evaluation of the external force term in the discrete boltzmann equation a representation of the forcing term in the boltzmann equation based on a hermite expansion of the boltzmann distribution function in velocity phase space is derived based on this description of the forcing term a systematic comparison of previous lattice boltzmann models describing a nonideal gas behavior is given

prediction properties of aitken s iterated d 2 process of wynn s epsilon algorithm and of brezinski s iterated theta algorithm abstract the prediction properties of aitken s iterated 2 process wynn s epsilon algorithm and brezinski s iterated theta algorithm for formal power series are analyzed as a first step the defining recursive schemes of these transformations are suitably rearranged in order to permit the derivation of accuracy through order relationships on the basis of these relationships the rational approximants can be rewritten as a partial sum plus an appropriate transformation term a taylor expansion of such a transformation term which is a rational function and which can be computed recursively produces the predictions for those coefficients of the formal power series which were not used for the computation of the corresponding rational approximant

nonexistence of universally accelerating linear summability methods abstract let a amn b bmn be two regular toeplitz matrices suppose that for every convergent sequence sn its b transform b m where b m n 1bmnsn converges not slower than a m in the sens 0 e that b m s k a m s where s limn sn and k does not depend on m only on sn then a and b are equivalent i e they sum the same sequences even unbounded and moreover a and b essentially coincide there exists a number p such that amn bmn if m p

estimating perturbative coefficients in quantum field theory and statistical physics we present a method to estimate perturbative coefficients in quantum field theory and statistical physics we show that our method works in a large number of cases and in a wide variety of areas the method is so reliable that it has enabled us to find several errors in various publications

entropic lattice boltzmann method for microflows abstract a new method for the computation of flows at the micrometer scale is presented it is based on the recently introduced minimal entropic kinetic models both the thermal and isothermal families of minimal models are presented and the simplest isothermal entropic lattice bhatnagar gross krook elbgk is studied in detail in order to quantify its relevance for microflow simulations elbgk is equipped with boundary conditions which are derived from molecular models diffusive wall a map of three dimensional kinetic equations onto two dimensional models is established which enables two dimensional simulations of quasi two dimensional flows the elbgk model is studied extensively in the simulation of the two dimensional poiseuille channel flow results are compared to known analytical and numerical studies of this flow in the setting of the bhatnagar gross krook model the elbgk is in quantitative agreement with analytical results in the domain of weak rarefaction characterized by knudsen number kn the ratio of mean free path to the hydrodynamic scale up to kn 0 01 which is the domain of many practical microflows moreover the results qualitatively agree throughout the entire knudsen number range demonstrating knudsen s minimum for the mass flow rate at moderate values of kn as well as the logarithmic scaling at large kn the present results indicate that elbm can complement or even replace computationally expensive microscopic simulation techniques such as kinetic monte carlo and or molecular dynamics for low mach and low knudsen number hydrodynamics pertinent to microflows

discrete effects on boundary conditions for the lattice boltzmann equation in simulating microscale gas flows the lattice boltzmann equation lbe has shown its promise in the simulation of microscale gas flows one of the critical issues with this advanced method is to specify suitable slip boundary conditions to ensure simulation accuracy in this paper we study two widely used kinetic boundary conditions in the lbe the combination of the bounce back and specular reflection scheme and the discrete maxwell s scheme we show that i both schemes are virtually equivalent in principle and ii there exist discrete effects in both schemes a strategy is then proposed to adjust the parameters in the two kinetic boundary conditions such that an accurate slip boundary condition can be implemented the numerical results demonstrate that the corrected boundary conditions are robust and reliable

machine learning and nonparametric bandit theory in its most basic form bandit theory is concerned with the design problem of sequentially choosing members from a given collection of random variables so that the regret i e r sub n spl sigma sub j spl mu spl mu sub j et sub n j grows as slowly as possible with increasing n here spl mu sub j is the expected value of the bandit arm i e random variable indexed by j t sub n j is the number of times arm j has been selected in the first n decision stages and spl mu sup sup sub j spl mu sub j the present paper contributes to the theory by considering the situation in which observations are dependent to begin with the dependency is presumed to depend only on past observations of the same arm but later we allow that it may be with respect to the entire past and that the set of arms is infinite this brings queues and more generally controlled markov processes into our purview thus our black box methodology is suitable for the case when the only observables are cost values and in particular the probability structure and loss function are unknown to the designer the conclusion of the analysis is that under lenient conditions using algorithms prescribed herein risk growth is commensurate with that in the simplest i i d cases our methods represent an alternative to stochastic approximation perturbation analysis ideas for tuning queues

on the convergence of rows of vector pade approximants we state and prove a de montessus like theorem for vector valued meromorphic functions using vector pade approximants which are defined in the context of clifford algebras the proof is based on that used by saff for the scalar case the theorem may be regarded as an answer to a general problem posed by graves morris and saff

recursive solutions of estimating equations and adaptive spectral factorization a recursive spectral factorization algorithm was developed by v solo ibid vol 34 no 10 pp 1047 1051 oct 1989 to be used with recursive instrumental variables for consistent estimation of the parameters of an armax system but the convergence of the spectral factorization algorithm is questionable herein a modified adaptive spectral factorization algorithm is shown to converge and a general method for constructing convergent recursive solutions of nonlinear equations that are used to define parameter estimates is also presented

a survey of some results in stochastic adaptive control some results in discrete time stochastic adaptive control are surveyed the survey divides itself into two parts bayesian and non bayesian adaptive control in the former area the problems of converting an incompletely observed system into a completely observed one multi armed bandit processes bayesian adaptive control of markov chains and bayesian adaptive control of linear systems are exposed and surveyed in the latter area non bayesian adaptive control of markov chains and the self tuning regulator are dealt with proofs are given where appropriate to illustrate the methods involved

the summation of the ordinary and renormalized perturbation series for the ground state energy of the quartic sextic and octic anharmonic oscillators using nonlinear sequence transformations the strongly divergent rayleigh schrodinger perturbation expansion for the ground state energy of the quartic sextic and octic anharmonic oscillator and a corresponding renormalized perturbation expansion f vinette and j ci ek j math phys 32 3392 1991 are summed by pade approximants by levin s sequence transformation d levin int j comput math b 3 371 1973 and by a closely related sequence transformation which was suggested recently e j weniger comput phys rep 10 189 1989 it is shown that the renormalized perturbation expansion can be summed much more easily than the original perturbation expansion from which it was derived and that levin s sequence transformation diverges and is not able to sum the perturbation expansions the pade summation of the renormalized perturbation expansions gives relatively good results in the quartic and sextic case in the case of the octic anharmonic oscillator even the renormalized perturbation expansion is not pade summable the best results are clearly obtained by the new sequence transformation which for instance is able to sum the renormalized perturbation expansions for the infinite coupling limits of the quartic sextic and octic anharmonic oscillator and which produces at least in the quartic and sextic case extremely accurate results

adaptive control with recursive identification for stochastic linear systems this paper presents a stochastic adaptive control algorithm which is shown to possess the following properties when applied to a possibly unstable inverse stable linear stochastic system with unknown parameters whenever that system satisfies a certain positive real condition on its moving average noise dynamics 1 the adaptive control part of the algorithm stabilizes and asymptotically optimizes the behavior of the system in the sense that the limit of the sample mean square variation of the output around a given demand level equals that of a minimum variance control strategy implemented with known parameters this optimal behavior is subject to an offset 2where 2is the variance of a dither signal added to the control action in order to produce a continually disturbed control for mu 2 0 it is shown that the input output process satisfies a persistent excitation property and hence subject to a simple identifiability condition the next property holds 2 the observed input and output of the controlled system may be taken as inputs to an approximate maximum likelihood algorithm aml which generates strongly consistent estimates of the system s parameters results are presented for the scalar and multivariable cases

effective characteristic polynomials and two point pade approximants as summation techniques for the strongly divergent perturbation expansions of the ground state energies of anharmonic oscillators pad e approximants are able to sum effectively the rayleigh schr odinger perturbation series for the ground state energy of the quartic anharmonic oscillator as well as the corresponding renormalized perturbation expansion e j weniger j ifmmode check c else v c fi i ifmmode check z else v z fi ek and f vinette j math phys 34 571 1993 in the sextic case pad e approximants are still able to sum these perturbation series but convergence is so slow that they are computationally useless in the octic case pad e approximants are not powerful enough and fail on the other hand the inclusion of only a few additional data from the strong coupling domain e j weniger ann phys ny to be published greatly enhances the power of summation methods the summation techniques which we consider are two point pad e approximants and effective characteristic polynomials it is shown that these summation methods give good results for the quartic and sextic anharmonic oscillators and even in the case of the octic anharmonic oscillator which represents an extremely challenging summation problem two point pad e approximants give relatively good results textcopyright 1996 the american physical society

lower bounds for quartic anharmonic and double well potentials rigorous and remarkably accurate lower bounds to the lower eigenvalue spectrum of the schrodinger equation with quartic anharmonic and symmetric double well potentials of the form v a b ax2 2 bx4 b 0 are presented this procedure exploits some exactly soluble model potentials and appears to be of quite general utility

an acceleration theorem for the ρ algorithm the algorithm of wynn is an excellent device for accelerating the convergence of some logarithmically convergent sequences until now a convergence theorem and an acceleration theorem for the algorithm have not been obtained the purpose of this paper is to give an acceleration theorem for the algorithm moreover it is proved that the algorithm cannot accelerate linear convergence numerical examples are given

on the summation of some divergent hypergeometric series and related perturbation expansions abstract divergent hypergeometric series 2 f 0 1 occur frequently in poincare type asymptotic expansions of special functions these divergent series 2 f 0 can be used for the evaluation of the corresponding special functions if suitable summation technique are applied there is considerable evidence that levin s sequence transformation 1973 and in particular also some closely related sequence transformations weniger 1989 which were derived recently sum divergent series 2 f 0 much more efficiently than pade approximants similar summation problems occur also in the case of divergent rayleigh schrodinger perturbation expansions of elementary quantum mechanical systems a comparison of the perturbation series for the quartic anharmonic oscillator with the closely related asymptotic series for the complementary error function shows that levin s sequence transformation and the recently derived new sequence transformations are again more efficient than pade approximants however the superiority of levin s sequence transformation and of the new sequence transformations is less pronounced in the case of the perturbation expansion

introduction to the improved levin type algorithms for accelerating convergence of sequence we describe two new algorithms for accelerating the convergence of scalar sequence we express these new algorithms as levin transformation the definitions of these new algorithms namely the improved levin algorithm and the modified levin algorithm are given we examine the effectiveness of the improved levin algorithm and the modified levin algorithm together with the well established methods namely wynn s r algorithm and brezinski s q algorithm for approximating the partial sum of a given power series estimates of the partial sum produced by the improved levin algorithm are found to be substantially more accurate than the original levin transformation and the wynn s r algorithm and it is similar to the modified levin algorithm and the brezinski s q algorithm

simulating fluid flow over sinusoidal surfaces using the lattice boltzmann method the objective of this paper is to analyze the laminar flow of a newtonian fluid pressurized and sheared through a two dimensional channel with one sinusoidal wall the motivation of this work is the investigation of shear flows in lubrication the effects of reynolds number and the geometrical dimensions on the velocity distributions and on the flow factors are studied numerically using the lattice boltzmann method lbm the results are verified with analytical and other numerical solutions

stochastic truncation for the 2 1 d ising model the method of symmetrized stochastic truncation is applied to the 2 1 dimensional ising model to calculate the ground state energy mass gap magnetization and susceptibility on lattices of sizes up to 6 6 sites the method is shown to be a powerful and accurate monte carlo technique provided the lattice size is not too large and the model is suitably convergent finite size scaling estimates of the critical indices are nu 0 627 2 alpha 0 12 2 beta 0 324 3 and delta 1 23 1 an estimate of the combination gamma nu 2 beta nu 3 000 1 is obtained in outstanding agreement with the expected hyperscaling relation

introduction to the determinantal formulae for the levin type algorithms abstract there are two aims of this paper firstly we shall introduce the determinantal formulae for the new levin type algorithms and secondly we shall demonstrate the similarities between the new levin type algorithms and the well established methods namely the aitken 2 algorithm the brezinski theta algorithm and the lubkin transformation actually there are two groups of similarities first of the group demonstrates the similarity between the super enhanced levin algorithm the super modified levin algorithm the brezinski theta algorithm and the lubkin transformation the second group of similarity is between the modified levin algorithm the original levin transformation and the aitken 2 algorithm

development of the levin type algorithms for accelerating convergence of sequences abstract there are two aims of this paper firstly we shall introduce the determinantal representations of the new levin type algorithms and secondly we shall demonstrate further development of the levin type algorithms we consider the use of the levin type algorithms to accelerate the convergence of scalar sequence and their effectiveness for approximating the solution of a given power series is illustrated in process we shall demonstrate the convergence of each of the methods considered the approximate solution of the super enhanced levin algorithm and the efficient levin algorithm are found to be substantially more accurate than the cizek zamastil and skala transformation and the iterated aitken 2 algorithm

simulation of micro and nano scale flows via the lattice boltzmann method abstract we use the lattice boltzmann method lbm for analysis of high and moderate knudsen number phenomena simulation results are presented for microscale couette and poiseuille flows the slip velocity nonlinear pressure drop and mass flow rate are compared with previous numerical results and or experimental data the knudsen minimum is successfully predicted for the first time within the lbm framework these results validate the usage of the lbm based commercial arbitrary geometry code powerflow for simulating nanoscale problems

analytical and numerical studies of the convergence behavior of the j transformation abstract a new nonlinear sequence transformation the iterative j transformation was proposed recently by the author 1993 for this transformation a derivation based on the method of hierarchical consistency alternative recursive representations general properties an explicit expression for the kernel model sequences and its relation to other sequence transformations have been given the author 1994 the j transformation is of similar generality as the well known e algorithm brezinski 1980 havie 1979 in the present contribution some results on convergence acceleration properties of the j transformation are proved numerical test results are presented which show that the j transformation is a very powerful computational tool for convergence acceleration extrapolation and summation of divergent series

acceleration of some logarithmic sequences abstract delahaye and germain bonne 1982 proved that the set of logarithmic sequences log is not accelerable nevertheless some subsets of log can be accelerated most of the algorithms are based on the asymptotic expansion of the sequence to be accelerated in this paper the point of view is quite different since an analytic property of the whole sequence is used if cn n n is a sequence of moments with an integral representation in the interval 0 1 then we w prove that the jacobi modified moments accelerate cn n n section 2 in a particular case section 3 the modified sequence can be written as a moment sequence and a second modified sequence is formed leading to further acceleration the modified moments methods are particular summation processes using orthogonal polynomials which have been extensively studied by wimp 1974 1981

extrapolation methods for improving the convergence of oligomer calculations to the infinite chain limit of quasi one dimensional stereoregular polymers abstract quasi one dimensional stereoregular polymers as for example polyacetylene are currently of considerable interest both experimentally as well as theoretically there are basically two different approaches for doing electronic structure calculations one method the so called crystal orbital method uses periodic boundary conditions and is essentially based on concepts of solid state theory the other method is essentially a quantum chemical method since it approximates the polymer by oligomers consisting of a finite number of monomer units i e by molecules of finite size in this way the highly developed technology of quantum chemical molecular programs can be used unfortunately oligomers of finite size are not necessarily able to model those features of a polymer which crucially depend on its in principle infinite extension however in such a case extrapolation techniques can be extremely helpful for example one can perform electronic structure calculations for a sequence of oligomers with an increasing number of monomer units in the next step one then can try to determine the limit of this sequence for an oligomer of infinite length with the help of suitable extrapolation methods several different extrapolation methods are discussed which are able to accomplish an extrapolation of energies and properties of oligomers to the infinite chain limit calculations for the ground state energy of polyacetylene are presented which demonstrate the practical usefulness of extrapolation methods

analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice boltzmann bgk model in this paper we analytically solve the velocity of the lattice boltzmann bgk equation lbgk for several simple flows the analysis provides a framework to theoretically analyze various boundary conditions in particular the analysis is used to derive the slip velocities generated by various schemes for the nonslip boundary condition we find that the slip velocity is zero as long as f e 0 at boundaries no matter what combination of distributions is chosen the schemes proposed by nobleet al and by inamuroet al yield the correct zeroslip velocity while some other schemes such as the bounce back scheme and the equilibrium distribution scheme would inevitably generate a nonzero slip velocity the bounce back scheme with the wall located halfway between a flow node and a bounce back node is also studied for the simple flows considered and is shown to produce results of second order accuracy the momentum exchange at boundaries seems to be highly related to the slip velocity at boundaries to be specific the slip velocity is zero only when the momentum dissipated by boundaries is equal to the stress provided by fluids

prediction properties of the t transformation some prediction properties of the t transformation of levin are analyzed for two kinds of limiting processes it is shown under certain circumstances that when the t transformation is applied to a power series the coefficients of the maclaurin series of the rational approximation obtained approximate those of the given series with increasing accuracy theoretical and numerical examples are appended

on l 1 approximation preface 1 preliminaries 2 approximation from finite dimensional subspaces of l1 3 approximation from finite dimensional subspaces in c1 k 4 unicity subspaces and property a 5 one sided l1 approximation 6 discrete lm1 approximation 7 algorithms appendices references author index subject index

convergence acceleration of logarithmic fixed point sequences let x n be some sequence generated by x n 1 f x n where f x email protected i email protected p ix p i p 1 a p 1 0 small it converges to zero logarithmically i e lim n x n l x n 1 thus we need algorithms for accelerating its convergence using asymptotic expansions in the analysis of the d 2 and q 2 algorithms leads to modified iterated versions of the first one and to combinations with the iterated q 2 algorithm in particular some superconvergence phenomena can be explained in this framework a similar study can be made for other nonlinear algorithms known at present moreover the above algorithms are also good accelerators for large classes of slowly convergent integrals and series

convergence acceleration methods the past decade abstract the aim of this paper is to present a review of the most significant results obtained the past ten years on convergence acceleration methods it is divided into four sections dealing respectively with the theory of sequence transformations the algorithms for such transformations the practical problems related to their implementation and their applications to various subjects of numerical analysis

acceleration of convergence of vector sequences a general approach to the construction of convergence acceleration methods for vector sequences is proposed using this approach one can generate some known methods such as the minimal polynomial extrapolation the reduced rank extrapolation and the topological epsilon algorithm and also some new ones some of the new methods are easier to implement than the known methods and are observed to have similar numerical properties the convergence analysis of these new methods is carried out and it is shown that they are especially suitable for accelerating the convergence of vector sequences that are obtained when one solves linear systems of equations iteratively a stability analysis is also given and numerical examples are provided the convergence and stability properties of the topological epsilon algorithm are likewise given

vector sequence transformations for the acceleration of logarithmic convergence logarithmic convergence of scalar sequences is generalized to vector sequences some sequence transformations for logarithmically convergent scalar sequences are also extended to the vector case theoretical results and numerical examples are given

numerical comparisons of nonlinear convergence accelerators as part of a continuing program of numerical tests of convergence accelerators we have compared the iterated aitken s a2 method wynn s algorithm brezinski s 0 algorithm and levin s u transform on a broad range of test problems linearly convergence alternating monotone and irregular sign series logarithmically convergent series power method and bernoulli method sequences alternating and monotone asymptotic series and some perturba tion series arising in applications in each category either the algorithm or the u transform gives the best results of the four methods tested in some cases differences among methods are slight and in others they are quite striking 1 introduction we reported in 19 on a numerical test program that compared eleven different scalar convergence accelerators on a broad range of infinite series problems in an attempt to find an across the board accelerator that would handle both linear and logarithmic convergence and both alternating and monotone series or sequences the best candidates to emerge from that study were levin s u transform 14 and brezinski s 0 algorithm 6 we also provided some theoretical support for that conclusion in terms of the wide range of series summed exactly by these methods and by proving that 02 and the levin transforms all accelerate linear convergence the other levin transforms t and v produced essentially equivalent numerical results to u except that t fails on logarithmic convergence we have continued the test program started with 19 to include monotone linearly convergent series series with irregular sign patterns specifically fourier series and sequences whose terms do not depend explicitly on the previous term and or the index e g bernoulli s method since convergence accelerators are often known to transform divergent series to convergent ones we have also included in our test program divergent power series asymptotic series and a few perturbation series the methods considered in this paper are the u and 0 algorithms noted above and

an algorithm for a generalization of the richardson extrapolation process in this paper we present a recursive method designated the w m algorithm for implementing a generalization of the richardson extrapolation process that has been ntroduced in 8 compared to the direct solution of the linear systems of equations defining the extrapolation procedure this method requires a small number of arithmetic operations and very little storage the technique is also applied to solve recursively the coefficient problem associated with the rational approximations obtained by applying the d transformation of 6 13 to power series in the course of development a new recursive algorithm for implementing a very general extrapolation procedure is introduced which is similar to that given in 2 4 for solving the same problem a fortran program for the w m algorithm is also appended

application of post s formula to optical pulse propagation in dispersive media in this paper we consider the propagation of optical pulses in dielectric media with nontrivial dispersion relations in particular we implement post s laplace transform formula to invert in time the fourier laplace space coefficients which arise from the joint space solution of the optical dispersive wave equation due to the inefficiency of a direct application of this formula we have considered and present here two more efficient implementations in the first the gaver post method we utilize the well known gaver functionals but store intermediate calculations to improve efficiency the second the bell post method involves an analytic reformulation of post s formula such that knowledge of the dispersion relation and its derivatives are sufficient to invert the coefficients from laplace space to time unlike other approaches to the dispersive wave equation which utilize a debye lorentzian assumption for the dispersion relation our algorithm is applicable to general maxwell hopkinson dielectrics moreover we formulate the approach in terms of the fourier laplace coefficients which are characteristic of a dispersive medium but are independent of the initial pulse profile they thus can be precomputed and utilized when necessary in a real time system finally we present an illustration of the method applied to optical pulse propagation in a free space and in two materials with cole type dispersion relations room temperature ionic liquid rtil hexafluorophosphate and brain white matter from an analysis of these examples we find that both methods perform better than a standard post formula implementation the bell post method is the more robust of the two while the gaver post is more efficient at high precision and post formula approximation orders

scalar levin type sequence transformations sequence transformations are important tools for the convergence acceleration of slowly convergent scalar sequences or series and for the summation of divergent series the basic idea is to construct from a given sequence sn a new sequence s t sn where each sdepends on a finite number of elements sn1 snm often the sn are the partial sums of an infinite series the aim is to find transformations such that s converges faster than or sums sn transformations t sn n that depend not only on the sequence elements or partial sums sn but also on an auxiliary sequence of so called remainder estimates n are of levin type if they are linear in the sn and nonlinear in the n such remainder estimates provide an easy to use possibility to use asymptotic information on the problem sequence for the construction of highly efficient sequence transformations as shown first by levin it is possible to obtain such asymptotic information easily for large classes of sequences in such a way that the n are simple functions of a few sequence elements sn then nonlinear sequence transformations are obtained special cases of such levin type transformations belong to the most powerful currently known extrapolation methods for scalar sequences and series here we review known levin type sequence transformations and put them in a com mon theoretical framework it is discussed how such transformations may be constructed by either a model sequence approach or by iteration of simple transformations as illus tration two new sequence transformations are derived common properties and results on convergence acceleration and stability are given for important special cases extensions of the general results are presented also guidelines for the application of levin type sequence transformations are discussed and a few numerical examples are given keywords convergence acceleration extrapolation summation of divergent se ries stability analysis hierarchical consistency iterative sequence transformation levin type transformations algorithm linear convergence logarithmic con vergence fourier series power series rational approximation subject classifications ams mos 65b05 65b10 65b15 40a05 40a25 42c15

mathematical properties of a new levin type sequence transformation introduced by cižek zamastil and skala i algebraic theory ci ek zamastil and skala j math phys 44 962 968 2003 introduced in connection with the summation of the divergent perturbation expansion of the hydrogen atom in an external magnetic field a new sequence transformation which uses as input data not only the elements of a sequence sn n 0 of partial sums but also explicit estimates n n 0 for the truncation errors the explicit incorporation of the information contained in the truncation error estimates makes this and related transformations potentially much more powerful than for instance pade approximants special cases of the new transformation are sequence transformations introduced by levin int j comput math b 3 371 388 1973 and weniger comput phys rep 10 189 371 1989 secs 7 9 numer algor 3 477 486 1992 and also a variant of richardson extrapolation philos trans r soc london ser a 226 299 349 1927 the algebraic theory of these transformations explicit expressions recurrence formulas explicit expres

vector extrapolation methods applications and numerical comparison the present paper is a survey of the most popular vector extrapolation methods such as the reduced rank extrapolation rre the minimal polynomial extrapolation mpe the modified minimal polynomial extrapolation mmpe the vector e algorithm vea and the topological e algorithm tea using projectors we derive a different interpretation of these methods and give some theoretical results the second aim of this work is to give a numerical comparison of the vector extrapolation methods above when they are used for practical large problems such as linear and nonlinear systems of equations

error analysis of aitken s δ2 process abstract a sequence to sequence transformation called the 2 process by its developer aitken and recently analyzed by daniel shanks is successful in accelerating convergence in many convergent sequences and inducing convergence in some divergent ones it is shown here that the 2 process applied to sequences whose terms have cauchy distributions results in sequences whose terms still have the cauchy distribution and that repeated applications of the 2 process to a sequence with terms having uniform distribution simulating round off error and to sequences with terms having a normal distribution simulating measurement error yields in both cases sequences whose terms approach the cauchy distribution the result for the uniform distribution is proven that for the normal distribution is referenced

a convergence acceleration method for some logarithmically convergent sequences a convergence acceleration method with one parameter for a sequence s n satisfying s n sim s n theta left c 0 frac c 1 n frac c 2 n 2 cdots right quad text as n to infty is proposed when the parameter is equal to 1 the method agrees with wynn s rho algorithm hence the method is designated as a generalized rho algorithm when s n is the nth partial sum of a series whose nth term is asymptotic to an theta 1 where theta is negative the generalized rho algorithm with parameter theta agrees essentially with the modified wynn s rho algorithm taken up by drummond error estimates and numerical examples of the generalized rho algorithm with parameter theta are given the exponent theta in the expansion above can be estimated using the generalized rho algorithm with parameter 2 in practice by estimating the exponent theta the generalized rho algorithm can be automatically applied to

the set of logarithmically convergent sequences cannot be accelerated some theorems pennacchi germain bonne smith and ford state that methods of a certain form which are exact on geometric sequences accelerate linear convergence but no corresponding theorem is known for logarithmic convergence our study shows the reason why there is no algorithm which can accelerate all logarithmically convergent sequences we obtain this result with a generalization of remanence which is a sufficient property for a set of sequences to be unaccelerable

convergence acceleration by extraction of linear subsequences logarithmic sequences are usually difficult to accelerate in this paper we show how to extract a subsequence converging linearly this subsequence can be easily accelerated by classical algorithms theoretical results are given as well as an algorithmic extraction procedure numerical examples illustrate the results

numerical evidence that the perturbation expansion for a non hermitian pt symmetric hamiltonian is stieltjes recently several studies of non hermitian hamiltonians having pt symmetry have been conducted most striking about these complex hamiltonians is how closely their properties resemble those of conventional hermitian hamiltonians this paper presents further evidence of the similarity of these hamiltonians to hermitian hamiltonians by examining the summation of the divergent weak coupling perturbation series for the ground state energy of the pt symmetric hamiltonian h p2 14x2 i x3 recently studied by bender and dunne for this purpose the first 193 nonzero coefficients of the rayleigh schrodinger perturbation series in powers of 2 for the ground state energy were calculated pade summation and pade prediction techniques recently described by weniger are applied to this perturbation series the qualitative features of the results obtained in this way are indistinguishable from those obtained in the case of the perturbation series for the quartic anharmonic oscillator which is known to be a stieltjes se

accelerable subsets of logarithmic sequences abstract the aim of this paper is to study what the accelerable subsets of logsf of all logarithmically convergent sequences are three kinds of subsets of logsf and sequence transformations are given theoretical results and numerical examples are given

a simultaneous process for convergence acceleration and error control abstract let xn be a real sequence converging to the limit x and such that x n x n 1 x n n n v i 0 a i n i a0 0 0 for n large enough here we propose a repeated process using two transformations namely aitken s 2 process and brezinski s 2 algorithm which provide a sequence of intervals asymptotically containing x in particular we introduce for 1 a version of the modified iterated 2 process based on the convergence orders so generalizing a method proposed by sablonniere we establish that the order vp used in the pth iteration is vp 1 kp 1 1 kp 1 3 being an integer we give an estimation method for vp confirming the choice of kp 1

acceleration methods for vector sequences the purpose of this paper is to propose acceleration methods for certain logarithmically convergent vector sequences these methods are obtained by extending acceleration methods for scalars lubkin s w transform the algorithm and the algorithm applied to a system of singular equations new methods are compared with the vector algorithm the minimal polynomial extrapolation and the generalized algorithm of brezinski theoretical results are also given

the epsilon algorithm and related topics the epsilon algorithm is recommended as the best all purpose acceleration method for slowly converging sequences it exploits the numerical precision of the data to extrapolate the sequence to its limit we explain its connections with pade approximation and continued fractions which underpin its theoretical base then we review the most recent extensions of these principles to treat application of the epsilon algorithm to vector valued sequences and some related topics in this paper we consider the class of methods based on using generalised inverses of vectors and the formulation specifically includes the complex case wherever possible

a multiplicative decomposition of poiseuille number on rarefaction and roughness by lattice boltzmann simulation poiseuille number of rarefied gas flow in channels with designed roughness is studied and a multiplicative decomposition of poiseuille number on the effects of rarefaction and roughness is proposed the numerical methodology is based on the mesoscopic lattice boltzmann method in order to eliminate the effect of compressibility the incompressible lattice boltzmann model is used and the periodic boundary is imposed on the inlet and outlet of the channel the combined bounced condition is applied to simulate the velocity slip on the wall boundary numerical results reveal the two opposite effects that velocity gradient and friction factor near the wall increase as roughness effect increases meanwhile the increments of the rarefaction effect and velocity slip lead to a corresponding decrement of friction factor an empirical relation of poiseuille number which contains the two opposite effects and has a better physical meaning is proposed in the form of multiplicative decomposition and then is validated by available experimental and numerical results

parallel recursive algorithms in asymptotically efficient adaptive control of linear stochastic systems first a review of some recent developments in stochastic adaptive control of linear stochastic systems is given by integrating and refining several basic ideas in these developments a relatively complete asymptotic solution to the adaptive control problem is then provided for such systems the solution involves parallel implementation of a few basic recursive identification algorithms that serve to complement each other it also involves occasional use of a dither signal to probe the system for information

aitken sequences and generalized fibonacci numbers consider the sequence t generated by t i avn bv l l n 5i 2 where vi 1 t 2 a with a and b real of which the fibonacci sequence is a special case it is shown that if aitken acceleration is used on the sequence x defined by a v 1 v the resulting sequence is a subsequence of jc second if newton s method and the secant method are used with suitable starting values to solve the equation x2 ax b 0 then the sequences obtained from both of those methods are also subsequences of the original sequence

comparison of sequence accelerators forthe gaver method of numerical laplace transform inversion the sequence of gaver functionals is useful in the numerical inversion of laplace transforms the convergence behavior of the sequence is logarithmic therefore an acceleration scheme is required the accepted procedure utilizes salzer summation because in many cases the gaver functionals have the asymptotic behavior n t n t an 2 as n for fixed t it seems that no other acceleration schemes have been investigated in this area surely the popular nonlinear methods should be more effective however to our surprise only one nonlinear method was superior to salzer summation namely the wynn rho algorithm

comparison of four nonlinear transforms on some classes of logarithmic fixed point sequences abstract let log be the set of logarithmic sequences i e of convergent sequences sn satisfying lim e n 1 e n 1 where en sn s and s limsn this paper is devoted to the comparison of four nonlinear sequence transforms on some subsets lf r of log for any positive integer r lf r denotes the set of fixed point sequences sn whose associated error sequences en sn s s limsn have an asymptotic expansion of the following type en 1 en i 1 ien1 ir where 1 c 0 r 1 2 r 1 2 1 2 is different from zero for sn lf r there holds e n a 1 n 1 r o n 1 1 r log n with a 1 r 1 1 r the four are respectively modifications of the algorithm and of the iterated 2 transform the iterated lubkin s transform and the algorithm of brezinski all of them accelerate the convergence of sequences in lf r we give accurate results concerning the asymptotic behaviour of transformed sequences we also study two algorithms combining the second and third transforms numerical examples are given to illustrate theoretical results

acceleration of convergence of a family of logarithmically convergent sequences the asymptotic behavior of several sequence transformations is in vestigated as n oo when applied to a certain family of logarithmically convergent sequences the transformations considered are the iterations of the transformations e an of shanks and wn of lubkin the algorithm of brezinski the levin u and u transforms and generalizations of the p algorithm and the neville table computational results are given for both real and complex sequences

numerical analysis 2000 vol ii interpolation and extrapolation multivariate polynomial interpolation is a basic and fundamental subject in approximation theory and numerical analysis which has received and continues receiving not deep but constant attention in this short survey we review its development in the rst 75 years of this century including a pioneering paper by kronecker in the 19th century c 2000 elsevier science b v all rights reserved