fast calculation of laurent expansions for matrix inverses previously described algorithms for calculating the laurent expansion of the inverse of a matrix valued analytic function become impractical already for singularity orders as low as around p 6 since they require over o 28p matrix multiplications and correspondingly large amounts of memory in place of using mathematically exact recursions we show that for floating point calculations a rational approximation approach can avoid this cost barrier without any significant loss in accuracy

a meshfree approach to non newtonian free surface ice flow numerical models of glacier and ice sheet dynamics traditionally employ finite difference or finite element methods although these are highly developed and mature methods they suffer from some drawbacks such as inability to handle complex geometries finite differences or a costly assembly procedure for nonlinear problems finite elements additionally they are mesh based and therefore moving domains become a challenge in this paper we introduce a novel meshfree approach based on a radial basis function rbf method the meshfree nature of rbf methods enables efficient handling of moving margins and free ice surface rbf methods are also accurate easy to implement and allow for reduction the computational cost associated with the linear system assembly since stated in strong form to demonstrate the global rbf method we model the velocity field of ice flow in the haut glacier d arolla which is governed by the nonlinear stokes equations we test the method for different basal conditions and for a free moving surface we also compare the global rbf method with its localized counterpart the rbf partition of unity method rbf pum that allows for a significant gain in the computational efficiency both rbf methods are compared with the classical finite element method in terms of accuracy and efficiency we find that the rbf methods are more efficient than the finite element method and well suited for ice dynamics modeling especially the partition of unity approach

analytic perturbation of generalized inverses abstract we investigate the analytic perturbation of generalized inverses firstly we analyze the analytic perturbation of the drazin generalized inverse also known as reduced resolvent in operator theory our approach is based on spectral theory of linear operators as well as on a new notion of group reduced resolvent it allows us to treat regular and singular perturbations in a unified framework we provide an algorithm for computing the coefficients of the laurent series of the perturbed drazin generalized inverse in particular the regular part coefficients can be efficiently calculated by recursive formulae finally we apply the obtained results to the perturbation analysis of the moore penrose generalized inverse in the real domain

a numerical study of some radial basis function based solution methods for elliptic pdes abstract during the last decade three main variations have been proposed for solving elliptic pdes by means of collocation with radial basis functions rbfs in this study we have implemented them for infinitely smooth rbfs and then compared them across the full range of values for the shape parameter of the rbfs this was made possible by a recently discovered numerical procedure that bypasses the ill conditioning which has previously limited the range that could be used for this parameter we find that the best values for it often fall outside the range that was previously available we have also looked at piecewise smooth versus infinitely smooth rbfs and found that for pde applications with smooth solutions the infinitely smooth rbfs are preferable mainly because they lead to higher accuracy in a comparison of rbf based methods against two standard techniques a second order finite difference method and a pseudospectral method the former gave a much superior accuracy

theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions multivariate interpolation of smooth data using smooth radial basis functions is considered the behavior of the interpolants in the limit of nearly flat radial basis functions is studied both theoretically and numerically explicit criteria for different types of limits are given using the results for the limits the dependence of the error on the shape parameter of the radial basis function is investigated the mechanisms that determine the optimal shape parameter value are studied and explained through approximate expansions of the interpolation error

stable computation of multiquadric interpolants for all values of the shape parameter abstract spectrally accurate interpolation and approximation of derivatives used to be practical only on highly regular grids in very simple geometries since radial basis function rbf approxima ions permit this even for multivariate scattered data there has been much recent interest in practical algorithms to compute these approximations effectively several types of rbfs feature a free parameter e g c in the multiquadric mq case f r r 2 c 2 the limit of c increasingly flat basis functions has not received much attention because it leads to a severely ill conditioned problem we present here an algorithm which avoids this difficulty and which allows numerically stable computations of mq rbf interpolants for all parameter values we then find that the accuracy of the resulting approximations in some cases becomes orders of magnitude higher than was the case within the previously available parameter range our new method provides the first tool for the numerical exploration of mq rbf interpolants in the limit of c the method is in no way specific to mq basis functions and can without any change be applied to many other cases as well

kramer analytic kernels and first order boundary value problems this paper is concerned with the generation of kramer analytic kernels from first order linear ordinary boundary value problems these kernels are obtained from boundary value problems that are represented by self adjoint differential operators necessary and sufficient conditions are given to ensure that these differential operators have a discrete spectrum which then allows of the introduction of the associated kramer analytic kernel an example is considered which leads to the important shannon whittaker interpolation expansion theorem

moments in infinite channel queues we consider a new approach to the problem of evaluating the moments of the steady state number of customers in system b for an infinite channel queue having general input our new approach uses the rate conservation law as a consequence we show that for a variety of g gi queues var b as long as service times have finite first moment in addition we give some ordering relations for the variance of b when comparing two alternative systems with convex ordered inter arrival time distributions

the runge phenomenon and spatially variable shape parameters in rbf interpolation many studies mostly empirical have been devoted to finding an optimal shape parameter for radial basis functions rbf when exploring the underlying factors that determine what is a good such choice we are led to consider the runge phenomenon rp best known in cases of high order polynomial interpolation as a key error mechanism this observation suggests that it can be advantageous to let the shape parameter vary spatially rather than assigning a single value to it benefits typically include improvements in both accuracy and numerical conditioning still another benefit arises if one wishes to improve local accuracy by clustering nodes in selected areas this idea is routinely used when working with splines or finite element methods however local refinement with rbfs may cause rp type errors unless we use a spatially variable shape paremeter with this enhancement rbf approximations combine freedom from meshes with spectral accuracy on irregular domains and furthermore permit local node clustering to improve the resolution wherever this might be needed

a high order multivariate approximation scheme for scattered data sets we present a high order multivariate approximation scheme for scattered data sets each data point is represented as a taylor series and the high order derivatives in the taylor series are treated as random variables the approximation coefficients are then chosen to minimize an objective function at each point by solving an equality constrained least squares the approximation is an interpolation when the data points are given as exact or a nonlinear regression function when nonzero measurement errors are associated with the data points using this formulation the gradient information on each data point can be used to significantly reduce the approximation error all parameters of the approximation scheme can be computed automatically from the data points an uncertainty bound of the approximation function is also produced by the scheme numerical experiments demonstrate that although this method is more computationally intensive than traditional methods it produces more accurate approximation functions

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

incomplete lipschitz hankel integrals of macdonald functions abstract various representations for incomplete lipschitz hankel integrals of macdonald functions have been given in closed form using kampe de feriet double hypergeometric functions in addition reduction formulas for the kampe de feriet functions associated with these integrals have been derived in some cases

application of the rbf meshless method to the solution of the radiative transport equation in this paper we introduce a radial basis function collocation method for computing solutions to the time dependent radiative transfer equation for these computations we use finite differences to discretize the time coordinate a discrete ordinate method to discretize the directional variable and an expansion in radial basis functions to approximate the spatial dependence of the solution the main advantages of the rbf method are that it does not require any mesh or grid achieves spectral accuracy in multi dimensions for arbitrary node layouts and it is extremely simple to implement

an incomplete lipschitz hankel integral of k0 on etudie k e0 a z 0 z exp at k 0 t dt ou k est la fonction de macdonald ou fonction de bessel d argument imaginaire on montre que k e0 a z peut s ecrire sous forme close en termes de fonctions elementaires k 0 k 1 et des fonctions hypergeometriques doubles de kampe de feriet

a stable algorithm for flat radial basis functions on a sphere when radial basis functions rbfs are made increasingly flat the interpolation error typically decreases steadily until some point when runge type oscillations either halt or reverse this trend because the most obvious method to calculate an rbf interpolant becomes a numerically unstable algorithm for a stable problem in the case of near flat basis functions there will typically also be a separate point at which disastrous ill conditioning enters we introduce here a new method rbf qr which entirely eliminates such ill conditioning and we apply it in the special case when the data points are distributed over the surface of a sphere this algorithm works even for thousands of node points and it allows the rbf shape parameter to be optimized without the limitations imposed by stability concerns since interpolation in the flat rbf limit on a sphere is found to coincide with spherical harmonics interpolation new insights are gained as to why the rbf approach with nonflat basis functions often is the more accurate of the two methods

frequency optimized rbf fd for wave equations abstract we present a method to obtain optimal rbf fd formulas which maximize their frequency range of validity the optimization is based on the idea of keeping an error of interest dispersion phase or group velocity errors below a given threshold for a wavenumber interval as large as possible to find the weights of these optimal finite difference formulas we solve an optimization problem in a previous work we developed a method to optimize the frequency range of validity for finite difference weights that method required to solve a system of nonlinear equations with as many unknowns as half of the number of weights which is a very hard task when the number of nodes gets large the current method requires solving an optimization problem with only one parameter which makes finding a global minimum easier and thus can be used for bigger stencils we also study which of the standard rbf are more appropriate for this problem and introduce a new rbf that depends on two parameters this new rbf improves the resulting frequency response of the rbf fd methods while keeping the cost of the optimization problem low

interpolation in the limit of increasingly flat radial basis functions abstract many types of radial basis functions such as multiquadrics contain a free parameter in the limit where the basis functions become increasingly flat the linear system to solve becomes highly ill conditioned and the expansion coefficients diverge nevertheless we find in this study that limiting interpolants often exist and take the form of polynomials in the 1 d case we prove that with simple conditions on the basis function the interpolants converge to the lagrange interpolating polynomial hence differentiation of this limit is equivalent to the standard finite difference method we also summarize some preliminary observations regarding the limit in 2 d

impossibility of fast stable approximation of analytic functions from equispaced samples it is shown that no stable procedure for approximating functions from equally spaced samples can converge exponentially for analytic functions to avoid instability one must settle for root exponential convergence the proof combines a bernstein inequality of 1912 with an estimate due to coppersmith and rivlin in 1992

improved radial basis function methods for multi dimensional option pricing in this paper we have derived a radial basis function rbf based method for the pricing of financial contracts by solving the black scholes partial differential equation as an example of a financial contract that can be priced with this method we have chosen the multi dimensional european basket call option we have shown numerically that our scheme is second order accurate in time and spectrally accurate in space for constant shape parameter for other non optimal choices of shape parameter values the resulting convergence rate is algebraic we propose an adapted node point placement that improves the accuracy compared with a uniform distribution compared with an adaptive finite difference method the rbf method is 20 40 times faster in one and two space dimensions and has approximately the same memory requirements

stable computations with gaussian radial basis functions radial basis function rbf approximation is an extremely powerful tool for representing smooth functions in nontrivial geometries since the method is mesh free and can be spectrally accurate a perceived practical obstacle is that the interpolation matrix becomes increasingly ill conditioned as the rbf shape parameter becomes small corresponding to flat rbfs two stable approaches that overcome this problem exist the contour pade method and the rbf qr method however the former is limited to small node sets and the latter has until now been formulated only for the surface of the sphere this paper focuses on an rbf qr formulation for node sets in one two and three dimensions the algorithm is stable for arbitrarily small shape parameters it can be used for thousands of node points in two dimensions and still more in three dimensions a sample matlab code for the two dimensional case is provided

the asymptotic cardinal function of the multiquadratic ϕ r r2 c2 12as c a radial basis function approximation has the form s x j rd yj x xj 2 x rd where 0 r is some given function yj j zd are real coefficients and the centres xj j zd are points in rd it is known that radial basis function approximations using the multiquadric r r2 c2 12 possess many useful and interesting properties when the centres form an infinite regular lattice we analyse the limiting case as c and identify a class of functions that arise as uniform limits of the multiquadric interpolants in the univariate case we observe that the cardinal function for the multiquadratic becomes the sinc function as c the limit of the multivariate cardinal function is also identified

laurent expansion of the inverse of perturbed singular matrices in this paper we describe a numerical algorithm to compute the laurent expansion of the inverse of singularly perturbed matrices the algorithm is based on the resolvent formalism used in complex analysis to study the spectrum of matrices the input of the algorithm are the matrix coefficients of the power series expansion of the perturbed matrix the matrix coefficients of the laurent expansion of the inverse are computed using recursive analytical formulae we show that the computational complexity of the proposed algorithm grows algebraically with the size of the matrix but exponentially with the order of the singularity we apply this algorithm to several matrices that arise in applications we make special emphasis to interpolation problems with radial basis functions

an augmented rbf method for solving fractional sturm liouville eigenvalue problems in this work we construct numerical schemes using radial basis functions rbfs to solve sturm liouville eigenvalue problems in which the so called fractional derivatives are involved we take into account the fact that the solutions to such problems might be nonsmooth and consequently we augment the space generated by rbf base functions with some fractional polynomials to overcome this problem we show through some examples the effectiveness of our method

some observations and remarks on differential operators generated by first order boundary value problems this paper deals with the study of the set of all self adjoint differential operators which are generated from first order linear ordinary boundary value problems these operators are defined on a weighted hilbert function space and are examined as an application of the result obtained by everitt and markus in their paper in 1997 an investigation is given so that first order self adjoint boundary value problems are transformed to a study of the nature of the spectrum of associated self adjoint operators however the analysis of this paper is restricted to consideration of conditions under which the spectral properties of these operators yield a discrete spectrum and consequently to the determination of conditions under which the construction of kramer analytic kernels from the above boundary value problems can be accomplished

radial basis function interpolation in the limit of increasingly flat basis functions abstract we propose a new approach to study radial basis function rbf interpolation in the limit of increasingly flat functions the new approach is based on the semi analytical computation of the laurent series of the inverse of the rbf interpolation matrix described in a previous paper 3 once the laurent series is obtained it can be used to compute the limiting polynomial interpolant the optimal shape parameter of the rbfs used for interpolation and the weights of rbf finite difference formulas among other things

a generalized inverse method for asymptotic linear programming consider a linear program in which the entries of the coefficient matrix vary linearly with time to study the behavior of optimal solutions as time goes to infinity it is convenient to express the inverse of the basis matrix as a series expansion of powers of the time parameter we show that an algorithm of wilkinson 1982 for solving singular differential equations can be used to obtain such an expansion efficiently the resolvent expansions of dynamic programming are a special case of this method

an efficient basis update for asymptotic linear programming abstract for a linear program in which the constraint coefficients vary linearly with the time parameter we showed in a previous paper that a basic feasible solution can be evaluated using o k 1 m 3 arithmetic operations where m is the number of constraints and k is the index of the basis matrix pair here we show in the special case when k 1 for all basis matrix pairs and when one of the matrices in each pair has nearly full rank how the possibly singular matrix factorization can be updated with only o m 2 operations using rank one update techniques this makes the arithmetic complexity of updating the basis in asymptotic linear programming comparable to that of updating the inverse in ordinary linear programming in this case moreover we show that the result holds in particular when computing a blackwell optimal policy for markov decision chains in the unichain case or when all policies have only a small number of recurrent subchains

perturbation of null spaces with application to the eigenvalue problem and generalized inverses we consider properties of a null space of an analytically perturbed matrix in particular we obtain taylor expansions for the eigenvectors which constitute a basis for the perturbed null space furthermore we apply these results to the calculation of puiseux expansion of the perturbed eigenvectors in the case of general eigenvalue problem as well as to the calculation of laurent series expansions for the perturbed group inverse and pseudoinverse matrices

some observations regarding interpolants in the limit of flat radial basis functions abstract radial basis functions rbfs form a primary tool for multivariate interpolation some of the most commonly used radial functions feature a shape parameter allowing them to vary from being nearly flat e small to sharply peaked e large the former limit can be particularly accurate when interpolating a smooth function based on scattered data this study discusses theoretical and computational aspects of the e 0 limit and includes the conjecture that gaussian rbf interpolants will never diverge in this limit

a reduction process for perturbed markov chains given a perturbed markov chain with generator a varepsilon a 0 varepsilon a 1 varepsilon 2 a 2 cdots it is shown that to each time scale t t varepsilon t varepsilon 2 cdots is associated a reduced markov chain this process of reduction has only finitely many steps it is given in a recursive way and each reduced chain admits as state space the recurrent classes of the reduced chain defined at the previous stage it is shown that if lambda is an eigenvalue different from zero of the generator of the kth reduced chain with multiplicity p then a varepsilon admits p eigenvalues lambda varepsilon such that lambda varepsilon varepsilon k lambda o varepsilon k a basis of ker a 0 resp ker a 0 which is robust to the perturbation is constructed

fully meshless solution of the one dimensional multigroup neutron transport equation with the radial basis function collocation method abstract in this paper a fully meshless method is proposed for the numerical solution of the one dimensional multigroup neutron transport equation with anisotropic scattering both first order and even parity forms of the transport equation are studied the radial basis function collocation method is chosen for the spatial treatment and legendre polynomials are used to approximate the angular variable the selection of the legendre polynomials instead of discrete ordinates approach resulted with a fully meshless algorithm in both independent variables multiquadric is utilized as the radial function seven problems are considered to evaluate the performance of the method the results show that the method converges exponentially and it is possible to obtain high levels of accuracies for the multiplication factor and neutron flux with a good stability in both spatial and angular domains for the one group isotropic benchmark problem discrete ordinates solutions employing discontinuous linear finite elements for the spatial variable are also provided and a comparison of the methods revealed that the fully meshless method produced more accurate results than the discrete ordinates finite element scheme when the shape parameter is properly chosen

limit problems for interpolation by analytic radial basis functions interpolation problems for analytic radial basis functions like the gaussian and inverse multiquadrics can degenerate in two ways the radial basis functions can be scaled to become increasingly flat or the data points coalesce in the limit while the radial basis functions stay fixed both cases call for a careful regularization which if carried out explicitly yields a preconditioning technique for the degenerating linear systems behind these interpolation problems this paper deals with both cases for the increasingly flat limit we recover results by larsson and fornberg together with lee yoon and yoon concerning convergence of interpolants towards polynomials with slight modifications the same technique can also handle scenarios with coalescing data points for fixed radial basis functions the results show that the degenerating local lagrange interpolation problems converge towards certain hermite birkhoff problems this is an important prerequisite for dealing with approximation by radial basis functions adaptively using freely varying data sites

miscellaneous error bounds for multiquadric and related interpolators we establish several types of a a priori error bounds for multiquadric and related interpolators the results are stated and proven in the general multivariate case these estimates show for example that in many cases such interpolators converge very quickly and can be used in the recovery of band limited functions from discrete data we also include numerical experiments which illustrate the theoretical results

the parameter r2 in multiquadric interpolation the multiquadric mq method is an effective bivariate interpolant to three dimensional data xi yi zi where the xi yi are arbitrarily located in the plane the accuracy of the mq method is dependent on a user defined parameter r2 and most practitioners select r2 based upon the number of data points and the locations of the xi yi in the plane we observe that the optimal value of r2 is a strong function of the zi and that it is essentially independent of both the number and the locationsof the xi yi points contrary to some opinions we observe that the mq method can effectively interpolate to track data that is data sampled densely along tracks in the plane together with several other observations we present an algorithm that generally yields an effective value for r2

a sonine gegenbauer integral of the neumann function abstract motivated by a desire to express in closed form a certain potential occurring in the study of diffraction of a plane electromagnetic wave by a wedge miller and exton developed computable expressions for a large class of sonine gegenbauer type integrals in the present paper one of these integrals given previously in terms of generalized hypergeometric functions in three variables is now obtained in terms of hypergeometric functions in two variables thereby much reducing the complexity of the representation in the course of this investigation certain identities involving srivastava s f 3 function kampe de feriet functions and wright functions are deduced in addition evaluations of integrals of bessel functions related to the above analysis are provided

sonine gegenbauer type integrals abstract representations for complete and incomplete sonine gegenbauer type integrals are given in terms of generalized hypergeometric functions in two and three variables and in terms of neumann type series certain interesting reductions of these integrals are also given in addition an application to a problem in diffraction theory is provided