a rational spectral collocation method for solving a class of parameterized singular perturbation problems a new kind of numerical method based on rational spectral collocation with the sinh transformation is presented for solving parameterized singularly perturbed two point boundary value problems with one boundary layer by means of the sinh transformation the original chebyshev points are mapped onto the transformed ones clustered near the singular points of the problem the results from asymptotic analysis as regards the singularity of the solution are employed to determine the parameters in the transformation numerical experiments including several nonlinear cases illustrate the high accuracy and efficiency of our method

barycentric interpolation collocation methods for solving linear and nonlinear high dimensional fredholm integral equations abstract in this article two barycentric interpolation collocation methods are proposed for solving linear and nonlinear high dimensional f r e d h o l m integral equations of the second kind the approaches respectively utilize the modified weighted l a g r a n g e functions and the novel rational functions as the interpolation basis functions they are effective schemes for evaluating the multidimensional undetermined function through the numerical strategies and some composite quadrature formulas the linear and nonlinear f r e d h o l m integral equations are transformed into the corresponding linear and nonlinear algebraic equations further we prove that the discrete collocation methods are equivalent to the n y s t r o m quadrature methods then the convergence analysis is established by the collectively compact theory moreover the error estimation of the approximate solution and the exact solution are also provided numerical examples are presented to illustrate the capability and efficiency of the techniques by compared with the classic l a g r a n g e interpolation collocation method and other methodologies

injectivity sets for the radon transform over circles and complete systems of radial functions abstract a necessary and sufficient characterization is given that specifies which sets of sums of translations of radial functions are dense in the set of continuous functions in the plane this problem is shown to be equivalent to inversion for the radon transform on circles centered on restricted subsets of the plane the proofs rest on the geometry of zero sets for harmonic polynomials and the microlocal analysis of this circular radon transform a characterization of nodal sets for the heat and wave equation in the plane are consequences of our theorems and questions of pinkus and ehrenpreis are answered

superconvergence and a posteriori error estimates of the dg method for scalar hyperbolic problems on cartesian grids in this paper we analyze the discontinuous galerkin dg finite element method for the steady two dimensional transport reaction equation on cartesian grids we prove the l2 stability and optimal l2 error estimates for the dg scheme we identify a special numerical flux for which the l2 norm of the solution is of order p 1 when tensor product polynomials of degree at most p are used we further prove superconvergence towards a particular projection of the directional derivative the order of superconvergence is proved to be p 1 2 we also provide a very simple derivative recovery formula which is o h p 1 superconvergent approximation to the directional derivative moreover we establish an o h 2 p 1 global superconvergence for the solution flux at the outflow boundary of the domain these results are used to construct asymptotically exact a posteriori error estimates for the directional derivative approximation by solving a local problem on each element finally we prove that the proposed a posteriori dg error estimates converge to the true errors in the l2 norm at o h p 1 rate and that the global effectivity index converges to unity at o h rate our results are valid without the flow condition restrictions we perform numerical experiments to demonstrate that theoretical rates proved in this paper are optimal

fast and stable rational interpolation in roots of unity and chebyshev points a new method for interpolation by rational functions of prescribed numerator and denominator degrees is presented when the interpolation nodes are roots of unity or chebyshev points the algorithm is particularly simple and relies on discrete fourier transform matrices which results in a fast implementation using the fast fourier transform the method is generalized for arbitrary grids which requires the construction of polynomials orthogonal on the set of interpolation nodes the appearance of common factors in the numerator and denominator due to finite precision arithmetic is explained by the behavior of the singular values of the linear system associated with the rational interpolation problem the new algorithm has connections with other methods particularly the work of jacobi and kronecker berrut and mittelmann and egecioglu and koc short matlab codes and numerical experiments are included

the set of unattainable points for the rational hermite interpolation problem abstract we describe geometrically and algebraically the set of unattainable points for the rational hermite interpolation problem i e those points where the problem does not have a solution we show that this set is a union of equidimensional complete intersection varieties of odd codimension the number of them being equal to the minimum between the degrees of the numerator and denominator of the problem each of these equidimensional varieties can be further decomposed as a union of as many rational irreducible varieties as input data points we exhibit algorithms and equations defining all these objects

the linear rational collocation method we introduce the collocation method based on linear rational interpolation for solving general hyperbolic problems prove its stability and its convergence in weighted norms and give numerical examples for its use

efficient hypersingular line and surface integrals direct evaluation by complex variable differentiation method abstract we present an efficient numerical scheme to evaluate hypersingular integrals appeared in boundary element methods the hypersingular integrals are first separated into regular and singular parts in which the singular integrals are defined as limits around the singularity and their values are determined analytically by taking the finite part values the remaining regular integrals can be evaluated using rational interpolatory quadrature or complex variable differentiation cvdm for the regular function when machine precision like accuracy is required the proposed method is then generalised for evaluating hypersingular surface integrals in which the inner integral is treated as the hypersingular line integral via coordinate transformations the procedure is implemented into 8 node rectangular boundary element and 6 node triangular element for numerical evaluation finally several numerical examples are presented to demonstrate the efficiency of the present method to the best of our knowledge the proposed method is more accurate faster and more generalised than other methods available in the literature to evaluate hypersingular integrals

on the lebesgue constant of barycentric rational hermite interpolants at equidistant nodes abstract barycentric rational floater hormann interpolants compare favourably to classical polynomial interpolants in the case of equidistant nodes because the lebesgue constant associated with these interpolants grows logarithmically in this setting in contrast to the exponential growth experienced by polynomials in the hermite setting in which also the first derivatives of the interpolant are prescribed at the nodes the same exponential growth has been proven for polynomial interpolants and the main goal of this paper is to show that much better results can be obtained with a recent generalization of floater hormann interpolants after summarizing the construction of these barycentric rational hermite interpolants we study the behaviour of the corresponding lebesgue constant and prove that it is bounded from above by a constant several numerical examples confirm this result

barycentric formulae for cardinal sinc interpolants we present a barycentric representation of cardinal interpolants as well as a weighted barycentric formula for their efficient evaluation we also propose a rational cardinal function which in some cases agrees with the corresponding cardinal interpolant and in other cases is even more accurate in numerical examples we compare the relative accuracy of those various interpolants with one another and with a rational interpolant proposed in former work

new representations of pade pade type and partial pade approximants pade approximants are rational functions with a denominator which does not vanish at zero and whose series expansion match a given series as far as possible these approximants are usually written under a rational form in this paper we will show how to write them also under two different barycentric forms and under a partial fraction form depending on free parameters according to the choice of these parameters pade type approximants can be obtained under a barycentric or a partial fraction form

a fast algorithm for rational interpolation via orthogonal polynomials l algorithme propose genere un ensemble de polynomes orthogonaux par la relation de recurrence a trois termes et ensuite utilise l interpolation de newton pour determiner les polynomes au numerateur et au denominateur de la fonction rationnelle d interpolation

on a rational differential quadrature method in irregular domains for problems with boundary layers abstract a rational differential quadrature method in irregular domains rdqmid is investigated to deal with a kind of singularly perturbed problems with boundary layers through a transformation the boundary layer which may be not straight is transformed into a segment of a line parallel to one of the cartesian axes the rational differential quadrature method rdqm is applied to discretize the governing equation finally a direct expansion method of the boundary conditions dembc is raised to deal with the boundary conditions numerical experiments show that rdqmid is of high accuracy efficiency and easy to programme

stable polefinding and rational least squares fitting via eigenvalues a common way of finding the poles of a meromorphic function f in a domain where an explicit expression of f is unknown but f can be evaluated at any given z is to interpolate f by a rational function pq such that r i f i at prescribed sample points i i 1l and then find the roots of q this is a two step process and the type of the rational interpolant needs to be specified by the user many other algorithms for polefinding and rational interpolation or least squares fitting have been proposed but their numerical stability has remained largely unexplored in this work we describe an algorithm with the following three features 1 it automatically finds an appropriate type for a rational approximant thereby allowing the user to input just the function f 2 it finds the poles via a generalized eigenvalue problem of matrices constructed directly from the sampled values f i in a one step fashion and 3 it computes rational approximants p q in a numerically stable manner in that p p q q f with small p q at the sample points making it the first rational interpolation or approximation algorithm with guaranteed numerical stability our algorithm executes an implicit change of polynomial basis by the qr factorization and allows for oversampling combined with least squares fitting through experiments we illustrate the resulting accuracy and stability which can significantly outperform existing algorithms

polynomial interpolation lagrange versus newton we show that the lagrangian form of the interpolating polynomial can be calculated with the same number of arithmetic operations as the newtonian form

lebesgue constant minimizing linear rational interpolation of continuous functions over the interval abstract polynomial interpolation between large numbers of arbitrary nodes does notoriously not in general yield useful approximations of continuous functions following 1 we suggest to determine for each set of n 1 nodes another denominator of degree n and then to interpolate every continuous function by a rational function with that same denominator so that the resulting interpolation process remains a linear projection the optimal denominator is chosen so as to minimize the lebesgue constant for the given nodes it has to be computed numerically for that purpose the barycentric representation of rational interpolants which displays the linearity of the interpolation and reduces the determination of the denominator to that of the barycentric weights is used the optimal weights can then be computed by solving an optimization problem with simple bounds which could not be solved accurately by the first author in 1 we show here how to do so and we present numerical results

an iterative approach to barycentric rational hermite interpolation in this paper we study an iterative approach to the hermite interpolation problem which first constructs an interpolant of the function values at n 1 nodes and then successively adds m correction terms to fit the data up to the mth derivatives in the case of polynomial interpolation this simply reproduces the classical hermite interpolant but the approach is general enough to be used in other settings in particular we focus on the family of rational floater hormann interpolants which are based on blending local polynomial interpolants of degree d with rational blending functions for this family the proposed method results in rational hermite interpolants which depend linearly on the data with numerator and denominator of degree at most m 1 n 1 1 and m 1 n d respectively they converge at the rate of o h m 1 d 1 as the mesh size h converges to 0 after deriving the barycentric form of these interpolants we prove the convergence rate for m 1 and m 2 and show that the approximation results compare favourably with other constructions

multirate hybrid adaptive control a design method is proposed based upon the use of several hybrid adaptive controllers which work in parallel at different sampling rates the stability of the overall adaptive scheme is preserved even in the presence of uniformly bounded disturbances the sampling periods are determined adaptively to the tracking error variations

numerical algorithms based on analytic function values at roots of unity let f z be an analytic or meromorphic function in the closed unit disk sampled at the n th roots of unity based on these data how can we approximately evaluate f z or f m z at a point z in the disk how can we calculate the zeros or poles of f in the disk these questions exhibit in the purest form certain algorithmic issues that arise across computational science in areas including integral equations partial differential equations and large scale linear algebra we analyze some of the possibilities and emphasize the distinction between algorithms based on polynomial or rational interpolation and those based on trapezoidal rule approximations of cauchy integrals we then show how these developments apply to the problem of computing the eigenvalues in the disk of a matrix of large dimension finally we highlight the power of rational in comparison with polynomial approximations for some of these problems

design of barycentric interpolators for uniform and nonuniform sampling grids this paper proposes a method to convert a fixed instant interpolator for band limited signals into a variable instant one which has the form of a barycentric interpolator this interpolator makes it possible to approximate the signal and its derivatives with minimal complexity in a range that surrounds the initial instant and is applicable to uniform and nonuniform sampling grids the barycentric form of the interpolator is derived using two different procedures first by the repeated use of bernstein s inequality and taylor s theorem and second by truncating a lagrange type series these procedures show that the proposed method can be applied to existing fixed instant interpolators and that it can be accurate in long time intervals finally an evaluation procedure for barycentric interpolators and their derivatives is presented that minimizes the number of divisions solving at the same time the numerical problems associated with small denominators this paper includes several numerical examples

an extension of the floater hormann family of barycentric rational interpolants the barycentric rational interpolants introduced by floater and hormann in 2007 are blends of polynomial interpolants of fixed degree d in some cases these rational functions achieve approximation of much higher quality than the classical polynomial interpolants which e g are ill conditioned and lead to runge s phenomenon if the interpolation nodes are equispaced for such nodes however the condition of floater hormann interpolation deteriorates exponentially with increasing d in this paper an extension of the floater hormann family with improved condition at equispaced nodes is presented and investigated the efficiency of its applications such as the approximation of derivatives integrals and primitives of functions is compared to the corresponding results recently obtained with the original family of rational interpolants math subject classification 65d05 65l12 65d32 41a05 41a20 41a25

spectrally accurate quadratures for evaluation of layer potentials close to the boundary for the 2d stokes and laplace equations dense particulate flow simulations using integral equation methods demand accurate evaluation of stokes layer potentials on arbitrarily close interfaces in this paper we generalize techniques for close evaluation of laplace double layer potentials in j helsing and r ojala j comput phys 227 2008 pp 2899 2921 we create a globally compensated trapezoid rule quadrature for the laplace single layer potential on the interior and exterior of smooth curves this exploits a complex representation a product quadrature in the style of kress for the sawtooth function careful attention to branch cuts and second kind barycentric type formulae for cauchy integrals and their derivatives upon this we build accurate single and double layer stokes potential evaluators by expressing them in terms of laplace potentials we test their convergence for vesicle vesicle interactions for an extensive set of laplace and stokes problems and when applying the system matrix in a boundary value problem solver

a class of reduced order models in the theory of waves and stability this paper presents a class of approximations to a type of wave field for which the dispersion relation is transcendental the approximations have two defining characteristics i they give the field shape exactly when the frequency and wavenumber lie on a grid of points in the frequency wavenumber plane and ii the approximate dispersion relations are polynomials that pass exactly through points on this grid thus the method is interpolatory in nature but the interpolation takes place in frequency wavenumber space rather than in physical space full details are presented for a non trivial example that of antisymmetric elastic waves in a layer the method is related to partial fraction expansions and barycentric representations of functions an asymptotic analysis is presented involving stirling s approximation to the psi function and a logarithmic correction to the polynomial dispersion relation

matrices for the direct determination of the barycentric weights of rational interpolation abstract let x0 xn be n 1 interpolation points nodes and 0 n be n 1 interpolation data then every rational function r with numerator and denominator degress n interpolating these values can be written in its barycentric form completely determined by a vector u of n 1 barycentric weights uk finding u is therefore an alternative to the determination of the coefficients in the canonical form of r it is advantageous inasmuch as u contains information about unattainable points and poles in classical rational interpolation the numerator and the denominator of r are made unique up to a constant factor by restricting their respective degrees we determine here the corresponding vectors u by applying a stabilized elimination algorithm to a matrix whose kernel is the space spanned by the u s the method is of complexity o n3 in terms of the denominator degree n it seems on the other hand to be among the most stable ones

conversions between barycentric rkfun and newton representations of rational interpolants we derive explicit formulas for converting between rational interpolants in barycentric rational krylov rkfun and newton form we show applications of these conversions when working with rational approximants produced by the aaa algorithm y nakatsukasa o sete l n trefethen arxiv preprint 1612 00337 2016 within the rational krylov toolbox and for the solution of nonlinear eigenvalue problems

a rational spectral collocation method for third order singularly perturbed problems a new numerical method is developed for solving a class of third order singularly perturbed boundary value problems first of all the given problem is transformed into a system of two ordinary differential equations odes subject to suitable initial and boundary conditions then the rational spectral collocation method in barycentric form with sinh transformation is applied to solve the system of odes according to the asymptotic analysis the location and width of boundary layer of the given problem which are chosen as parameters in the sinh transformation can be determined ample numerical experiments are presented to illustrate the computational efficiency and accuracy of the our method

on the denominator values and barycentric weights of rational interpolants we improve upon the method of zhu and zhu a method for directly finding the denominator values of rational interpolants j comput appl math 148 2002 341 348 for finding the denominator values of rational interpolants reducing considerably the number of arithmetical operations required for their computation in a second stage we determine the points if existent which can be discarded from the rational interpolation problem furthermore when the interpolant has a linear denominator we obtain a formula for the barycentric weights which is simpler than the one found by berrut and mittelmann matrices for the direct determination of the barycentric weights of rational interpolation j comput appl math 78 1997 355 370 subsequently we give a necessary and sufficient condition for the rational interpolant to have a pole

the barycentric weights of rational interpolation with prescribed poles abstract we introduce a method for calculating rational interpolants when some but not necessarily all of their poles are prescribed the algorithm determines the weights in the barycentric representation of the rationals it simply consists in multiplying each interpolated value by a certain number computing the weights of a rational interpolant without poles and finally multiplying the weights by those same numbers the supplementary cost in comparison with interpolation without poles is about v 2 n where v is the number of poles and n the number of interpolation points we also give a condition under which the computed rational interpolation really shows the desired poles