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

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

best approximation of alternating power series abstract this paper contains results in which the author extends the work of fiedler jurkat and himself from series of the form 1 n a n to series of the form x n a n where a n is moment sequence and 0 x l p approximation to the sum of the series for p 2 1 specifically find the error in each case calculate the actual terms of the matrices involved explicitly and find the gaussian distributions of the matrices

acceleration of convergence of general linear sequences by the shanks transformation the shanks transformation is a powerful nonlinear extrapolation method that is used to accelerate the convergence of slowly converging and even diverging sequences a n it generates a two dimensional array of approximations a j n to the limit or anti limit of a n defined as solutions of the linear systems a l a j n sum n k 1 bar beta k delta a l k 1 j leq l leq j n where bar beta k are additional unknowns in this work we study the convergence and stability properties of a j n as j with n fixed derived from general linear sequences a n where a n sim a sum m k 1 zeta k n sum infty i 0 beta ki n gamma k i as n where k 1 are distinct and 1 m and k 0 1 2 here a is the limit or the anti limit of a n such sequences arise for example as partial sums of fourier series of functions that have finite jump discontinuities and or algebraic branch singularities we show that definitive results are obtained with those values of n for which the integer programming problems begin array ll quad quad quad quad max limits s 1 ldots s m sum limits k 1 m left re gamma k s k s k s k 1 right rm subject to s 1 geq0 ldots s m geq0 quad rm and quad sum limits k 1 m s k n end array have unique integer solutions for s 1 s m a special case of our convergence result concerns the situation in which re gamma 1 cdots re gamma m alpha and n m with 1 2 for which the integer programming problems above have unique solutions and it reads a j n a o theta j j alpha 2 nu as j when compared with a j a o j j as j this result shows that the shanks transformation is a true convergence acceleration method for the sequences considered in addition we show that it is stable for the case being studied and we also quantify its stability properties the results of this work are the first ones pertaining to the shanks transformation on general linear sequences with m 1

complexity issues in natural gradient descent method for training multilayer perceptrons the natural gradient descent method is applied to train an n m 1 multilayer perceptron based on an efficient scheme to represent the fisher information matrix for an n m 1 stochastic multilayer perceptron a new algorithm is proposed to calculate the natural gradient without inverting the fisher information matrix explicitly when the input dimension n is much larger than the number of hidden neurons m the time complexity of computing the natural gradient is o n

a new fuzzy model of pattern recognition and hitch diagnoses of complex systems in this paper we give a new fuzzy model of pattern recognition and hitch diagnoses of complex systems which are based on afs structures ei algebras and eii algebras this model can be applied to the recognition and the diagnoses of complex systems whose relative sets are countable

an accurate numerical inversionof laplace transforms based on the location of their poles we introduce an efficient and easily implemented numerical method for the inversion of laplace transforms using the analytic continuation of integrands of bromwich s integrals after deforming the bromwich s contours so that it consists of the union of small circles around singular points we evaluate the bromwich s integrals by quadrature rules we prove that the error bound of our method has spectral accuracy of type n eps p 0 1 and provide several numerical examples

numerical integration of related hankel transforms by quadrature and continued fraction expansion an algorithm is presented for the accurate evaluation of hankel or bessel transforms of algebraically related kernel functions defined here as the non bessel function portion of the integrand that is more widely applicable than the standard digital filter methods without enormous increases in computational burden the algorithm performs the automatic integration of the product of the kernel and bessel functions between the asymptotic zero crossings of the latter and sums the series of partial integrations using a continued fraction expansion equivalent to an analytic continuation of the series the integrands may be saved to allow the rapid computation of related transforms without recalculating the kernel or bessel functions and the integration steps use interlacing quadrature formulas so that no function evaluations are wasted when it is necessary to increase the order of the quadrature rule the continued fraction algorithm allows very slowly divergent or even formally divergent integrals to be computed quite easily the local error is controlled at each step in the algorithm and accuracy is limited largely by machine resolution the algorithm is written in fortran and is listed in an appendix along with a driver program that illustrates its features the driver program and subroutine are available from the seg business offtce

compatibility conditions for time dependent partial differential equations and the rate of convergence of chebyshev and fourier spectral methods abstract compatibility conditions for partial differential equations pdes are an infinite set of relations between the initial conditions the pde and the boundary conditions which are necessary and sufficient for the solution to be c that is infinitely differentiable everywhere on the computational domain including the boundaries since the performance of chebyshev spectral and spectral element methods is dramatically reduced when the solution is not c one would expect that the compatibility conditions would be a major theme in the spectral literature instead it has been completely ignored therefore we pursue three goals here first we present a proof of the compatibility conditions in a simplified form that does not require functional analysis second we analyze the connection between the compatibility conditions and the rate of convergence of chebyshev methods lastly we describe strategies for slightly adjusting initial conditions so that the compatibility conditions are satisfied

sum accelerated pseudospectral methods finite differences and sech weighted differences this work continues our previous studies of algorithms for accelerating the convergence of pseudospectral derivative series in order to obtain new differentiation schemes which have the sparsity and low cost of finite differences but the accuracy of spectral methods we develop a general theoretical framework for difference schemes finite differences are close to optimum but can be bettered by a new scheme we have dubbed sech weighted differences through numerical examples we show that sech weighted differences are effective in contrast non linear accelerations like pade approximants and the levin u transform so popular in other applications are inferior for approximating derivatives to linear accelerations like finite differences sech weighted differences and the euler method

best approximation of positive power series this paper extends work of fiedler jurkat and the present author to series of the form la x where a is a moment sequence and 0 and f ty 1 ja jri in this paper we consider series of the form e anx with 0 x 1 where an is as above the results obtained here are similar to those in 4 we show the best approximation explicitly for the cases in which it can be calculated in particular in the notation of 4 we show that the error of the best l approximation is given by

new convergence results on the generalized richardson extrapolation process grep 1 for logarithmic sequences let a t a t i 0 iti as t 0 where a t and t are known for 0 0 but a and the i are not known the generalized richardson extrapolation process grep 1 is used in obtaining good approximations to a the limit or antilimit of a t as t 0 the approximations an j to a obtained via grep 1 are defined by the linear systems a tl an j tl i 0n 1 itli l j j 1 j n where tl l 0 is a decreasing positive sequence with limit zero the study of grep 1 for slowly varying functions a t was begun in two recent papers by the author for such a t we have t t as t 0 with possibly complex and 0 1 2 in the present work we continue to study the convergence and stability of grep 1 as it is applied to such a t with different sets of collocation points tl that have been used in practical situations in particular we consider the cases in which i tl are arbitrary ii liml tl 1 tl 1 iii tl cl q as l for some c q 0 iv tl 1 tl 0 1 for all l v liml tl 1 tl 0 1 and vi tl 1 tl 0 1 for all l

the numerical evaluation of very oscillatory infinite integrals by extrapolation recently the author has given two modifications of a nonlinear extrapolation method due to levin and sidi which enable one to accurately and economically compute certain infinite integrals whose integrands have a simple oscillatory behavior at infinity in this work these modifications are extended to cover the case of very oscillatory infinite integrals whose integrands have a complicated and increasingly rapid oscillatory behavior at infinity the new method is applied to a number of complicated integrals among them the solution to a problem in viscoelasticity some convergence results for this method are presented

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

the weighted averages algorithm revisited the classic weighted averages wa algorithm for the evaluation of sommerfeld like integrals is reviewed and reappraised as a result a new version of the wa algorithm called generalized wa is introduced the new version can be considered as a generalization of the well established holder and cesaro means used to sum divergent series generalized wa exhibits a more compact formulation devoid of iterative and recursive steps and a wider range of applications it is more robust as it provides a unique formulation valid for monotonic and oscillating functions the implementation of the new version is easier and more economical in terms of basic operations preliminary numerical examples show that generalized wa also outperforms in terms of accuracy the classic wa algorithm which is currently recognized as the most competitive algorithm to evaluate sommerfeld integral tails

a numerical method for electromagnetic scattering by 3 d dielectric objects buried under 2 d locally rough surfaces in this work a numerical method is developed to solve a forward scattering problem of three dimensional 3 d objects buried into two half space media with two dimensional 2 d locally rough interface the proposed method exploits the electric field integral equation efie formulation of the scattered field which involves the dyadic green s function of the background medium since the required green s function has no analytical expression in general buried object approach boa is applied to calculate it numerically method of moments mom is then used to solve the efie the presented method is numerically validated by comparing it with finite difference time domain fdtd method furthermore an elaborative numerical analysis of the proposed method is presented to show its feasibility for various roughness profiles and buried objects

convergence speeding convergence and summability abstract four general principles are proposed for use in constructing convergence speeding methods five old methods and about 15 more recent non linear methods are discussed with reference to their range of applicability and rate of convergence speeding when applied to alternating monotonic slightly erratic fourier and alternating divergent series several of these methods are proved to give analytic continuations while even asymptotic series with zero radius of convergence can be summed accurately divergent series of positive terms may also in some cases be interpreted with moderate accuracy using four different methods convergence tests are reviewed in the light of these convergence speeding methods and analytic continuation

analysis of a plane wave excited subwavelength circular aperture in a planar conducting screen illuminating a multilayer uniaxial sample a spectral domain analysis is presented of a plane wave excited subwavelength circular aperture in a planar perfectly conducting screen in close proximity to a multilayer stack which may comprise uniaxially anisotropic slices the formulation employs the bethe bouwkamp aperture model for the general case of oblique incidence and it combines the hankel transform method with a transmission line network analogue of the medium which facilitates the inclusion of any number of material layers the electromagnetic fields are expressed in terms of one dimensional 1 d hankel transform integrals which are efficiently evaluated by a numerical quadrature along a suitably selected path with convergence acceleration by extrapolation sample numerical results are presented to demonstrate the validity of the method and its utility in the modeling of multilayer near field lenses and aperture type probes of near field optical microscopy

singularities affect dynamics of learning in neuromanifolds the parameter spaces of hierarchical systems such as multilayer perceptrons include singularities due to the symmetry and degeneration of hidden units a parameter space forms a geometrical manifold called the neuromanifold in the case of neural networks such a model is identified with a statistical model and a riemannian metric is given by the fisher information matrix however the matrix degenerates at singularities such a singular structure is ubiquitous not only in multilayer perceptrons but also in the gaussian mixture probability densities arma time series model and many other cases the standard statistical paradigm of the cramer rao theorem does not hold and the singularity gives rise to strange behaviors in parameter estimation hypothesis testing bayesian inference model selection and in particular the dynamics of learning from examples prevailing theories so far have not paid much attention to the problem caused by singularity relying only on ordinary statistical theories developed for regular nonsingular models only recently have researchers remarked on the effects of singularity and theories are now being developed this article gives an overview of the phenomena caused by the singularities of statistical manifolds related to multilayer perceptrons and gaussian mixtures we demonstrate our recent results on these problems simple toy models are also used to show explicit solutions we explain that the maximum likelihood estimator is no longer subject to the gaussian distribution even asymptotically because the fisher information matrix degenerates that the model selection criteria such as aic bic and mdl fail to hold in these models that a smooth bayesian prior becomes singular in such models and that the trajectories of dynamics of learning are strongly affected by the singularity causing plateaus or slow manifolds in the parameter space the natural gradient method is shown to perform well because it takes the singular geometrical structure into account the generalization error and the training error are studied in some examples

a lag averaged generalization of euler s method for accelerating series abstract euler s method is an efficient accelerator for an alternating series that is one whose terms a j are such that sign a j sign a j 1 however its effectiveness deteriorates rapidly as the period p of the oscillations in degree j increases to reduce this nonuniformity we introduce an algorithm which begins with the recursive computation of a triangular array of numbers the diagonal elements define a generalization of euler s method each horizontal row defines a delayed euler method which is even more effective the key element is that the array is generated by repeatedly averaging partial sums of the original series with a lag that is a distance in degree between the partial sums that are combined which optimally is half the period p of the oscillations in the series we illustrate the algorithm with the fourier series for a function with a shock wave jump discontinuity the terms a j 1 j sin jx vary from strictly alternating p 2 at x 0 to monotonic p as x the lag averaged euler scheme with and without delay is much more efficient than its classical counterpart especially near the discontinuities we present an analytical convergence theory for both the standard and generalized euler methods

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

applications of axiomatic fuzzy set clustering method on management strategic analysis abstract in this paper first the axiomatic fuzzy set afs clustering method x d liu w wang t y chai the fuzzy clustering analysis based on afs theory ieee transactions on systems man and cybernetics part b 35 5 2005 1013 1027 is investigated further by improving the algorithm to be more applicable then it is used to analyze the evaluation results of 30 companies which have been studied and analyzed by g s liang et al g s liang t y chou t c han cluster analysis based on fuzzy equivalence relation european journal of operational research 166 2005 160 171 compared with liang s algorithm the proposed method is more transparent understandable and interpretable this method can be applied to the management strategic analysis based on the datasets described by mixed features such as real numbers boolean logical values linguistic descriptions the illustrative examples show that the interpretations of the clustering results of the 30 companies are almost consistent with the expert s intuitions

the fuzzy theory based on afs algebras and afs structure abstract in this paper we define a more general algebra class called afs algebras which includes ei algebra and eii algebra published by the author which are completely distributive molecular lattices we establish the fuzzy theory using afs algebras and afs structure published by the author and propose the concept of fuzzy variable and a simple controlling model based on afs theory

perturbative solution for the generalised anharmonic oscillators serious difficulties have been encountered in the past when attempting the straightforward application of the borel method to the summation of perturbation series in quantum mechanics and quantum field theory because of an analytic continuation of the borel transform series which is usually performed by means of conformal mapping techniques the author shows that this continuation may be accomplished by combining the borel method with the first confluent form of the epsilon algorithm of wynn 1960 1968 for the calculation of the ground state energy eigenvalue of the generalised anharmonic oscillators two different approaches are presented both of which lead to accurate results

application of dcim to mpie mom analysis of 3d pec objects in multilayered media the formulation c green s functions for multilayered media are reformulated in this paper to extract z dependent part when the source and observation points are in different layers the discrete complex image method dcim is applied to obtain complex images independent on z but dependent on z spl acute for simulation of three dimensional 3 d perfectly electrical conduction pec objects penetrating two or more layers tables of complex images are built in a z spl acute grid during the matrix filling stage of the mixed potential integral equation together with method of moments mpie mom the spatial domain green s functions are interpolated from values in the z spl acute grid while these values are calculated from complex images in the z spl acute grid this interpolation scheme is more efficient than those directly storing the spatial domain green s functions numerical examples are given for analyzing a dipole in a two layered media

some new convergence acceleration methods abstract the algorithm is a general extrapolation method which includes most of thesequence transformations actually known some new convergence acceleration methods arederived from the f algorithm by applying the so called procedure the algorithms thusobtained are studied some theoretical results are proved and numerical examples are given 1 introduction the general idea for accelerating the convergence of a slowlyconvergent sequence is to transform it into another sequence which will undercertain assumptions converge faster to the same limit among the most famoussequence transformations are linear summation processes richardson polynomialextrapolation aitken s a2 process 1 and shanks transformation 15 wynn s algorithm 17 an extensive study of these methods can be found in 3 whilenumerical examples applications and fortran subroutines are given in 4 each of these methods usually works in some particular cases since a universalmethod accelerating the convergence of wide classes of sequences cannot exist 10 however numerical examples 11 16 and also some theoretical results 6 8 revealthat the best sequence transformations that is those working in most of theexamples seem to be levin s u and v transforms 14 and the algorithm 2 the algorithm is a method derived from the rule of the e algorithm by a procedurewhich will be recalled in the next section recently a general sequence transformation as well as the corresponding recursivealgorithm was obtained by havie 13 and brezinski 5 this transformation calledthe transformation includes most of the convergence acceleration methods actu ally known thus it seemed interesting to apply to the algorithm the sameprocedure which was used to derive the algorithm from the e algorithm and tostudy the new convergence acceleration methods obtained in that way it is the aimof this paper 2 the procedure 8 let us first begin by some reminiscences about shanks transformation and the e algorithm let sn be a sequence of real or complex numbers shanks transformationconsists in transforming sn into a set of sequences ek sn given by

computation of the sommerfeld integral tails using the matrix pencil method the oscillating infinite domain sommerfeld integrals si are difficult to integrate using a numerical procedure when dealing with structures in a layered media even though several researchers have attempted to do that generally integration along the real axis is used to compute the si however significant computational effort is required to integrate the oscillating and slowly decaying function along the tail extrapolation methods are generally applied to accelerate the rate of convergence of these integrals however there are difficulties with the extrapolation methods such as locations for the breakpoints in this paper we illustrate a simplified approach for accurate and efficient calculation of the integrals dealing with the tails of the si in this paper we fit the tail by a sum of finite usually 10 to 20 complex exponentials using the matrix pencil method mpm the integral of the tail of the si is then simply calculated by summing some complex numbers no numerical integration is needed in this process as the integrals can be done analytically good accuracy is achieved with a small number of evaluations for the integral kernel 60 points for the mpm as compared with hundreds or thousands of functional evaluations using the traditional extrapolation methods along the tails of the si simulation results show that to obtain the similar accuracy in the evaluation of the si the mpm is approximately 10 times faster than the traditional extrapolation methods moreover since the mpm is robust to the effects of noise this method is more stable especially for large values of the horizontal distances the method proposed in this paper is thus a new and better technique to obtain accurate results for the computation of the green s function for a layered media in the spatial domain

a new technique for the derivation of closed form electromagnetic green s functions for unbounded planar layered media a closed form electromagnetic green s function for unbounded planar layered media is derived in terms of a finite sum of hankel functions the derivation is based on the direct inverse hankel transform of a pole residue representation of the spectral domain form of the green s function such a pole residue form is obtained through the solution of the spectral domain form of the governing green s function equation numerically through a finite difference approximation rather than analytically the proposed methodology can handle any number of layers including the general case where the planar media exhibit arbitrary variation in their electrical properties in the vertical direction the numerical implementation of the proposed methodology is straightforward and robust and does not require any preprocessing of the spectrum of the green s function for the extraction of surface wave poles or its quasi static part the number of terms in the derived closed form expression is chosen adaptively with the distance between source and observation point as parameter the development of the closed form green s function is presented for both vertical and horizontal dipoles its accuracy is verified through a series of numerical examples and comparisons with results from other established methods

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

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

statistical dynamics of on line independent component analysis the learning dynamics of on line independent component analysis is analysed in the limit of large data dimension we study a simple hebbian learning algorithm that can be used to separate out a small number of non gaussian components from a high dimensional data set the de mixing matrix parameters are confined to a stiefel manifold of tall orthogonal matrices and we introduce a natural gradient variant of the algorithm which is appropriate to learning on this manifold for large input dimension the parameter trajectory of both algorithms passes through a sequence of unstable fixed points each described by a diffusion process in a polynomial potential choosing the learning rate too large increases the escape time from each of these fixed points effectively trapping the learning in a sub optimal state in order to avoid these trapping states a very low learning rate must be chosen during the learning transient resulting in learning time scales of o n2 or o n3 iterations where n is the data dimension escape from each sub optimal state results in a sequence of symmetry breaking events as the algorithm learns each source in turn this is in marked contrast to the learning dynamics displayed by related on line learning algorithms for multilayer neural networks and principal component analysis although the natural gradient variant of the algorithm has nice asymptotic convergence properties it has an equivalent transient dynamics to the standard hebbian algorithm

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

an algorithm for a special case of a generalization of the richardson extrapolation process a special case of a generalization of the richardson extrapolation process is considered and its complete solution is given in closed form using this an algorithm for implementing the extrapolation is devised it is shown that this algorithm needs a very small amount of arithmetic operations and very little storage convergence and stability properties for some cases are also considered

efficient evaluation of the finite part of pole free sommerfeld integrals in half space problems with predefined accuracy this communication presents a novel simple and robust approach for the computation of the finite part of pole free sommerfeld integrals sis in half space problems with high and controllable accuracy over a large range of source observer distances the approach includes the following techniques 1 cancellation of the branch point singularities based on the square root change of variables for numerical integration 2 approximation of real axis integration path in order to enhance the singularity cancellation for arbitrary low loss dielectrics 3 thresholds for truncation of the interval of integration for given accuracy which improve the efficiency of computation and 4 prediction formulas that estimate the required number of integration points for a given accuracy up to 1000 wavelengths of source observer distance the proposed approach is verified through numerical examples and comparison to reference methods

numerical computation of diffraction catastrophes with codimension eight a computational strategy aimed at evaluating diffraction catastrophes belonging to the x 9 family is presented the approach proposed is based on the use of power series expansions suitably derived for giving meaningful representation of the whole 0 x 9 subfamily jointly with a powerful sequence transformation algorithm the so called weniger transformation the convergence features of the above series expansions are investigated and several numerical experiments are carried out to assess the effectiveness of the retrieving action of the weniger transformation as well as the ease of implementation of the whole approach

multilayered media green s functions in integral equation formulations a compact representation is given of the electric and magnetic type dyadic green s functions for plane stratified multilayered uniaxial media based on the transmission line network analog along the aids normal to the stratification furthermore mixed potential integral equations are derived within the framework of this transmission line formalism for arbitrarily shaped conducting or penetrable objects embedded in the multilayered medium the development emphasizes laterally unbounded environments but an extension to the case of a medium enclosed by a rectangular shield is also included

a domain integral equation approach for simulating two dimensional transverse electric scattering in a layered medium with a gabor frame discretization abstract we solve the 2d transverse electrically polarized domain integral equation in a layered background medium by applying a gabor frame as a projection method this algorithm employs both a spatial and a spectral discretization of the electric field and the contrast current in the direction of the layer extent in the spectral domain we use a representation on the complex plane that avoids the poles and branchcuts found in the green function because of the special choice of the complex plane path in the spectral domain and because of the choice to use a gabor frame to represent functions on this path fast algorithms based on ffts are available to transform to and from the spectral domain yielding an o n log n scaling in computation time

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

methods of numerical integration of oscillatory functions by the de formula with the richardson extrapolation abstract in 1978 a method for numerical integration of oscillatory functions by means of the de formula and the richardson extrapolation was introduced by toda and ono and yet its usefulness was proved only experimentally in this paper theoretical considerations are bestowed on the usefulness of this toda and ono s method particularly a class of integration problems is given for which toda and ono s method is efficient an alternative method is also proposed along the same line as toda and ono s method the usefulness of the proposed method is considered theoretically finally it is demonstrated through numerical experiments that the proposed method surpasses toda and ono s method in efficiency

dynamics of learning in multilayer perceptrons near singularities the dynamical behavior of learning is known to be very slow for the multilayer perceptron being often trapped in the plateau it has been recently understood that this is due to the singularity in the parameter space of perceptrons in which trajectories of learning are drawn the space is riemannian from the point of view of information geometry and contains singular regions where the riemannian metric or the fisher information matrix degenerates this paper analyzes the dynamics of learning in a neighborhood of the singular regions when the true teacher machine lies at the singularity we give explicit asymptotic analytical solutions trajectories both for the standard gradient sgd and natural gradient ngd methods it is clearly shown in the case of the sgd method that the plateau phenomenon appears in a neighborhood of the critical regions where the dynamical behavior is extremely slow the analysis of the ngd method is much more difficult because the inverse of the fisher information matrix diverges we conquer the difficulty by introducing the blow down technique used in algebraic geometry the ngd method works efficiently and the state converges directly to the true parameters very quickly while it staggers in the case of the sgd method the analytical results are compared with computer simulations showing good agreement the effects of singularities on learning are thus qualitatively clarified for both standard and ngd methods

analysis of natural gradient descent for multilayer neural networks natural gradient descent ngd is an on line algorithm for redefining the steepest descent direction an analysis of ngd for training a multilayer neural network is presented using statistical mechanics the performance can be significantly improved using ngd algorithm and can be used for both the transient and asymptotic stages of learning

cluster expansions for the ising and heisenberg spin models in hamiltonian lattice field theory cluster expansion methods are applied to calculate high temperature series for the vacuum energy the susceptibility and the mass gap for the ising model and the o 2 and o 3 heisenberg models in 1 1 dimensions and 2 1 dimensions critical points and critical indices are estimated for the line the square and the triangular lattices the results demonstrate universality with the normal euclidean versions of these models within errors

natural gradient works efficiently in learning when a parameter space has a certain underlying structure the ordinary gradient of a function does not represent its steepest direction but the natural gradient does information geometry is used for calculating the natural gradients in the parameter space of perceptrons the space of matrices for blind source separation and the space of linear dynamical systems for blind source deconvolution the dynamical behavior of natural gradient online learning is analyzed and is proved to be fisher efficient implying that it has asymptotically the same performance as the optimal batch estimation of parameters this suggests that the plateau phenomenon which appears in the backpropagation learning algorithm of multilayer perceptrons might disappear or might not be so serious when the natural gradient is used an adaptive method of updating the learning rate is proposed and analyzed

indefinite integration of oscillatory functions by the chebyshev series expansion abstract an automatic quadrature scheme is presented for evaluating the indefinite integral of oscillatory function x 0 t e i t dt 0 x 1 of a given function t which is usually assumed to be smooth the function t is expanded in the chebyshev series to make an efficient evaluation of the indefinite integral combining the automatic quadrature method obtained and sidi s extrapolation method makes an effective quadrature scheme for oscillatory infinite integral a x cos x d x for which numerical examples are also presented

fourier series and the δ 2 process we discuss the effect of a particular sequence acceleration method the d 2 process on the partial sums of fourier series we show that for a very general class of functions this method fails on a dense set of points not only does it not speed up convergence it turns the sequence of partial sums into a sequence with multiple limit points

theory of topological molecular lattices abstract the aim of this paper is to bring the theories of point set topology fuzzy topology and l fuzzy topology into a unified theory the theory of topological molecular lattices for this purpose we first introduce the concepts of molecule for completely distributive complete lattice molecular lattice for short the cotopology on it and the concept of remote neighborhoods of molecules therefrom we develop the theory of pointwise topology on molecular lattices

convergence acceleration of some logarithmic sequences abstract let s n be some real sequence defined as s n 1 s n for n where x x i 1 p i x s p i with p n p 0 and p 1 0 given for s 0 well chosen it converges to s and lim n s n 1 s s n s 1 logarithmic convergence by asymptotic analysis we show that different algorithms modified iterated versions of aitken s 2 process iterated 2 algorithm modified algorithms and algorithm accelerate the convergence of this type of sequences and we estimate the errors on the transformed sequences for sequences of type 1 with p 1 2 we give more precise results and an efficient algorithm combining the modified 2 and 2 algorithm finally we apply these algorithms to some series and integrals

a convergence and stability study of the iterated lubkin transformation and the θ algorithm in this paper we analyze the convergence and stability of the iterated lubkin transformation and the algorithm as these are being applied to sequences an whose members behave like an a n n r i 0 in i as n where and are complex scalars and r is a nonnegative integer we study the three different cases in which i r 0 1 and 0 1 logarithmic sequences ii r 0 and 1 linear sequences and iii r 1 2 factorial sequences we show that both methods accelerate the convergence of all three types of sequences we show also that both methods are stable on linear and factorial sequences and they are unstable on logarithmic sequences on the basis of this analysis we propose ways of improving accuracy and stability in problematic cases finally we provide a comparison of these results with analogous results corresponding to the levin u transformation

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

parameter identifiability in statistical machine learning a review this review examines the relevance of parameter identifiability for statistical models used in machine learning in addition to defining main concepts we address several issues of identifiability closely related to machine learning showing the advantages and disadvantages of state of the art research and demonstrating recent progress first we review criteria for determining the parameter structure of models from the literature this has three related issues parameter identifiability parameter redundancy and reparameterization second we review the deep influence of identifiability on various aspects of machine learning from theoretical and application viewpoints in addition to illustrating the utility and influence of identifiability we emphasize the interplay among identifiability theory machine learning mathematical statistics information theory optimization theory information geometry riemann geometry symbolic computation bayesian inference algebraic geometry and others finally we present a new perspective together with the associated challenges

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

exact closed form expression of a sommerfeld integral for the impedance plane problem a sommerfeld integral for an impedance half plane is one of the classical problems in electromagnetic theory in this paper the integral is evaluated into two series representations which are expressed in terms of the exponential integral and the lommel function respectively it is shown that the series expansions absolutely converge for any parameters such as distance or surface impedance then based on the lommel function expansion an exact closed form expression of the integral is formulated this expression is written in terms of incomplete weber integrals which are directly related to incomplete cylindrical functions such as incomplete lipschitz hankel integrals additionally a complete asymptotic series of the integral is obtained based on the exponential integral expansion with conventional methods such as steepest descent method it is cumbersome to derive the divergent series the validity of all the formulations derived in this paper is demonstrated through a comparison with a numerical integration of the integral for various situations

a novel analysis of microstrip structures using the gaussian green s function method a novel closed form expression is derived for spatial green s functions of microstrip structures by expanding the spectral green s function into a gaussian series this innovative method is called the gaussian green s function ggf method due to the gaussian form in the closed form green s function representation the main advantage of the ggf method lies in its precision as well as rapid convergence to demonstrate the versatility of this method the current distribution of a microstrip antenna is achieved via the combination of the method of moments mom and ggf method it is shown that this method can be computationally very efficient with less than 1 error compared to the numerical integration of the spectral integral also the results of the ggf method have been compared to the results of the commercial full wave software of agilent ads

approaches to the representations and logic operations of fuzzy concepts in the framework of axiomatic fuzzy set theory ii in this paper the representations of fuzzy concepts based on raw data have been investigated within the framework of afs axiomatic fuzzy set theory first a brief review of afs theory is presented and a completely distributive lattice the e i algebra is proposed secondly two kinds of e i algebra representations of fuzzy concepts are derived in detail in order to represent the membership functions of fuzzy concepts in the interval 0 1 the norm of afs algebra is defined and studied finally the relationships of various representations with their advantages and drawbacks are analyzed

new hybrid non linear transformations of divergent perturbation series for quadratic zeeman effects the problem of hydrogen atoms in an external uniform magnetic field quadratic zeeman effect is studied by means of perturbation theory the power series for the ground state energy in terms of magnetic field strength b is divergent nevertheless it is possible to induce convergence of this divergent series by applying various non linear transformations such as those of shanks e levin t u v as well as the present omega algorithm these transformations of originally divergent perturbation series yield new sequences which then converge the induced convergence is however quite slow a new hybrid shanks levin non linear transform is devised here for accelerating these slowly converging series and sequences significant improvement in the convergence rate is obtained agreement with the exact results is excellent

the topology of afs structure and afs algebras abstract in this paper we will study the topological structure of the afs structures published by the author discuss the topological molecular lattice structures on ei ei eii eii algebras published by the author and give the relations of these topological structures as applications we study the topological produced by a family of fuzzy concepts in the afs algebras and apply these to analyze relations among fuzzy concepts using these we believe that we can study the law of human thinking the most important fact is that all these can be operated by computers we are also sure afs theory will play an important role in information processing

extrapolation methods for divergent oscillatory infinite integrals that are defined in the sense of summability in a recent work by the author an extrapolation method the w transformation was developed by which a large class of oscillatory infinite integrals can be computed very efficiently the results of this work are extended to a class of divergent oscillatory infinite integrals in the present paper it is shown in particular that these divergent integrals exist in the sense of abel summability and that the w transformation can be applied to them without any modifications convergence results are stated and numerical examples given

evaluation of efficient closed form green s function in a cylindrically stratified medium an efficient technique is proposed for the removal of the spatial singularity for a z directed current source in a cylindrically layered structure by decomposing the spectral domain green s function into the particular and homogeneous parts the particular part with the singularity is converted into the spatial domain using sommerfeld s identity while the dcim with general pencil of function technique is applied on the spectral domain homogeneous part the use of the ratio form of the reflection transmission matrices the debye approximation and the subsequent envelope extraction for the treatment of the spatial domain singularity is thus avoided in addition a very significant reduction in the number of modes required for convergence is also achieved the technique is applied for evaluating the input impedance of several antennas in the cylindrical multilayer structure together with a coupling configuration for the excitation of the multilayer structure

convergence acceleration of fourier series by analytical and numerical application of poisson s formula a form of the poisson summation formula appropriate for the transformation of infinite fourier series is derived and shown to be applicable to slowly converging series arising in the solution of laplace s equation in a rectangular electrochemical cell the result is a rapidly convergent fourier series which is particularly useful in giving the potential close to the electrode surface results obtained from the analytical expression are found to be in close agreement with those resulting from numerical evaluation of the fourier integrals using simpson s rule and the euler transformation and direct summation by using goertzel s 1958 algorithm

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

extrapolation methods for sommerfeld integral tails a review is presented of the extrapolation methods for accelerating the convergence of sommerfeld type integrals i e semi infinite range integrals with bessel function kernels which arise in problems involving antennas or scatterers embedded in planar multilayered media attention is limited to partition extrapolation procedures in which the sommerfeld integral is evaluated as a sum of a series of partial integrals over finite subintervals and is accelerated by an extrapolation method applied over the real axis tail segment spl alpha spl infin of the integration path where spl alpha 0 is selected to ensure that the integrand is well behaved an analytical form of the asymptotic truncation error or the remainder which characterizes the convergence properties of the sequence of partial sums and serves as a basis for some of the most efficient extrapolation methods is derived several extrapolation algorithms deemed to be the most suitable for the sommerfeld integrals are described and their performance is compared it is demonstrated that the performance of these methods is strongly affected by the horizontal displacement of the source and field points spl rho and by the choice of the subinterval break points furthermore it is found that some well known extrapolation techniques may fail for a number of values of spl rho and ways to remedy this are suggested finally the most effective extrapolation methods for accelerating sommerfeld integral tails are recommended

concept analysis via rough set and afs algebra formal concept analysis fca was originally proposed by wille 1982 which is an important theory for data analysis and knowledge discovery afs axiomatic fuzzy set algebra was proposed by liu x liu the fuzzy theory based on afs algebras and afs structure journal of mathematical analysis and applications 217 1998 459 478 x liu the topology on afs algebra and afs structure journal of mathematical analysis and applications 217 1998 479 489 which is a semantic methodology relating to the fuzzy theory combining above two theories we propose afs formal concept which can be viewed as the generalization and development of monotone concept proposed by deogun and saquer 2003 moreover we show that the set of all afs formal concepts forms a complete lattice afs formal concept can be applied to represent the logic operations of queries in information retrieval furthermore we give an approach to find the afs formal concepts whose intents extents approximate any element of afs algebra by virtue of rough set theory

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

finite size scaling in the transverse ising model on a square lattice energy eigenvalues and order parameters are calculated by exact diagonalization for the transverse ising model on square lattices of up to 6 6 sites finite size scaling is used to estimate the critical parameters of the model confirming universality with the three dimensional classical ising model critical amplitudes are also estimated for both the energy gap and the ground state energy

complete closed form expression of dyadic green s function and its far and near field approximations for an impedance half plane a closed form expression of the dyadic green s function is formulated for an impedance half plane which is written in terms of the incomplete cylindrical function of poisson form due to the branch cut of the logarithm function that is required to calculate the input argument of the incomplete cylindrical function the closed form representation consists of two formulations since the closed form expression contains a singularity at 0 the small argument expansion of the expression is also derived to rigorously characterize the behavior of the function at 0 the previously reported complete asymptotic expansion for the sommerfeld integral for an impedance half plane is not accurate for practically important cases such as near earth propagation and or when the surface is highly conductive hence in this paper a new complete asymptotic series of the sommerfeld integral are derived for the case that the existing asymptotic series is not accurate the two asymptotic series not only allow efficient numerical computation but also provide more accurate results for virtually all propagation scenarios based on the two asymptotic series the complete asymptotic series of the dyadic green s function is derived all derived formulations are numerically verified and their accuracies are investigated

a user friendly extrapolation method for oscillatory infinite integrals developpement d une methode d extrapolation flexible pour les integrales infinies oscillatoires

on the plane wave excited subwavelength circular aperture in a thin perfectly conducting flat screen the electromagnetic field transmitted by a subwavelength circular aperture in a perfectly conducting screen of infinitesimal thickness and illuminated by an obliquely incident plane wave is analyzed based on the bethe bouwkamp quasi static model of the field in the aperture plane the problem is formulated in the spectral domain and the solution is expressed in terms of a minimum number of one dimensional hankel transform integrals involving products of bessel functions and spherical bessel functions an efficient computational scheme for these integrals is implemented based on numerical quadrature with convergence acceleration by extrapolation from the rigorous spectral domain formulation closed form expressions are also derived for the transmitted near zone and far zone fields sample numerical results are presented to demonstrate the validity of the method and its utility in the modeling of aperture type near field optical probes

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

evaluating infinite integrals involving bessel functions of arbitrary order abstract the evaluation of intergrals of the form i n 0 x j n x d x is considered in the past the method of dividing an oscillatory integral at its zeros forming a sequence of partial sums and using extrapolation to accelerate convergence has been found to be the most efficient technique available where the oscillation is due to a trigonometric function or a bessel function of order n 0 1 here we compare various extrapolation techniques as well as choices of endpoints in dividing the integral and establish the most efficient method for evaluating infinite integrals involving bessel functions of any order n not just zero or one we also outline a simple but very effective technique for calculating bessel function zeros