet efficient family of three point iterative methods for solving nonlinear equations each method of the family requires four evaluations namely three functions and one derivative per full iteration and possesses eighth order of convergence thus the family is optimal in the sense of kung traub conjecture and has the efficiency index 1 682 which is better than that of newton s and many other higher order methods various numerical examples are considered to check the performance and to verify the theoretical results computational results including the elapsed cpu time confirm the efficient and robust character of proposed technique moreover the presented basins of attraction also confirm better performance of the methods as compared to other established methods in literature

eigenvalue approximation by mixed and hybrid methods note univ maryland dept math college pk md 20742 ecole polytech ctr math appl f 91128 palaiseau france univ paris 6 f 75230 paris 05 france mercier b cea limeil f 94190 villeneuve france isi document delivery no lq584times cited 29cited reference count 44 reference asn article 1981 005doi 10 2307 2007651 record created on 2006 08 24 modified on 2017 05 12

on a family of halley like methods to find simple roots of nonlinear equations there are many methods for solving a nonlinear algebraic equation here we introduce a family of halley like methods and show that euler chebyshev and bsc are just members of the family we discuss the conjugacy maps and the effect of the extraneous roots on the basins of attraction

on the convergence of an optimal fourth order family of methods and its dynamics in this paper we present the study of the semilocal and local convergence of an optimal fourth order family of methods moreover the dynamical behavior of this family of iterative methods applied to quadratic polynomials is studied some anomalies are found in this family be means of studying the dynamical behavior parameter spaces are shown and the study of the stability of all the fixed points is presented

a biparametric extension of king s fourth order methods and their dynamics a class of two point quartic order simple zero finders and their dynamics are investigated in this paper by extending king s fourth order family of methods with the introduction of an error corrector having a weight function dependent on a function to function ratio higher order convergence is obtained through a variety of test equations numerical experiments strongly support the theory developed in this paper in addition relevant dynamics of the proposed methods is successfully explored for a prototype quadratic polynomial as well as parameter spaces and dynamical planes

a posteriori error estimates for nonconforming approximations of steklov eigenvalue problems this paper deals with a posteriori error estimators for the non conforming crouzeix raviart finite element approximations of the steklov eigenvalue problem first we define an error estimator of the residual type which can be computed locally from the approximate eigenpair and we prove the equivalence between this estimator and the broken energy norm of the error with constants independent of the corresponding eigenvalue next we prove that edge residuals dominate the volumetric part of the residual and that the volumetric part of the residual terms dominate the normal component of the jumps of the discrete fluxes across interior edges finally based on these results we introduce two simpler equivalent error estimators the analysis shows that these a posteriori error estimates are optimal up to higher order terms and that may be used for the design of adaptive algorithms

analysis of finite element approximation of evolution problems in mixed form this paper deals with the finite element approximation of evolution problems in mixed form following d boffi f brezzi and l gastaldi math comp 69 2000 pp 121 140 we handle separately two types of problems a model for the first case is the heat equation in mixed form while the time dependent stokes problem fits within the second one for either case we give sufficient conditions for a good approximation in the natural functional spaces the results are not obvious in the first situation in this case the well known conditions for the well posedness and convergence of the corresponding steady problem are not sufficient for the good approximation of the time dependent problem this issue is demonstrated with a numerical counter example and justified analytically

simpson s rule to approximate hilbert integral and its application abstract in this paper the computation of hilbert singular integral with generalized composite simpson s rule for is discussed when singular points coincide with some a priori known point the convergence rate of simpson s rule higher than global one we obtain the pointwise superconvergence phenomenon which means the especial function equal zero the superconvergence points are got then choosing the superconvergence point as the collocation points we get a collocation scheme for solving the relevant hilbert integral equation at last some numerical examples are presented to validate the theoretical analysis

fourth order iterative methods free from second derivative in this paper we present a family of new iterative methods with order of convergence four for solving non linear equations per iteration these methods require one evaluation of the function and two of its first derivative analysis of efficiency in term of function evaluations shows that this family of methods has great superiority which is also demonstrated by numerical examples

finite element vibration analysis of fluid solid systems without spurious modes this paper deals with the finite element approximation of the vibration modes of a problem with fluid structure interaction displacement variables are used for both the fluid and the solid to avoid the typical spurious modes of this formulation we introduce a nonconforming discretization error estimates for the approximation of eigenvalues and eigenvectors are given

a study of dynamics via mobius conjugacy map on a family of sixth order modified newton like multiple zero finders with bivariate polynomial weight functions abstract a generic family of sixth order modified newton like multiple zero finders have been proposed in geum et al 2016 among them we select a specific family of iterative methods with uniparametric bivariate polynomial weight functions and study their dynamics by investigating the relevant parameter spaces and dynamical planes via mobius conjugacy map applied to a prototype polynomial of the form z a m z b m the resulting dynamics is best illustrated through a variety of parameter spaces as well as dynamical planes

guaranteed and robust a posteriori bounds for laplace eigenvalues and eigenvectors a unified framework this paper develops a general framework for a posteriori error estimates in numerical approximations of the laplace eigenvalue problem applicable to all standard numerical methods guaranteed and computable upper and lower bounds on an arbitrary simple eigenvalue are given as well as on the energy error in the approximation of the associated eigenvector the bounds are valid under the sole condition that the approximate i th eigenvalue lies between the exact i 1 th and i 1 th eigenvalue where the relative gaps are sufficiently large we give a practical way how to check this the precision of the resulting estimates depends on these relative gaps our bounds feature no unknown solution regularity or polynomial degree dependent constant are optimally convergent efficient and polynomial degree robust under a further explicit a posteriori minimal resolution condition the multiplicative constant in our estimates can be reduced by a fixed factor moreover when an elliptic regularity assumption on the corresponding source problem is satisfied with known constants this multiplicative constant can be brought to the optimal value of 1 with mesh refinement applications of our framework to nonconforming discontinuous galerkin and mixed finite element approximations of arbitrary polynomial degree are provided along with numerical illustrations our key ingredient are equivalences between the i th eigenvalue error the associated eigenvector energy error and the dual norm of the residual we extend them in an appendix to the generic class of bounded below self adjoint operators with compact resolvent

rational eigenvalue problems and applications to photonic crystals we establish new analytic results for a general class of rational spectral problems they arise e g in modelling photonic crystals whose capability to control the flow of light depends on specific

spectral approximation of variationally posed eigenvalue problems by nonconforming methods this paper deals with the nonconforming spectral approximation of variationally posed eigenvalue problems it is an extension to more general situations of known previous results about nonconforming methods as an application of the present theory convergence and optimal order error estimates are proved for the lowest order crouzeix raviart approximation of the eigenpairs of two representative second order elliptical operators

linearization techniques for band structure calculations in absorbing photonic crystals band structure calculations for photonic crystals require the numerical solution of eigenvalue problems in this paper we consider crystals composed of lossy materials with frequency dependent per

spectral approximations by the hdg method we consider the numerical approximation of the spectrum of a secondorder elliptic eigenvalue problem by the hybridizable discontinuous galerkin hdg method we show for problems with smooth eigenfunctions that the approximate eigenvalues and eigenfunctions converge at the rate 2k 1 and k 1 respectively here k is the degree of the polynomials used to approximate the solution its flux and the numerical traces our numerical studies show that a rayleigh quotient like formula applied to certain locally postprocessed approximations can yield eigenvalues that converge faster at the rate 2k 2 for the hdg method as well as for the brezzidouglas marini bdm method we also derive and study a condensed nonlinear eigenproblem for the numerical traces obtained by eliminating all the other variables

discontinuous galerkin computation of the maxwell eigenvalues on simplicial meshes this paper is concerned with the discontinuous galerkin approximation of the maxwell eigenproblem after reviewing the theory developed in a buffa i perugia discontinuous galerkin approximation of the maxwell eigenproblem technical report 24 pv imati cnr pavia italy 2005 we present a set of numerical experiments which both validate the theory and provide further insight regarding the practical performance of discontinuous galerkin methods particularly in the case when non conforming meshes characterized by the presence of hanging nodes are employed

an analysis of a new family of eighth order optimal methods a new family of eighth order optimal methods is developed and analyzed numerical experiments show that our family of methods perform well and in many cases some members are superior to other eighth order optimal methods it is shown how to choose the parameters to widen the basin of attraction

an analysis of a family of maheshwari based optimal eighth order methods in this paper we analyze an optimal eighth order family of methods based on maheshwari s fourth order method this family of methods uses a weight function we analyze the family using the information on the extraneous fixed points two measures of closeness of an extraneous points set to the imaginary axis are considered and applied to the members of the family to find its best performer the results are compared to a modified version of wang liu method

an optimal reconstruction of chebyshev halley type methods for nonlinear equations having multiple zeros abstract establishment of a new optimal higher order iterative scheme for multiple zeros with known multiplicity m 1 of univariate function is one of the hard and demanding task in the area of computational mathematics and numerical analysis we write this paper with the aim to propose a new higher order reconstruction of chebyshev halley type iteration functions for multiple zeros in a simple way minimum order of convergence of our scheme is seven and maximum reaches to eight without using any extra functional evaluation with regard to computational cost each member of our scheme needs four functional evaluations at each step so the maximum efficiency index of our scheme is 1 6818 for 1 in addition we thoroughly study the convergence analysis which confirms the theoretical order of convergence moreover we check the efficiency and compare them to the existing ones on four real life problems namely fractional conversion chemical engineering continuous stirred tank reactor and eigen value problems and two well known academic problems finally we confirm on the basis of obtained results that our methods have better residual error errors between two consecutive iterations and stable computational order of convergence as compared to the other existing ones

basins of attraction for optimal eighth order methods to find simple roots of nonlinear equations several optimal eighth order methods to obtain simple roots are analyzed the methods are based on two step fourth order optimal methods and a third step of modified newton the modification is performed by taking an interpolating polynomial to replace either f z n or f z n in six of the eight methods we have used a hermite interpolating polynomial the other two schemes use inverse interpolation we discovered that the eighth order methods based on jarratt s optimal fourth order methods perform well and those based on king s or kung traub s methods do not in all cases tested the replacement of f z by hermite interpolation is better than the replacement of the derivative f z

a schwarz generalized eigen oscillation spectral element method gesem for 2 d high frequency electromagnetic scattering in dispersive inhomogeneous media in this paper we propose a parallel schwarz generalized eigen oscillation spectral element method gesem for 2 d complex helmholtz equations in high frequency wave scattering in dispersive inhomogeneous media this method is based on the spectral expansion of complex generalized eigen oscillations for the electromagnetic fields and the schwarz non overlapping domain decomposition iteration method the gesem takes advantages of a special real orthogonality property of the complex eigen oscillations and a new radiation interface condition for the system of equations for the spectral expansion coefficients numerical results validate the high resolution and the flexibility of the method for various materials

the superconvergence of the newton cotes rule for cauchy principal value integrals we consider the general composite newton cotes method for the computation of cauchy principal value integrals and focus on its pointwise superconvergence phenomenon which means that the rate of convergence of the newton cotes quadrature rule is higher than what is globally possible when the singular point coincides with some a priori known point the necessary and sufficient conditions satisfied by the superconvergence point are given moreover the superconvergence estimate is obtained and the properties of the superconvergence points are investigated finally some numerical examples are provided to validate the theoretical results

complex dynamics of derivative free methods for nonlinear equations the dynamical behavior of two iterative derivative free schemes steffensen and m4 methods is studied in case of quadratic and cubic polynomials the parameter plane is analyzed for both procedures on quadratic polynomials different dynamical planes are showed when the mentioned methods are applied on particular cubic polynomials with real or complex coefficients the property of immersion of the basins of attraction in all cases is analyzed

non conforming high order approximations of the elastodynamics equation in this paper we formulate and analyze two non conforming high order strategies for the approximation of elastic wave problems in heterogeneous media namely the mortar spectral element method and the discontinuous galerkin spectral element method starting from a common variational formulation we make a full comparison of the two techniques from the points of view of accuracy convergence grid dispersion and stability

discontinuous galerkin approximation of the laplace eigenproblem in this paper we analyse the problem of computing eigenvalues and eigenfunctions of the laplace operator by means of discontinuous galerkin dg methods it results that several dg methods actually provide a spectrally correct approximation of the laplace operator we present here the convergence theory which applies to a wide class of dg methods as well as numerical tests demonstrating the theoretical results

comparing the basins of attraction for kanwar bhatia kansal family to the best fourth order method there are relatively few optimal fourth order methods for solving nonlinear algebraic equations having roots of known multiplicity m in a previous paper we have compared 5 such methods two of which require the evaluation of the m 1 st root we have used the basin of attraction idea to recommend the best optimal fourth order method here we compare the family of methods developed by kanwar et al 2013 to the best known method we will also point out some mistake in deriving that family

constructing a family of optimal eighth order modified newton type multiple zero finders along with the dynamics behind their purely imaginary extraneous fixed points abstract an optimal family of eighth order multiple zero finders and the dynamics behind their basins of attraction are proposed by considering modified newton type methods with multivariate weight functions extensive investigation of purely imaginary extraneous fixed points of the proposed iterative methods is carried out for the study of the dynamics associated with corresponding basins of attraction numerical experiments strongly support the underlying theory pursued in this paper an exploration of the relevant dynamics of the proposed methods is presented along with illustrative basins of attraction for various polynomials

accurate pressure post process of a finite element method for elastoacoustics this paper deals with a post process to obtain a more accurate approximation of the fluid pressure from a finite element computation of the vibration modes of a fluid structure coupled system the underlying finite element method based on a displacement formulation for both media consists of using raviart thomas elements for the fluid combined with standard continuous elements for the solid an easy to compute post process of the pressure is derived the relation between this post process and an alternative finite element approximation of the problem based on discretizing the fluid pressure by enriched crouzeix raviart elements is studied higher order estimates for the l2 norm of the post processed pressure are proved by exploiting this relation as a by product higher order l2 estimates for the solid displacements obtained with the original method are also proved

an efficient optimal family of sixteenth order methods for nonlinear models abstract the principle aim of this manuscript is to propose a general scheme that can be applied to any optimal iteration function of order eight whose first substep employ newton s method to further develop new interesting optimal scheme of order sixteen this scheme requires four evaluations of the involved function and one evaluation of its first order derivative at each step so it is being optimally consistent with the conjecture of kung traub in addition theoretical and computational properties are fully investigated along with a main theorem describing the order of convergence moreover the conjugacy maps and the strange fixed points of some iterative methods are discussed their basins of attractions are also given to show their dynamical behavior around the simple roots from the numerical experiments we find that our methods perform better than the existing ones when we checked the performance on a concrete variety of nonlinear equations

basins of attraction for several methods to find simple roots of nonlinear equations there are many methods for solving a nonlinear algebraic equation the methods are classified by the order informational efficiency and efficiency index here we consider other criteria namely the basin of attraction of the method and its dependence on the order we discuss several third and fourth order methods to find simple zeros the relationship between the basins of attraction and the corresponding conjugacy maps will be discussed in numerical experiments the effect of the extraneous roots on the basins is also discussed

a discontinuous hp finite element method for convection diffusion problems this paper presents a new method which exhibits the best features of both finite volume and finite element techniques special attention is given to the issues of conservation flexible accuracy and stability the method is elementwise conservative the order of polynomial approximation can be adjusted element by element and the stability is not based on the introduction of artificial diffusion but on the use of a very particular finite element formulation with discontinuous basis functions the method supports h p and hp approximations and can be applied to any type of meshes including non matching grids a priori error estimates and numerical experiments on representative model problems indicate that the method is robust and capable of delivering high accuracy

stability analysis of a parametric family of iterative methods for solving nonlinear models a one parametric family of fourth order iterative methods for solving nonlinear systems is presented proving the fourth order of convergence of all members in this family except one of them whose order is five the methods in our family are numerically compared with other known methods in terms of the classical efficiency index order of convergence and number of functional evaluations and in terms of the operational efficiency index which also takes into account the total number of product quotients per iteration in order to analyze its stability and its dynamical properties the parameter space for quadratic polynomials is shown the stability of the strange fixed points is studied in this case we note that even for this particular case the family presents a very interesting dynamical behavior the analysis of the parameter plane allows us to find values for the involved parameter with good stability properties as well as other values with bad numerical behavior finally amongst the first ones there is a special value of the parameter related to a fifth order method in the family

superconvergence of the local discontinuous galerkin method for elliptic problems on cartesian grids in this paper we present a superconvergence result for the local discontinuous galerkin ldg method for a model elliptic problem on cartesian grids we identify a special numerical flux for which the l2 norm of the gradient and the l2 norm of the potential are of orders k 1 2 and k 1 respectively when tensor product polynomials of degree at most k are used for arbitrary meshes this special ldg method gives only the orders of convergence of k and k 1 2 respectively we present a series of numerical examples which establish the sharpness of our theoretical results

an optimal family of eighth order simple root finders with weight functions dependent on function to function ratios and their dynamics underlying extraneous fixed points we extend in this paper an optimal family of three step eighth order methods developed by duni etal 2011 with higher order weight functions employed in the second and third sub steps and investigate their dynamics under the relevant extraneous fixed points among which purely imaginary ones are specially treated for the analysis of the rich dynamics their theoretical and computational properties are fully investigated along with a main theorem describing the order of convergence and the asymptotic error constant as well as proper choices of special cases a wide variety of relevant numerical examples are illustrated to confirm the underlying theoretical development in addition this paper investigates the dynamics of selected existing optimal eighth order iterative maps with the help of illustrative basins of attraction for various polynomials

a uniformly accurate finite element method for the reissner mindlin plate a simple finite element method for the reissner mindlin plate model in the prim itive variables is presented and analyzed the method uses nonconforming linear finite elements for the transverse displacement and conforming linear finite elements enriched by bubbles for the rotation with the computation of the element stiffness matrix modified by the inclusion of a simple elementwise averaging it is proved that the method converges with optimal order uniformly with respect to thickness

a two grid method of the non conforming crouzeix raviart element for the steklov eigenvalue problem abstract this paper discusses a high efficient scheme for the steklov eigenvalue problem a two grid discretization scheme of nonconforming crouzeix raviart element is established with this scheme the solution of a steklov eigenvalue problem on a fine grid h is reduced to the solution of the eigenvalue problem on a much coarser grid h and the solution of a linear algebraic system on the fine grid h by using spectral approximation theory and nitsche lascaux lesaint technique in space h 1 2 we prove that the resulting solution obtained by our scheme can maintain an asymptotically optimal accuracy by taking h h and the numerical experiments indicate that when the eigenvalues k h of nonconforming crouzeix raviart element approximate the exact eigenvalues from below the approximate eigenvalues k h obtained by the two grid discretization scheme also approximate the exact ones from below and the accuracy of k h is higher than that of k h

choosing weight functions in iterative methods for simple roots weight functions with a parameter are introduced into an iteration process to increase the order of the convergence and enhance the behavior of the iteration process the parameter can be chosen to restrict extraneous fixed points to the imaginary axis and provide the best basin of attraction the process is demonstrated on several examples

contrasts in the basins of attraction of structurally identical iterative root finding methods a numerical comparison is performed between three methods of third order with the same structure namely bsc halley s and euler chebyshev s methods as the behavior of an iterative method applied to a nonlinear equation can be highly sensitive to the starting points the numerical comparison is carried out allowing for complex starting points and for complex roots on the basins of attraction in the complex plane several examples of algebraic and transcendental equations are presented

on the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form in the approximation of linear elliptic operators in mixed form it is well known that the so called inf sup and ellipticity in the kernel properties are sufficient and in a sense to be made precise necessary in order to have good approximation properties and optimal error bounds one might think in the spirit of mercier osborn rappaz raviart and in consideration of the good behavior of commonly used mixed elements like raviart thomas or brezzi douglas marini elements that these conditions are also sufficient to ensure good convergence properties for eigenvalues in this paper we show that this is not the case in particular we present examples of mixed finite element approximations that satisfy the above properties but exhibit spurious eigenvalues such bad behavior is proved analytically and demonstrated in numerical experiments we also present additional assumptions fulfilled by the commonly used mixed methods already mentioned which guarantee optimal error bounds for eigenvalue approximations as well

mixed discontinuous galerkin approximation of the elasticity eigenproblem we introduce a discontinuous galerkin method for the mixed formulation of the elasticity eigenproblem with reduced symmetry the analysis of the resulting discrete eigenproblem does not fit in the standard spectral approximation framework since the underlying source operator is not compact and the scheme is nonconforming we show that the proposed scheme provides a correct approximation of the spectrum and prove asymptotic error estimates for the eigenvalues and the eigenfunctions finally we provide several numerical tests to illustrate the performance of the method and confirm the theoretical results

discontinuous galerkin methods of the non selfadjoint steklov eigenvalue problem in inverse scattering abstract in this paper we apply discontinuous galerkin methods to the non selfadjoint steklov eigenvalue problem arising in inverse scattering the variational formulation of the problem is non selfadjoint and does not satisfy h1 elliptic condition by using the spectral approximation theory of compact operators we prove the spectral approximation and optimal convergence order for the eigenvalues finally some numerical experiments are reported to show that the proposed numerical schemes are efficient for real and complex steklov eigenvalues

an a posteriori estimator of eigenvalue eigenvector error for penalty type discontinuous galerkin methods we provide an abstract framework for analyzing discretization error for eigenvalue problems discretized by discontinuous galerkin methods such as the local discontinuous galerkin method and symmetric interior penalty discontinuous galerkin method the analysis applies to clusters of eigenvalues that may include degenerate eigenvalues we use asymptotic perturbation theory for linear operators to analyze the dependence of eigenvalues and eigenspaces on the penalty parameter we first formulate the dg method in the framework of quadratic forms and construct a companion infinite dimensional eigenvalue problem with the use of the companion problem the eigenvalue vector error is estimated as a sum of two components the first component can be viewed as a non conformity error that we argue can be neglected in practical estimates by properly choosing the penalty parameter the second component is estimated a posteriori using auxiliary subspace techniques and this constitutes the practical estimate

a comparison of some quadrature methods for approximating cauchy principal value integrals cauchy principal value integrals are evaluated by the imt quadrature scheme which like the tanh quadrature scheme is essentially a trapezoidal scheme after making a transformation of the variable of integration numerical results for some test problems demonstrate that the imt scheme is superior to the tanh scheme while both these methods are comparable to or even better than the standard methods like the gaussian or the chebyshev schemes in terms of accuracy and simplicity

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

improved chebyshev halley methods with sixth and eighth order convergence we present a second derivative free family of modified chebyshev halley methods with increasing order of convergence for solving nonlinear equations the idea is based on the recent development by li et al 2014 analysis of convergence shows that the family possesses at least sixth order of convergence in a particular case even eighth order of convergence is achieved per iteration each method of the family requires four evaluations namely three functions and one first derivative that means the eighth order method is optimal in the sense of kung traub hypothesis numerical tests are performed which confirm the theoretical results regarding order of convergence and efficiency of the new methods

basin attractors for various methods for multiple roots abstract there are several methods for approximating the multiple zeros of a nonlinear function when the multiplicity is known the methods are classified by the order informational efficiency and efficiency index here we consider other criteria namely the basin of attraction of the method and its dependence on the order we discuss all known methods of orders two to four and present the basin of attraction for several examples

a stable family with high order of convergence for solving nonlinear equations recently li et al 2014 have published a new family of iterative methods without memory with order of convergence five or six which are not optimal in the sense of kung and traub s conjecture therefore we attempt to modify this suggested family in such a way that it becomes optimal to this end we consider the same two first steps of the mentioned family and furthermore we introduce a better approximation for f z in the third step based on interpolation idea as opposed to the taylor s series used in the work of li et al theoretical dynamical and numerical aspects of the new family are described and investigated in details

stability study of eighth order iterative methods for solving nonlinear equations in this paper we study the stability of the rational function associated to a known family of eighth order iterative schemes on quadratic polynomials the asymptotic behavior of the fixed points corresponding to the rational function is analyzed and the parameter space is shown in which we find choices of the parameter for which there exists convergence to cycles or even chaotical behavior showing the complexity of the family moreover some elements of the family with good stability properties are obtained

on the convergence of gander s type family of iterative methods for simultaneous approximation of polynomial zeros abstract in this paper we propose a fifth order family of iterative methods for approximation of all zeros of a polynomial simultaneously the new family is developed by combining gander s third order family of iterative methods with the second order weierstrass root finding method the aim of the paper is to state initial conditions that provide local and semilocal convergence of the proposed methods as well as a priori and a posteriori error estimates in the case of semilocal convergence the initial conditions and error estimates are computationally verifiable which is of practical importance

a composite fourth order iterative method for solving non linear equations in this paper we present a new iterative method with order of convergence four for solving non linear equations per iteration the new method requires two evaluations of the function and one evaluation of its first derivative analysis of efficiency in term of function evaluations shows that the new method is superior its practical utility is demonstrated by numerical examples

immersed finite element method for eigenvalue problem abstract we consider the approximation of elliptic eigenvalue problem with an interface the main aim of this paper is to prove the stability and convergence of an immersed finite element method ifem for eigenvalues using crouzeix raviart p 1 nonconforming approximation we show that spectral analysis for the classical eigenvalue problem can be easily applied to our model problem we analyze the ifem for elliptic eigenvalue problems with an interface and derive the optimal convergence of eigenvalues numerical experiments demonstrate our theoretical results

the order of convergence of eigenfrequencies in finite element approximations of fluid structure interaction problems in this paper we prove a double order for the convergence of eigenfrequencies in fluid structure vibration problems we improve estimates given recently for compressible and incompressible fluids to do this we extend classical results on finite element spectral approximation to nonconforming methods for noncompact operators

an error analysis of the cauchy principal value integral evaluation by the imt scheme we provide an error analysis for the cauchy principal value integral evaluation by the imt method for analytic functions we get the discretization error e d in terms of a contour integral and we derive an asymptotic estimate for this error integral which shows that e d o e n 2 n 3 4 we give two numerical examples where the asymptotically estimated errors are compared with known actual values

two grid discretizations and a local finite element scheme for a non selfadjoint stekloff eigenvalue problem abstract in this paper for a new stekloff eigenvalue problem which is non selfadjoint and not h 1 elliptic we establish and analyze two kinds of two grid discretization scheme and a local finite element scheme we present the error estimates of approximations of two grid discretizations we also prove a local error estimate which is suitable for the case that the local refined region contains singular points lying on the boundary of domain numerical experiments are reported finally to show the efficiency of our schemes

on the convergence of a higher order family of methods and its dynamics in this paper we present the study of the local convergence of a higher order family of methods moreover the dynamical behavior of this family of iterative methods applied to quadratic polynomials is studied some anomalies are found in this family by means of studying the associated rational function parameter spaces are shown and the study of the stability of all the fixed points is presented

a sixth order family of three point modified newton like multiple root finders and the dynamics behind their extraneous fixed points a class of three point sixth order multiple root finders and the dynamics behind their extraneous fixed points are investigated by extending modified newton like methods with the introduction of the multivariate weight functions in the intermediate steps the multivariate weight functions dependent on function to function ratios play a key role in constructing higher order iterative methods extensive investigation of extraneous fixed points of the proposed iterative methods is carried out for the study of the dynamics associated with corresponding basins of attraction numerical experiments applied to a number of test equations strongly support the underlying theory pursued in this paper relevant dynamics of the proposed methods is well presented with a variety of illustrative basins of attraction applied to various test polynomials

a new optimal eighth order family of iterative methods for the solution of nonlinear equations in this paper a new optimal family of methods for solving nonlinear equations is presented it is proved by analysis of convergence that each member of the family which requires three evaluations of the function and one evaluation of its first derivative per iteration is of optimal order eight numerical comparisons are made with several other existing eighth order methods by using the basins of attraction and through numerical computations to demonstrate the efficiency and the performance of the presented methods

basins of attraction for several third order methods to find multiple roots of nonlinear equations there are several third order methods for solving a nonlinear algebraic equation having roots of a given multiplicity m here we compare a recent family of methods of order three to euler cauchy s method which is found to be the best in the previous work there are fewer fourth order methods for multiple roots but we will not include them here

on generalized biparametric multipoint root finding methods with memory a general family of biparametric n point methods with memory for solving nonlinear equations is proposed using an original accelerating procedure with two parameters this family is based on derivative free classes of n point methods without memory of interpolatory type and steffensen like method with two free parameters the convergence rate of the presented family is considerably increased by self accelerating parameters which are calculated in each iteration using information from the current and previous iteration and newton s interpolating polynomials with divided differences the improvement of convergence order is achieved without any additional function evaluations so that the proposed family has a high computational efficiency numerical examples are included to confirm theoretical results and demonstrate convergence behavior of the proposed methods

generating function method for constructing new iterations abstract in this paper we propose a generating function method for constructing new two and three point iterations with p 3 p 8 order of convergence this approach allows us to derive a new family of the optimal order iterative methods that include well known methods as special cases the necessary and sufficient conditions for pth order convergence of the proposed iterations are given in terms of parameters n and n we also propose some generating functions for n and n we give the extension of a class of optimal fourth order jarratt s type iterations with a 2 3 we develop a unified representation of all optimal eighth order methods several numerical results are given to demonstrate the efficiency and the performance of the presented methods and compare them with some other existing methods

an analysis of a khattri s 4th order family of methods in this paper we analyze an optimal fourth order family of methods suggested by khattri and babajee 2013 we analyze the family using the information on the extraneous fixed points two measures of closeness to the imaginary axis of the set of extraneous points are considered and applied to the members of the family to find its best performer the results are compared to three best members of king s family of methods

on a two step optimal steffensen type method relaxed local and semi local convergence analysis and dynamical stability abstract in this paper we study on the local and semi local convergence and dynamic of an iterative optimal two step steffensen s method we study local convergence by using lipschitz conditions and continuity of differentiable function f we also obtain semi local convergence by using majorizing sequences by this approach we can obtain the convergence of the method by minimum of auxiliary sequences and few initial conditions moreover we analyze the stability of the method via julia sets and parameter planes that is a novelty for steffensen type methods we extend this method for obtaining the sign function of a matrix numerical examples are given to confirm our theoretical results

comparison of several families of optimal eighth order methods several families of optimal eighth order methods to find simple roots are compared to the best known eighth order method due to wang and liu 2010 we have tried to improve their performance by choosing the free parameters of each family using two different criteria

basin attractors for various methods abstract there are many methods for the solution of a nonlinear algebraic equation the methods are classified by the order informational efficiency and efficiency index here we consider other criteria namely the basin of attraction of the method and its dependence on the order we discuss several methods of various orders and present the basin of attraction for several examples it can be seen that not all higher order methods were created equal newton s halley s murakami s and neta johnson s methods are consistently better than the others in two of the examples neta s 16th order scheme was also as good

different anomalies in a jarratt family of iterative root finding methods abstract in this paper the behavior of a jarratt family of iterative methods applied to quadratic polynomials is studied some anomalies are found in this family be means of studying the dynamical behavior of this fourth order family of methods parameter spaces are shown and the study of the stability of all the fixed points is presented dynamical planes for members with good and bad dynamical behavior are also provided

on the dynamics of a triparametric family of optimal fourth order multiple zero finders with a weight function of the principal mth root of a function to function ratio abstract under the assumption of known root multiplicity m n a triparametric family of two point optimal quartic order methods locating multiple zeros are investigated in this paper by introducing a weight function dependent on a function to function ratio special cases of weight functions with selected free parameters are considered and studied through various test equations and numerical experiments to support the theory developed in this paper in addition we explore the relevant dynamics of proposed methods via mobius conjugacy map when applied to a prototype polynomial z a m z b m the results of such dynamics are visually illustrated through a variety of parameter spaces as well as dynamical planes

on a certain quadrature formula traduction anglaise d un article en japonais publie en aout 1970 dans la serie kokyuroku vol 91 pp 82 118

asymptotic expansions and extrapolations of eigenvalues for the stokes problem by mixed finite element methods this paper derives a general procedure to produce an asymptotic expansion for eigenvalues of the stokes problem by mixed finite elements by means of integral expansion technique the asymptotic error expansions for the approximations of the stokes eigenvalue problem by bernadi raugel element and q 2 p 1 element are given based on such expansions the extrapolation technique is applied to improve the accuracy of the approximations

on a family of laguerre methods to find multiple roots of nonlinear equations there are several methods for solving a nonlinear algebraic equation having roots of a given multiplicity m here we compare a family of laguerre methods of order three as well as two others of the same order and show that euler cauchy s method is best we discuss the conjugacy maps and the effect of the extraneous roots on the basins of attraction