a generalization of the e algorithm abstract salam a on a generalization of the algorithm journal of computational and applied mathematics 46 1993 455 464 the algorithm is a quite powerful algorithm for accelerating the convergence of some classes of sequences it was generalized in two different ways some years ago in this paper a determinantal formula for its second generalization is obtained then its kernel the set of sequences which are transformed into a constant sequence is given as the solution of a difference equation

a super discrete variational identity and its application for constructing super discrete hamiltonian systems in the paper a super discrete variational identity on lie super algebras is established first it provides an approach for constructing super discrete hamiltonian structures of super evolution lattice equations with discrete zero curvature representation when the super spectral matrix u is selected appropriately as an application super discrete hamiltonian structures of super toda lattice hierarchy are developed and this method can be used to construct super discrete bi hamiltonian structures of more super discrete integrable evolutive hierarchies super discrete variational identity is devoted to deduce super integrability and solutions of super discrete integrable equations

decoding algorithm as a moment problem related to the extended lotka volterra system the moment problem related to the extended lotka volterra system sometimes also called the hungry lotka volterra system or the narita itoh bogoyavlensky lattice over finite fields is introduced it turns out the moment problem could be used to design an algorithm for decoding multiple bch goppa codes over the same finite field simultaneously when multiple codes have the same error locations the algorithm requires fewer known syndromes and has lower computational complexity than running the decoding algorithm in nakamura 1996 phys lett a 223 75 81 multiple times

about several classes of bi orthogonal polynomials and discrete integrable systems by introducing some special bi orthogonal polynomials we derive the so called discrete hungry quotient difference dhqd algorithm and a system related to the qd type discrete hungry lotka volterra qd type dhlv system together with their lax pairs these two known equations can be regarded as extensions of the qd algorithm when this idea is applied to a higher analogue of the discrete time toda hadt equation and the quotient quotient difference qqd scheme proposed by spicer nijhoff and van der kamp two extended systems are constructed we call these systems the hungry forms of the higher analogue discrete time toda hhadt equation and the quotient quotient difference hqqd scheme respectively in addition the corresponding lax pairs are provided

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

the richardson extrapolation process with a harmonic sequence of collocation points let a y sim a sum infty k 1 alpha ky sigma k as y rightarrow 0 where y is a discrete or continuous variable assume that sigma k are known numbers that may be complex in general and that a y is known for y in 0 b for some b 0 the aim is to find or approximate a the limit or antilimit of a y as y rightarrow0 one very effective way of approximating a is by the richardson extrapolation process that is defined via the linear systems a y l a j n sum n k 1 overline alpha ky sigma k l j leq l leq j n here a j n are approximations to a and overline alpha k are additional unknowns the yl are picked such that y0 y1 y2 0 and lim l rightarrow infty y l 0 in this paper we give a detailed analysis of the convergence and stability of the column sequences a j n infty j 0 with n fixed when y l c l eta q for some positive c eta and q specifically we prove that convergence takes place as j to infty and give the precise rate at which it does we also prove that the process is unstable and quantify its instability asymptotically this instability may be dealt with numerically by using high precision floating point arithmetic

a derivation of extrapolation algorithms based on error estimates in this paper we shall emphasize the role played by error estimates and annihilation difference operators in the construction of extrapolation processes it is shown that this approach leads to a unified derivation of many extrapolation algorithms and related devices to general results about their kernels and that it opens the way to many new algorithms their convergence and acceleration properties could also be studied within this framework

convergence acceleration for boundary value problems with singularities using the e algorithm in the present work we use the e algorithm to accelerate the convergence of finite difference schemes for ordinary differential equations as a model a boundary value problem for a second order differential equation is considered where the right hand side may have different kinds of singularities an asymptotic error expansion is obtained enabling the use of the e algorithm to accelerate the convergence the applicability and the efficiency of the e algorithm are discussed and illustrated by numerical examples

convergence properties of the sbar d transformation and a fast and accurate numerical evaluation of molecular integrals the present work concerns the study of the convergence properties of the nonlinear s transformation a fast and accurate numerical evaluation of highly oscillatory semi infinite integrals involved in the analytic expressions of molecular integrals over slater type orbitals is obtained using the s transformation these semi infinite integrals involve spherical and reduced bessel functions the convergence properties are analysed and they show that the approximations obtained using the s approach converge to the exact values of the semi infinite integrals without any constraint the numerical tables show that the s method gives unprecedented accuracy with speed up by a factor 2 over the h method and a factor 10 over the transformation this illustrates the superiority of this new approach

a new method for approximating improper integrals in this paper a new transformation referred to as the g n m transform is introduced for the purpose of increasing the rate of convergence of a sequence to its limit in particular if the sequence is a sequence of partial integrals f x then g n m f x is shown to converge super fast to f infty under very general conditions the g n m transform is shown to be closely related to levin and sidi s d transform appl math comp 9 1981 pp 175 215 and the b n transform introduced by gray atchinson and mcwilliams siam j numer anal 8 1971 pp 365 381 several examples are given

calculating laplace transforms in terms of the toda molecule a method for calculating a continued fraction expansion of laplace transforms is presented in terms of the tau function of the continuous time toda molecule equation and its integrable discretization the z transform and a discrete analogue of the laplace transform of a set of finite samples are also formulated

the generalized richardson extrapolation process grep 1 and computation of derivatives of limits of sequences with applications to the d 1 transformation abstract let sm be an infinite sequence whose limit or antilimit s can be approximated very efficiently by applying a suitable extrapolation method e0 to sm assume that the sm and hence also s are differentiable functions of some parameter d d s being the limit or antilimit of d d sm and that we need to approximate d d s a direct way of achieving this would be by applying again a suitable extrapolation method e1 to the sequence d d sm and this approach has often been used efficiently in various problems of practical importance unfortunately as has been observed at least in some important cases when d d sm and sm have essentially different asymptotic behaviors as m the approximations to d d s produced by this approach despite the fact that they are good do not converge as quickly as those obtained for s and this is puzzling in a recent paper a sidi extrapolation methods and derivatives of limits of sequences math comp 69 2000 305 323 we gave a rigorous mathematical explanation of this phenomenon for the cases in which e0 is the richardson extrapolation process and e1 is a generalization of it and we showed that the phenomenon has nothing to do with numerics following that we proposed a very effective procedure to overcome this problem that amounts to first applying the extrapolation method e0 to sm and then differentiating the resulting approximations to s as a practical means of implementing this procedure we also proposed the direct differentiation of the recursion relations of the extrapolation method e0 used in approximating s we additionally provided a thorough convergence and stability analysis in conjunction with the richardson extrapolation process from which we deduced that the new procedure for d d s has practically the same convergence properties as e0 for s finally we presented an application to the computation of integrals with algebraic logarithmic endpoint singularities via the romberg integration in this paper we continue this research by treating sidi s generalized richardson extrapolation process grep 1 in detail we then apply the new procedure to various infinite series of logarithmic type whether convergent or divergent in conjunction with the d 1 transformation of levin and sidi both the theory and the numerical results of this paper too indicate that this approach is the preferred one for computing derivatives of limits of infinite sequences and series

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

schur flow for orthogonal polynomials on the unit circle and its integrable discretization a one parameter deformation of the measure of orthogonality for orthogonal polynomials on the unit circle is considered the corresponding dynamics of the schur parameters of the orthogonal polynomials is shown to be characterized by the complex semi discrete modified kdv equation namely the schur flow a discrete analogue of the miura transformation is found an integrable discretization of the schur flow enables us to compute a pade approximation of the caratheodory functions or equivalently to compute a perron caratheodory continued fraction in a polynomial time

extended procedures for extrapolation to the limit in this paper new procedures for the extrapolation to the limit of slowly convergent sequences and functions are proposed they are based on the notions of error estimates and annihilation operators we obtain generalizations of the discrete and confluent e transformation which are the most general sequence and function transformations known so far it is shown that many transformations studied in the literature are included in our formalism particular cases of these procedures are discussed then several extensions of the e algorithm are given finally the procedure q is applied to our procedures to produce new procedures for extrapolation to the limit

on a periodic soliton cellular automaton we propose a box and ball system with a periodic boundary condition periodic box and ball system pbbs the time evolution rule of the pbbs is represented as a boolean recurrence formula an inverse ultradiscretization of which is shown to be equivalent to the algorithm of the calculus for the 2nth root the relations to the pbbs of the combinatorial r matrix of u q an 1 are also discussed

extrapolation methods and derivatives of limits of sequences let s m be an infinite sequence whose limit or antilimit s can be approximated very efficiently by applying a suitable extrapolation method e 0 to s m assume that the s m and hence also s are differentiable functions of some parameter d d s being the limit or antilimit of d d s m and that we need to approximate d d s a direct way of achieving this would be by applying again a suitable extrapolation method e 1 to the sequence d d s m and this approach has often been used efficiently in various problems of practical importance unfortunately as has been observed at least in some important cases when d d s m and s m have essentially different asymptotic behaviors as m the approximations to d d s produced by this approach despite the fact that they are good do not converge as quickly as those obtained for s and this is puzzling in this paper we first give a rigorous mathematical explanation of this phenomenon for the cases in which e 0 is the richardson extrapolation process and e 1 is a generalization of it thus showing that the phenomenon has very little to do with numerics following that we propose a procedure that amounts to first applying the extrapolation method e 0 to s m and then differentiating the resulting approximations to s and we provide a thorough convergence and stability analysis in conjunction with the richardson extrapolation process it follows from this analysis that the new procedure for d d s has practically the same convergence properties as e 0 for s we show that a very efficient way of implementing the new procedure is by actually differentiating the recursion relations satisfied by the extrapolation method used and we derive the necessary algorithm for the richardson extrapolation process we demonstrate the effectiveness of the new approach with numerical examples that also support the theory we discuss the application of this approach to numerical integration in the presence of endpoint singularities we also discuss briefly its application in conjunction with other extrapolation methods

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

algorithms associated with arithmetic geometric and harmonic means and integrable systems abstract gauss algorithm for arithmetic geometric mean agm can be regarded as a discrete time integrable dynamical system having an elliptic theta function solution and a conserved quantity in this paper we consider algorithms associated with arithmetic geometric and harmonic means from a viewpoint of integrable systems first a max plus limit and its inverse limit of the agm algorithm are discussed these mean operations are shown to be connected to each other by the max plus limit secondly continous time dynamical systems associated with the arithmetic harmonic mean ahm algorithm are found thirdly it is shown that the ahm algorithm in indefinite case has a chaotic dynamics and is a generator of numbers which obey the cauchy distribution finally an extension of the ahm algorithm to the space of positive definite symmetric matrices is considered

conserved quantities of the discrete finite toda equation and lower bounds of the minimal singular value of upper bidiagonal matrices some numerical algorithms are known to be related to discrete time integrable systems where it is essential that quantities to be computed for example eigenvalues and singular values of a matrix poles of a continued fraction are conserved quantities in this paper a new application of conserved quantities of integrable systems to numerical algorithms is presented for an n n n 2 real upper bidiagonal matrix b where all the diagonals and the upper subdiagonals are positive conserved quantities tr btb m 1 m 1 2 of the discrete finite toda equation give a sequence of lower bounds of the minimal singular value of b recurrence relations for computing higher order conserved quantities tr btb m 1 are also derived

on the euler lagrange equation for functionals of the calculus of variations without upper growth conditions for a class of functionals of the calculus of variations we prove that each minimum of the functional satisfies the associated euler lagrange equation the integrand is assumed to be convex but no upper growth condition is imposed

a complete convergence and stability theory for a generalized richardson extrapolation process let a y sim a sum k 1 infty q k log y y sigma k as y rightarrow 0 where y is a discrete or continuous variable and q k xi are polynomials in xi it is assumed that sigma k and the degree of q k xi or an upper bound for it are known for each k and that a y is known for all possible y in 0 b the aim is to find a whether it is the limit or antilimit of a y for y rightarrow 0 a very effective way of doing this is by the generalized richardson extrapolation in this paper this procedure is described and a very efficient recursive algorithm for its implementation is given when the set of extrapolation points is y l y 0 omega l l 0 1 ldots for some omega in 0 1 in addition a complete theory of convergence and stability for the columns and the diagonals of the corresponding extrapolation table is provided finally two applications are considered in detail one of which is to generalized romberg integration of functions with algebraic and logarithmic endpoint singularities

the toda molecule equation and the 3 algorithm one of the well known convergence acceleration methods the e algorithm is investigated from the viewpoint of the toda molecule equation it is shown that the error caused by the algorithm is evaluated by means of solutions for the equation the acceleration algorithm based on the discrete toda molecule equation is also presented

a combinatorial method for the vanishing of the poisson brackets of an integrable lotka volterra system the combinatorial method is useful to obtain conserved quantities for some nonlinear integrable systems as an alternative to the lax representation method here we extend the combinatorial method and introduce an elementary geometry to show the vanishing of the poisson brackets of the hamiltonian structure for a lotka volterra system of competing species we associate a set of points on a circle with a set of species of the lotka volterra system where the dominance relations between points are given by the dominance relations between the species we associate each term of the conserved quantities with a subset of points on the circle which simplifies to show the vanishing of the poisson brackets

asymptotic analysis of a generalized richardson extrapolation process on linear sequences in this work we give a detailed convergence and stability analysis for the author s generalized richardson extrapolation process grep m as this is being applied to linearly convergent or divergent infinite sequences a n where a n a m k 1 n k i 0 ki n k i as n k 1 being distinct the quantity we would like to compute is a whether it is the limit or antilimit of a n such sequences arise for example as partial sums of power series and of fourier series of functions that have algebraic and or logarithmic branch singularities specifically we define the grep m approximation a n m j to a with n n 1 n m and 0 via the linear systems a l a n m j m k 1 l k n i 0 1 ki l k 1 j l j m k 1 n k where ki are additional unknowns we study the convergence and stability properties of a n m j as j we show in particular that a n m j a m k 1 o j k j k 2nk as j when compared with a j a m k 1 o j k j k as j this result shows that grep m is a true convergence acceleration method for the sequences considered in addition we show that grep m 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 grep m with m 1

new integrable differential difference systems we report a new and relatively simple procedure for finding new integrable differential difference equations the procedure starts from a rather general differential difference equation taken in bilinear form and then searches for appropriate backlund transformations for it in this way three particular new integrable differential difference systems are found and their corresponding backlund transformations presented

backlund transformations and nonlinear superposition formulae of a differential difference kdv equation in this paper a differential difference kdv equation is considered we present two backlund transformations and corresponding nonlinear superposition formulae for it as an application of the obtained results n soliton solutions first obtained by ohta and hirota are rederived furthermore a sequence of rational solutions are also obtained thus the integrability of the differential difference kdv equation is further confirmed

an extrapolation method for a class of boundary integral equations boundary value problems of the third kind are converted into boundary integral equations of the second kind with periodic logarithmic kernels by using green s formulas for solving the induced boundary integral equations a nystrom scheme and its extrapolation method are derived for periodic fredholm integral equations of the second kind with logarithmic singularity asymptotic expansions for the approximate solutions obtained by the nystrom scheme are developed to analyze the extrapolation method some computational aspects of the methods are considered and two numerical examples are given to illustrate the acceleration of convergence

convergence analysis for a generalized richardson extrapolation process with an application to the d 1 transformation on convergent and divergent logarithmic sequences in an earlier work by the author the generalized richardson extrapolation process grep was introduced and some of its convergence and stability properties were discussed in a more recent work by the author a special case of grep which we now call grep 1 was considered and its properties were reviewed with emphasis on oscillatory sequences in the first part of the present work we give a detailed convergence and stability analysis of grep 1 as it applies to a large class of logarithmic sequences both convergent and divergent in particular we prove several theorems concerning the columns and the diagonals of the corresponding extrapolation table these theorems are very realistic in the sense that they explain the remarkable efficiency of grep 1 in a very precise manner in the second part we apply this analysis to the levin sidi d 1 transformation as the latter is used with a new strategy to accelerate the convergence of infinite series that converge logarithmically or to sum the divergent extensions of such series this is made possible by the observation that when the proper analogy is drawn the d 1 transformation is in fact a grep 1 we append numerical examples that demonstrate the theory

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

bell polynomial approach and n soliton solution for the extended lotka volterra equation in plasmas symbolically investigated in this paper is the extended lotka volterra elv equation which can govern the kinetics of the discrete peaks of the weak langmuir turbulence in plasmas without the linear damping and random noise binary bell polynomials are applied to the bilinearization of the discrete system bilinear backlund transformation of the elv equation is constructed n soliton solution in terms of the extended casorati determinant is also presented and verified propagation and interaction behaviors of the langmuir turbulence are analyzed it is demonstrated that the number of the interacting langmuir waves can influence the soliton velocity and amplitude as well as the collision phase shift graphic illustrations of the solitonic collisions show that the repulsion effects and nonlinear interactions are also associated with the number of the interacting langmuir waves

on the validity of the euler lagrange equation in a nonlinear case abstract we prove the validity of the euler lagrange equation for the class of functionals of the type l u g x u d x where u g x u is a concave function our result is based on a recent paper by degiovanni and marzocchi 2009 6 where no growth assumptions on l are needed

application of a differential transformation method to strongly nonlinear damped q difference equations the theory of approximate solution lacks research on the area of nonlinear q difference equations this article explores the possibility of using the differential transformation method to find an approximate solution for strongly nonlinear damped q difference equations the time response of the nonlinear equation is presented under different parameter conditions and the results are then compared with those derived from the numerical method to verify the accuracy of the proposed method

cross rules and non abelian lattice equations for the discrete and confluent non scalar e algorithms in this paper we give the cross rules of the discrete and confluent vector topological and matrix e algorithms then from the rules of these confluent algorithms we derive non abelian lattice equations in particular some extensions of the lotka volterra system in the style of the equation related to the confluent form of the scalar e algorithm

equivalence and normal forms for the restricted and bounded fixpoint in the nested algebra the nested model is an extension of the traditional flat relational model in which relations can also have relation valued entries its default query language the nested algebra is rather weak unfortunately since it is only a conservative extension of the traditional flat relational algebra and thus can express only a small fraction of the polynomial time queries therefore it was proposed to extend the nested algebra with a fixpoint construct but the resulting language turned out to be too powerful many inherently exponential queries could also be expressed two polynomial time restrictions of the fixpoint closure of the nested algebra were proposed the restricted fixpoint closure by gyssens and van gucht and the bounded fixpoint closure by suciu here we prove two results first we show that both restrictions are equivalent in expressive power the proof technique relies on known encodings of nested relations into flat ones and on a novel technique called type substitution by which we reduce the equivalence of the two restrictions to its obvious counterpart in the flat relational model second we prove that both the bounded fixpoint queries and the restricted fixpoint queries admit normal forms in which the fixpoint occurs exactly once the proof technique relies on a novel encoding method of nested relations into flat ones

determinantal solutions of solvable chaotic systems it is shown that two solvable chaotic systems the arithmetic harmonic mean ahm algorithm and the ulam von neumann uvn map have determinantal solutions an additional formula for certain determinants and riccati difference equations play a key role in both cases two infinite hierarchies of solvable chaotic systems are presented which have determinantal solutions

q difference and confluent forms of the lattice boussinesq equation and the relevant convergence acceleration algorithms a q difference lattice boussinesq equation and its confluent form are proposed with the help of determinant identity their molecule solutions are constructed it is shown that the q difference lattice boussinesq equation can be used as a numerical convergence acceleration algorithm to compute the approximation limt f t numerical examples with the application of this algorithm are presented

an extended multistep shanks transformation and convergence acceleration algorithm with their convergence and stability analysis the molecule solution of an extended discrete lotka volterra equation is constructed from which a new sequence transformation is proposed a convergence acceleration algorithm for implementing this sequence transformation is found it is shown that our new sequence transformation accelerates some kinds of linearly convergent sequences and factorially convergent sequences with good numerical stability some numerical examples are also presented

on a generalization of the richardson extrapolation process a convergence result for a generalized richardson extrapolation process is improved upon considerably and additional results of interest are proved an application of practical importance is also given finally some known results concerning the convergence of levin s t transformation are reconsidered in light of the results of the present work

an application of the discrete time toda lattice to the progressive algorithm by lanczos and related problems the finite or the semi infinite discrete time toda lattice has many applications to various areas in applied mathematics the purpose of this paper is to review how the toda lattice appears in the lanczos algorithm through the quotient difference algorithm and its progressive form pqd then a multistep progressive algorithm mpa for solving linear systems is presented the extended lanczos parameters can be given not by computing inner products of the extended lanczos vectors but by using the pqd algorithm with highly relative accuracy in a lower cost the asymptotic behavior of the pqd algorithm brings us some applications of mpa related to eigenvectors

an extremely efficient and rapid algorithm for numerical evaluation of three centre nuclear attraction integrals over slater type functions the present work concerns the development of an extremely accurate and rapid algorithm for the numerical evaluation of the three centre nuclear attraction integrals over slater type functions these integrals are numerous they occur in many millions of terms even for small molecules and they require rapid and accurate evaluation the new algorithm is based on nonlinear transformation methods on numerical quadrature and on properties of the sine and bessel functions the section with numerical results shows the high accuracy and the substantial gain in calculation time realized using the new algorithm the complete expressions of the three centre nuclear attraction integrals over b functions and over slater type functions are evaluated for different values of the quantum numbers to show the efficiency of the new approach numerical results obtained with linear and nonlinear systems and comparisons with numerical results from the literature are listed

peakons strings and the finite toda lattice as is well known the toda lattice flow may be realized as an isospectral flow of a jacobi matrix a bijective map from a discrete string problem with positive weights to jacobi matrices allows the pure peakon flow of the camassa holm equation to be realized as an isospectral jacobi flow as well this gives a unified picture of the toda jacobi and multipeakon flows and leads to explicit solutions of the jacobi flows via stieltjes determination of the continued fraction expansion of a stieltjes transform a simple modification produces a bijection from generalized strings with positive and negative weights to singular jacobi matrices and thus brings peakon antipeakon flows into the same picture c

an extension of the levin sidi class of nonlinear transformations for accelerating convergence of infinite integrals and series in this paper we extend levin and sidi s nonlinear d and d transformations for accelerating convergence of infinite integrals and series the simple extension is made by viewing all these transformations as generalized jackknifes the new transformations appear to be more accurate in general and are as easily implemented as d and d transformations the new transformations have also some computational advantages as a result more accurate approximations can be obtained in many cases as is shown in the examples which are included

on the validity of the euler lagrange equation abstract under some regularity assumptions on the boundary datum u0 assumptions automatically satisfied in the classical case when the growth of the integrand is bounded by a b p we prove the validity of the euler lagrange equation for the functional f u x g x u x dx under general growth assumptions on f for instance for f p ek with p a nontrivial polynomial and k 0 or for f

the staircase method integrals for periodic reductions of integrable lattice equations we show in full generality that the staircase method papageorgiou et al 1990 phys lett a 147 106 14 quispel et al 1991 physica a 173 243 66 provides integrals for mappings and correspondences obtained as traveling wave reductions of systems of integrable partial difference equations we apply the staircase method to a variety of equations including the korteweg de vries equation the five point bruschi calogero droghei equation the quotient difference qd algorithm and the boussinesq system we show that in all these cases if the staircase method provides r integrals for an n dimensional mapping with then one can introduce q 2r variables which reduce the dimension of the mapping from n to q these dimension reducing variables are obtained as joint invariants of k symmetries of the mappings our results support the idea that often the staircase method provides sufficiently many integrals for the periodic reductions of integrable lattice equations to be completely integrable we also study reductions on other quad graphs than the regular lattice and we prove linear growth of the multi valuedness of iterates of high dimensional correspondences obtained as reductions of the qd algorithm

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

multicentre two electron coulomb and exchange integrals over slater functions evaluated using a generalized algorithm based on nonlinear transformations when calculating molecular electronic energies the contributions involving the coulomb operator for bielectronic terms are required rapidly and to high chemically significant accuracy the atomic orbital basis functions chosen in the present work are slater type functions stfs these functions can be expressed as finite linear combinations of b functions which are suitable to apply the fourier transform method the difficulties of the numerical evaluation of the analytic expressions of the integrals of interest arise mainly from the presence of two or three dimensional integral representations in this work we present a generalized algorithm for a precise and fast numerical evaluation of molecular integrals over stfs numerical results obtained with c2h2 c2h4 and ch4 molecules show the efficiency of the approach presented in this work comparisons with the existing codes are also listed

the higher integrability and the validity of the euler lagrange equation for solutions to variational problems we prove higher integrability properties of solutions to the problem of minimizing int omega l x u x nabla u x rm d x where xi mapsto l x u xi is a convex function satisfying some additional conditions as an application we prove the validity of the euler lagrange equation for a class of functionals with growth faster than exponential

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

a comparison between algebraic query languages for flat and nested databases abstract recently much attention has been paid to query languages for nested relations in the present paper we consider the nested algebra and the powerset algebra and compare them both mutually as well as to the traditional flat algebra we show that either nest or difference can be removed as a primitive operator in the powerset algebra while the redundancy of the nest operator might have been expected the same cannot be said of the difference basically this result shows that the presence of one nonmonotonic operator suffices in the powerset algebra as an interesting consequence of this result the nested algebra without the difference remains complete in the sense of bancilhon and paredaens finally we show there are both similarities and fundamental differences between the expressiveness of query languages for nested relations and that of their counterparts for flat relations

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

a q difference version of the ϵ algorithm in this paper a q difference version of the algorithm is proposed by using determinant identities the solutions of an initial value problem thus arisen can be expressed as ratios of hankel determinants it is shown that in numerical analysis this algorithm can be used to compute the approximation limt f t and in the field of integrable systems it could be viewed as the q difference version of the modified toda molecule equation

some new results on the triple humped soliton equation backlund transformation superposition principle and new explicit solutions a triple humped soliton equation is considered a corresponding backlund transformation for it is obtained and a nonlinear superposition formula is presented as an application of the results obtained soliton solutions first found by narita to the system are rederived a sequence of rational solutions are found and other types of solutions which are a mix of exponentials and rational expressions are deduced

symmetries conserved quantities and hierarchies for some lattice systems with soliton structure basic invariants such as conserved quantities symmetries mastersymmetries and recursion operators are explicitly constructed for the following nonlinear lattice systems the modified korteweg de vries lattice the ablowitz ladik lattice the brusci ragnisco lattice the ragnisco tu lattice and some cases of the class of integrable systems introduced by bogoyavlensky the algorithmic basis for obtaining these quantities is described and the interrelation between the underlying mastersymmetry approach and the lax pair analysis is discussed by explicit presentation of the higher order members of the corresponding hierarchies new completely integrable lattice flows are found for all systems multi hamiltonian formulations are given

exact solutions of non integrable lattice equations the method of generalized conditional symmetry used by fokas and liu for deriving nonintegrable evolution equations which possess exact analytic solutions is here extended to the construction of differential difference equations supporting two kink and two soliton solutions in particular we build the discrete analogue of a burgers type equation with reaction term investigated in the continuous case by satsuma and describing the coalescence of two travelling waves we also derive the discrete form of a nondispersive evolution equation possessing like the korteweg de vries equation a two soliton solution

an integrable poisson map generated from the eigenvalue problem of the lotka volterra hierarchy a 3 3 discrete eigenvalue problem associated with the lotka volterra hierarchy is studied and the corresponding nonlinearized one an integrable poisson map with a lie poisson structure is also presented moreover a 2 2 nonlinearized eigenvalue problem which also begets the lotka volterra hierarchy is proved to be a reduction of the poisson map on the leaves of the symplectic foliation

initial value problems for lattice equations we describe how to pose straight band initial value problems for lattice equations defined on arbitrary stencils in finitely many directions we arrive at discrete goursat problems and in the remaining directions we find cauchy problems next we consider s1 s2 periodic initial value problems in the goursat directions the periodic solutions are generated by correspondences in the cauchy directions assuming the equation to be multi linear the periodic solution can be obtained uniquely by iteration of a particularly simple mapping whose dimension is a piecewise linear function of s1 s2

backlund transformation and nonlinear superposition formula of an extended lotka volterra equation an extended lotka volterra equation is considered and a backlund transformation and a nonlinear superposition formula given for it multiple soliton solutions are derived step by step as an application of the obtained results thus the n soliton conjecture by narita is confirmed

weights whose biorthogonal polynomials admit a rodrigues formula abstract let 0 and x x let w be a non negative integrable function on an interval i let p n be a polynomial of degree n determined by the biorthogonality conditions i p n j w 0 j 0 1 n 1 we determine for which weights w p n admits an analogue of the classical rodrigues formula for orthogonal polynomials and present the formula whenever it exists we also provide generating functions and fairly explicit representations for p n

continued fractions and integrable systems stieltjes solution of the classical moment problem is the forerunner of inverse spectral theory in the modern theory of completely integrable systems the method of inverse scattering is used to obtain explicit solutions of a class of hamiltonian systems which arise as isospectral deformations of a linear operator in this lecture we explain how stieltjes s formulas are used to obtain explicit solutions to a number of completely integrable systems including discrete reductions of some nonlinear partial differential equations and the isospectral deformations of jacobi matrices a k a toda flows

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

special solutions of the toda chain and combinatorial numbers we classify all solutions of the restricted toda chain from the ansatz that all moments are expressed in terms of polynomials of some variable we show that each such solution corresponds to the sheffer class orthogonal polynomials it is shown that corresponding hankel determinants are related to some well known combinatorial numbers

one dimensional variational problems whose minimizers do not satisfy the euler lagrange equation in this paper we consider the problem of minimizing i u text int a b text f x u x u x dx 1 1 in the set a of absolutely continuous function u a b satisfying the end conditions u left a right text alpha text quad u left b right text beta text 1 2 where and are given constant in 1 1 a b is a finite interval denotes frac d dx and the integrand f f x u p is assumed to be smooth nonnegative and to satisfy the regularity condition f pp text 0 text 1 3

necessary conditions for solutions to variational problems we prove necessary conditions for a solution u to the problem of minimizing int omega f nabla v x g x v x dx in the form of a pontryagin maximum principle for f convex and satisfying a growth assumption but without assuming differentiability

sequences of transformations and triangular recursion schemes with applications in numerical analysis abstract in many fields of numerical analysis there appear transformations of the form t k v v k i v k i v t i when v varies a sequence of transformations is obtained this approach covers for example the e and algorithms the recursion formulae for b splines bernstein polynomials and orthogonal polynomials pade approximants the divided difference scheme and projection methods in this paper it will be proved that such transformations can be written as a ratio of determinants and can be recursively computed by a triangular recursion scheme the reciprocal of these results also holds furthermore we will show that t v k can be represented in terms of a complex contour integral throughout the paper we will study several examples in some detail and it will turn out that the application of our general theory leads to interesting new results in the special cases among others we will derive a new determinantal representation formula for b splines a recurrence relation for generalized bernstein polynomials a generalization of the e algorithm and we will prove that the algorithm can be represented as a quotient of determinants

computation of conservation laws for nonlinear lattices an algorithm to compute polynomial conserved densities of polynomial nonlinear lattices is presented the algorithm is implemented in mathematica and can be used as an automated integrability test with the code diffdens m conserved densities are obtained for several well known lattice equations for systems with parameters the code allows one to determine the conditions on these parameters so that a sequence of conservation laws exist

relational queries computable in polynomial time we characterize the polynomial time computable queries as those expressible in relational calculus plus a least fixed point operator and a total ordering on the universe we also show that even without the ordering one application of fixed point suffices to express any query expressible with several alternations of fixed point and negation this proves that the fixed point query hierarchy suggested by chandra and harel collapses at the first fixed point level it is also a general result showing that in finite model theory one application of fixed point suffices

bounded fixpoints for complex objects we study a query language for complex object databases which is designed to 1 express only tractable queries and 2 be as expressive over flat relations as first order logic with fixpoints the language is obtained by extending the nested relational algebra nra with a bounded fixpoint operator similar to results for flat relations all tractable queries over ordered databases are expressible in this language the main result consists in proving that this language is a conservative extension of the first order logic with fixpoints or of the while queries depending on the interpretation of the bounded fixpoint inflationary or partial that is a query from flat relations to flat relations is expressible in our language if and only if it is expressible in first order logic with fixpoints or in the while queries respectively the proof technique for this theorem uses indexes to encode complex objects into flat relations it can serve as basis for an implementation method of complex objects databases in terms of relational databases which works well for queries expressed both with fixpoints and with bounded fixpoint we also define a complex object logical calculus with fixpoints and prove that its range restricted fragment is equivalent to nra with bounded fixpoints

combinatorial expressions of the solutions to initial value problems of the discrete and ultradiscrete toda molecules combinatorial expressions are presented of the solutions to initial value problems of the discrete and ultradiscrete toda molecules for the discrete toda molecule a subtraction free expression of the solution is derived in terms of non intersecting paths for which two results in combinatorics flajolet s interpretation of continued fractions and gessel viennot s lemma on determinants are applied by ultradiscretizing the subtraction free expression the solution to the ultradiscrete toda molecule is obtained it is finally shown that the initial value problem of the ultradiscrete toda molecule is exactly solved in terms of shortest paths on a specific graph the behavior of the solution is also investigated in comparison with the box ball system

the euler equation in the multiple integrals calculus of variations for a multiple integrals problem in the calculus of variations we establish the validity of the euler equation when the lagrangian l satisfies a mild growth assumption from below at infinity we do not assume that the map l is differentiable or convex

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 integration of functions with poles near the interval of integration abstract an automatic quadrature scheme is presented for approximating integrals of functions that are analytic in the interval of integration but contain pole or poles of order 2 i e a double pole on the real axis or a complex conjugate pair of double poles near the interval of integration the present scheme is based on product integration rules of interpolatory type using function values of the abscissae only in the interval of integration the integral is approximated and evaluated by using recurrence relations and some extrapolation method after the smooth part of the integrand is expanded in terms of the chebyshev polynomials the fast fourier transform fft technique is used to generate efficiently the sequence of the finite chebyshev series expansions until an approximation of the integral satisfying the required tolerance is obtained with an adequate estimate of the error numerical examples are included to illustrate the performance of the method

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