measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

convergence analysis of a high order nystrom integral equation method for surface scattering problems in this paper we present a convergence analysis for the nystrom method proposed in j comput phys 169 1 80 110 2001 for the solution of the combined boundary integral equation formulations of sound soft acoustic scattering problems in three dimensional space this fast and efficient scheme combines fft techniques and a polar change of variables that cancels out the kernel singularity we establish the stability of the algorithms in the l 2 norm and we derive convergence estimates in both the l 2 and l infty norms in particular our analysis establishes theoretically the previously observed super algebraic convergence of the method in cases in which the right hand side is smooth

transient behavior of the m g 1 workload process in this paper we describe the time dependent moments of the workload process in the m g 1 queue the kth moment as a function of time can be characterized in terms of a differential equation involving lower moment functions and the time dependent server occupation probability for general initial conditions we show that the first two moment functions can be represented as the difference of two nondecreasing functions one of which is the moment function starting at zero the two nondecreasing components can be regarded as probability cumulative distribution function cdf s after appropriate normalization the normalized moment functions starting empty are called moment cdf s the other normalized components are called moment difference cdf s we establish relations among these cdf s using stationary excess relations we apply these relations to calculate moments and derivatives at the origin of these cdf s we also obtain results for the covariance function of the stationary workload process it is inte

an efficient implicit fem scheme for fractional in space reaction diffusion equations fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity however the presence of a fractional differential operator causes memory time fractional or nonlocality space fractional issues that impose a number of computational constraints in this paper we develop efficient scalable techniques for solving fractional in space reaction diffusion equations using the finite element method on both structured and unstructured grids via robust techniques for computing the fractional power of a matrix times a vector our approach is show cased by solving the fractional fisher and fractional allen cahn reaction diffusion equations in two and three spatial dimensions and analyzing the speed of the traveling wave and size of the interface in terms of the fractional power of the underlying laplacian operator

a projection method for generalized eigenvalue problems using numerical integration in this paper we propose a method for finding certain eigenvalues of a generalized eigenvalue problem that lie in a given domain of the complex plane the proposed method projects the matrix pencil onto a subspace associated with the eigenvalues that are located in the domain via numerical integration

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

laplace s equation and the dirichlet neumann map a new mode for mikhlin s method mikhlin s method for solving laplace s equation in domains exterior to a number of closed contours is discussed with particular emphasis on the dirichlet neumann map in the literature there already exist two computational modes for mikhlin s method here a new mode is presented the new mode is at least as stable as the previous modes furthermore its computational complexity in the number of closed contours is better as a result highly accurate solutions in domains exterior to tens of thousands of closed contours can be obtained on a simple workstation

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

numerical evaluation of two and three parameter mittag leffler functions the mittag leffler ml function plays a fundamental role in fractional calculus but very few methods are available for its numerical evaluation in this work we present a method for the efficient computation of the ml function based on the numerical inversion of its laplace transform lt an optimal parabolic contour is selected on the basis of the distance and the strength of the singularities of the lt with the aim of minimizing the computational effort and reducing the propagation of errors numerical experiments are presented to show accuracy and efficiency of the proposed approach the application to the three parameter ml also known as prabhakar function is also presented

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

rapid solution of the wave equation in unbounded domains in this paper we propose and analyze a new fast method for the numerical solution of time domain boundary integral formulations of the wave equation we employ lubich s convolution quadrature method for the time discretization and a galerkin boundary element method for the spatial discretization the coefficient matrix of the arising system of linear equations is a triangular block toeplitz matrix possible choices for solving the linear system arising from the above discretization include the use of fast fourier transform fft techniques and the use of data sparse approximations by using fft techniques the computational complexity can be reduced substantially while the storage cost remains unchanged and is typically high using data sparse approximations the gain is reversed i e the computational cost is approximately unchanged while the storage cost is substantially reduced the method proposed in this paper combines the advantages of these two approaches first the discrete convolution related to the block toeplitz system is transformed into the discrete fourier image thereby arriving at a decoupled system of discretized helmholtz equations with complex wave numbers a fast data sparse e g fast multipole or panel clustering method can then be applied to the transformed system additionally significant savings can be achieved if the boundary data are smooth and time limited in this case the right hand sides of many of the helmholtz problems are almost zero and hence can be disregarded finally the proposed method is inherently parallel we analyze the stability and convergence of these methods thereby deriving the choice of parameters that preserves the convergence rates of the unperturbed convolution quadrature we also present numerical results which illustrate the predicted convergence behavior

fast and accurate computation of gauss legendre and gauss jacobi quadrature nodes and weights an efficient algorithm for the accurate computation of gauss legendre and gauss jacobi quadrature nodes and weights is presented the algorithm is based on newton s root finding method with initial guesses and function evaluations computed via asymptotic formulae the n point quadrature rule is computed in mathcal o n operations to an accuracy of essentially double precision for any n geq 100

robust pade approximation via svd pade approximation is considered from the point of view of robust methods of numerical linear algebra in particular the singular value decomposition this leads to an algorithm for practical computation that bypasses most problems of solution of nearly singular systems and spurious pole zero pairs caused by rounding errors for which a matlab code is provided the success of this algorithm suggests that there might be variants of pade approximation that are pointwise convergent as the degrees of the numerator and denominator increase to infty unlike traditional pade approximants which converge only in measure or capacity

explicit solutions for transcendental equations a simple method to formulate an explicit expression for the roots of any analytic transcendental function is presented the method is based on cauchy s integral theorem and uses only basic concepts of complex integration one convenient method for numerically evaluating the exact expression is presented the application of both the formulation and evaluation of the exact expression is illustrated for several classical root finding problems

numerical calculation of incomplete gamma functions by the trapezoidal rule the trapezoidal rule is applied to the numerical calculation of a known integral representation of the complementary incomplete gamma function a x in the regiona 0 since this application of the rule is not standard a careful investigation of the remainder terms using the euler maclaurin formula is carried out the outcome is a simple numerical procedure for obtaining values of incomplete gamma functions with surprising accuracy in the stated region

an integral method for solving nonlinear eigenvalue problems we propose a numerical method for computing all eigenvalues and the corresponding eigenvectors of a nonlinear holomorphic eigenvalue problem that lie within a given contour in the complex plane the method uses complex integrals of the resolvent operator applied to at least k column vectors where k is the number of eigenvalues inside the contour the theorem of keldysh is employed to show that the original nonlinear eigenvalue problem reduces to a linear eigenvalue problem of dimension k no initial approximations of eigenvalues and eigenvectors are needed the method is particularly suitable for moderately large eigenvalue problems where k is much smaller than the matrix dimension we also give an extension of the method to the case where k is larger than the matrix dimension the quadrature errors caused by the trapezoid sum are discussed for the case of analytic closed contours using well known techniques it is shown that the error decays exponentially with an exponent given by the product of the number of quadrature points and the minimal distance of the eigenvalues to the contour

computing fresnel integrals via modified trapezium rules in this paper we propose methods for computing fresnel integrals based on truncated trapezium rule approximations to integrals on the real line these trapezium rules modified to take into account poles of the integrand near the real axis our starting point is a method for computation of the error function of complex argument due to matta and reichel j math phys 34 298 307 1956 and hunter and regan math comp 26 539 541 1972 we construct approximations which we prove are exponentially convergent as a function of n n the number of quadrature points obtaining explicit error bounds which show that accuracies of 10 15 10 15 uniformly on the real line are achieved with n 12 n 12 this confirmed by computations the approximations we obtain are attractive additionally in that they maintain small relative errors for small and large argument are analytic on the real axis echoing the analyticity of the fresnel integrals and are straightforward to implement

lattice sums for the helmholtz equation a survey of different representations for lattice sums for the helmholtz equation is made these sums arise naturally when dealing with wave scattering by periodic structures one of the main objectives is to show how the various forms depend on the dimension d of the underlying space and the lattice dimension d lambda lattice sums are related to and can be calculated from the quasi periodic green s function and this object serves as the starting point of the analysis

a circular interpretation of the euler maclaurin formula the present work makes the case for viewing the euler maclaurin formula as an expression for the effect of a jump on the accuracy of riemann sums on circles and draws some consequences thereof e g when the integrand has several jumps on the way we give a construction of the bernoulli polynomials tailored to the proof of the formula and we show how extra jumps may lead to a smaller quadrature error

a fast and well conditioned spectral method for singular integral equations abstract we develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing chebyshev and ultraspherical polynomials to reformulate the equations as almost banded infinite dimensional systems this is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels the resulting system can be solved in o m 2 n operations using an adaptive qr factorization where m is the bandwidth and n is the optimal number of unknowns needed to resolve the true solution the complexity is reduced to o m n operations by pre caching the qr factorization when the same operator is used for multiple right hand sides stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity applications considered include the faraday cage and acoustic scattering for the helmholtz and gravity helmholtz equations including spectrally accurate numerical evaluation of the far and near field solution the julia software package singularintegralequations jl implements our method with a convenient user friendly interface

dynamic transient and stationary behavior of the m gi 1 queue via martingales l auteur associe une martingale exponentielle a la chaine de markov du nombre de clients d une file d attente m gi 1 il demontre a l aide de la theorie de renouvellement que cette martingale fournit un cadre probabiliste unifie pour deduire differentes fonctions generatrices usuelles pour les files d attente m gi 1

a spectral order method for inverting sectorial laplace transforms laplace transforms which admit a holomorphic extension to some sector strictly containing the right half plane and exhibiting a potential behavior are considered a spectral order parallelizable method for their numerical inversion is proposed the method takes into account the available information about the errors arising in the evaluations several numerical illustrations are provided

equivalence of nystrom s method and fourier methods for the numerical solution of fredholm integral equations on demontre un resultat d equivalence de la methode de nystrom et de la methode de fourier avec des fonctions orthogonales pour la solution numerique des equations integrales de fredholm on en deduit des estimations d erreur pour la methode de nystrom on decrit exactement la regle de quadrature pour l integration numerique

on mittag leffler type functions in fractional evolution processes abstract we review a variety of fractional evolution processes so defined being governed by equations of fractional order whose solutions turn out to be related to mittag leffler type functions the chosen equations are the simplest of the fractional calculus and include the abel integral equations of the second kind which are relevant in typical inverse problems and the fractional differential equations which govern generalized relaxation and oscillation phenomena

on the evaluation of layer potentials close to their sources when solving elliptic boundary value problems using integral equation methods one may need to evaluate potentials represented by a convolution of discretized layer density sources against a kernel standard quadrature accelerated with a fast hierarchical method for potential field evaluation gives accurate results far away from the sources close to the sources this is not so cancellation and nearly singular kernels may cause serious degradation this paper presents a new scheme based on a mix of composite polynomial quadrature layer density interpolation kernel approximation rational quadrature high polynomial order corrected interpolation and differentiation temporary panel mergers and splits and a particular implementation of the gmres solver criteria for which mix is fastest and most accurate in various situations are also supplied the paper focuses on the solution of the dirichlet problem for laplace s equation in the plane in a series of examples we demonstrate the efficiency of the new scheme for interior domains and domains exterior to up to 2000 close to touching contours densities are computed and potentials are evaluated rapidly and accurate to almost machine precision at points that lie arbitrarily close to the boundaries

diagnosis of dynamical systems with fluctuating parameters many time evolutions occurring in nature may be considered as non autonomous but dependent on parameters that vary slowly with time it is argued here that some but not all of the tools used to understand chaotic dynamics remain useful in this situation

h matrix approximation for the operator exponential with applications we develop a data sparse and accurate approximation to parabolic solution operators in the case of a rather general elliptic part given by a strongly p positive operator 4

the kink phenomenon in fejer and clenshaw curtis quadrature the fejer and clenshawcurtis rules for numerical integration exhibit a curious phenomenon when applied to certain analytic functions when n the number of points in the integration rule increases the error does not decay to zero evenly but does so in two distinct stages for n less than a critical value the error behaves like o 2n where is a constant greater than 1 for these values of n the accuracy of both the fejer and clenshawcurtis rules is almost indistinguishable from that of the more celebrated gausslegendre quadrature rule for larger n however the error decreases at the rate o n i e only half as fast as before convergence curves typically display a kink where the convergence rate cuts in half in this paper we derive explicit as well as asymptotic error formulas that provide a complete description of this phenomenon

extension of a class of periodizing variable transformations for numerical integration class s m variable transformations with integer m for numerical computation of finite range integrals were introduced and studied by the author in the paper a sidi a new variable transformation for numerical integration numerical integration iv 1993 h brass and g hammerlin eds pp 359 373 a representative of this class is the sin m transformation that has been used with lattice rules for multidimensional integration these transformations periodize the integrand functions in a way that enables the trapezoidal rule to achieve very high accuracy especially with even m in the present work we extend these transformations to arbitrary values of m and give a detailed analysis of the resulting transformed trapezoidal rule approximations we show that with suitable m they can be very useful in different situations we prove for example that if the integrand function is smooth on the interval of integration add vanishes at the endpoints then results of especially high accuracy are obtained by taking 2m to be an odd integer such a situation can be realized in general by subtracting from the integrand the linear interpolant at the endpoints of the interval of integration we also illustrate some of the results with numerical examples via the extended sin m transformation

algorithms for parameter selection in the weeks method for inverting the laplace transform the weeks method is one of the most efficient numerical techniques for inverting the laplace transform provided two free parameters in the laguerre expansion on which the method is based are chosen judiciously however there appear to be no theoretical estimates or numerical algorithms for computing these parameters the two algorithms presented in this paper fill that gap both algorithms aim to minimize a theoretical error estimate and both are implemented with the fft the first algorithm is restricted to a certain class of transforms and the user is expected to provide information on the singularities of the transform the second algorithm though more expensive is completely general no singularity information is required in challenging numerical tests both algorithms successfully predicted values of the parameters that are near optimal

a modification of the weeks method for numerical inversion of the laplace transform the weeks method for calculating numerical values of f t from its laplace transform f s involves the numerical evaluation of a set of contour integrals in this paper we describe a powerful modification by means of which results of comparable accuracy can be obtained using approximately half the number of complex function values this and other possible minor modifications are discussed at any values of s we like we wish to construct an algorithm which obtains or uses this information to evaluate f t at a given selection of values of t real to a specified accuracy one may assume that f t is real in which case f s f in almost all methods one needs to know or one has to find out first the numerical value of the laplace convergence abscissa ao this may be defined as the limit of the set of values of re s for which this integral converges or alternatively as the maximum of the real parts of the singular points sj of f s a good brief description of the formulas on which the weeks method 2 is based is given in piessens and branders 1 in order to provide a proper background we repeat some of these here briefly we expand f t in terms of laguerre functions thus we assume that

extension of chebfun to periodic functions algorithms and underlying mathematics are presented for numerical computation with periodic functions via approximations to machine precision by trigonometric polynomials including the solution of linear and nonlinear periodic ordinary differential equations differences from the nonperiodic chebyshev case are highlighted

convergence analysis of a periodic like waveform relaxation method for initial value problems via the diagonalization technique we present in this paper a time parallel algorithm for dot u f t u with initial value u 0 u 0 by using the waveform relaxation wr technique and the diagonalization technique with a suitable parameter alpha the wr technique generates a functional sequence u k t via the dynamic iterations dot u k f t u k u k 0 alpha u k t alpha u k 1 t u 0 and at convergence we get u infty t u t each wr iterate represents a periodic like differential equation which is very suitable for applying the diagonalization technique yielding direct parallel in time computation the parameter alpha controls both the roundoff error arising from the diagonalization procedure and the convergence factor of the wr iterations and we perform a detailed analysis for the influence of the parameter alpha on the method we show that the roundoff error is proportional to epsilon 2n 1 max alpha 2 alpha 2 n t vardelta t and epsilon is the machine precision and the convergence factor can be bounded by alpha e tl 1 alpha e tl where l ge 0 is the one sided lipschitz constant of f we also perform a convergence analysis at the discrete level and the effect of temporal discretizations is explored our analysis includes the heat and wave equations as special cases numerical results are given to support our findings

the caratheodory fejer method for real rational approximation a caratheodory fejer method is presented for near best real rational approximation on intervals based on the eigenvalue or singular value analysis of a hankel matrix of chebyshev coefficients in approximation of a smooth function f the cf approximant r cf frequently differs from the best approximation r by only one part in millions or billions to account for this we show here under weak assumptions that if f is approximated on varepsilon varepsilon then as varepsilon to 0 f r o varepsilon m n 1 while r cf r o varepsilon 3m 2n 3 in contrast the latter figure would be o varepsilon m n 2 for the chebyshev economization approximant of maehly or the chebyshev pade approximant of gragg it follows that as varepsilon to 0 best approximation error curves approach the real parts of m n 1 winding rational functions of constant modulus to within o varepsilon 3m 2n 3 numerical examples are given

harmonic density interpolation methods for high order evaluation of laplace layer potentials in 2d and 3d abstract we present an effective harmonic density interpolation method for the numerical evaluation of singular and nearly singular laplace boundary integral operators and layer potentials in two and three spatial dimensions the method relies on the use of green s third identity and local taylor like interpolations of density functions in terms of harmonic polynomials the proposed technique effectively regularizes the singularities present in boundary integral operators and layer potentials and recasts the latter in terms of integrands that are bounded or even more regular depending on the order of the density interpolation the resulting boundary integrals can then be easily accurately and inexpensively evaluated by means of standard quadrature rules a variety of numerical examples demonstrate the effectiveness of the technique when used in conjunction with the classical trapezoidal rule to integrate over smooth curves in two dimensions and with a chebyshev type quadrature rule to integrate over surfaces given as unions of non overlapping quadrilateral patches in three dimensions

efficient numerical solution of acoustic scattering from doubly periodic arrays of axisymmetric objects we present a high order accurate boundary based solver for three dimensional 3d frequency domain scattering from a doubly periodic grating of smooth axisymmetric sound hard or transmission obstacles we build the one obstacle solution operator using separation into p azimuthal modes via the fft the method of fundamental solutions with n proxy points lying on a curve and dense direct least squares solves the effort is o n 3 p with a small constant periodizing then combines fast multipole summation of nearest neighbors with an auxiliary global helmholtz basis expansion to represent the distant contributions and enforcing quasiperiodicity and radiation conditions on the unit cell walls eliminating the auxiliary coefficients and preconditioning with the one obstacle solution operator leaves a well conditioned square linear system that is solved iteratively the solution time per incident wave is then o n p at fixed frequency our scheme avoids singular quadratures periodic green s functions and lattice sums and its convergence rate is unaffected by resonances within obstacles we include numerical examples such as scattering from a grating of period 13 13 comprising highly resonant sound hard cups each needing n p 64800 surface unknowns to 10 digit accuracy in half an hour on a desktop

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

a contour integral method for the black scholes and heston equations a contour integral method recently proposed by weideman ima j numer anal 30 2010 pp 334 350 for integrating semidiscrete advection diffusion pdes is improved and extended for application to some of the important equations of mathematical finance using estimates for the numerical range of the spatial operator optimal contour parameters are derived theoretically and tested numerically an improvement on the existing method is the use of krylov methods for the shifted linear systems the solution of which represents the major computational cost of the algorithm a parallel implementation is also considered test examples presented are the black scholes pde in one space dimension and the heston pde in two dimensions for both vanilla and barrier options in the heston case efficiency is compared to adi splitting schemes and experiments show that the contour integral method is superior for the range of medium to high accuracy requirements

periodizing transformations for numerical integration the order of the trapezoidal rule can be raised by making a substitution that transforms the integrand into a new integrand which has a smooth periodic extension the exponent in the rate at which the transformed weights approach zero near the endpoints is called the damping power of the transformation currently used polynomial transformations of damping power m have order m 1 if m is odd and order m 2 if m is even we prove that the highest possible order is m 1 if m is odd but 2m 2 if m is even and give new polynomial transformations that achieve the optimal order

parabolic and hyperbolic contours for computing the bromwich integral some of the most eectiv e methods for the numerical inversion of the laplace transform are based on the approximation of the bromwich contour integral the accuracy of these methods often hinges on a good choice of contour and several such contours have been proposed in the literature here we analyze two recently proposed contours namely a parabola and a hyperbola using a representative model problem we determine estimates for the optimal parameters that dene these contours an application to a fractional diusion equation is presented

on the convergence rates of gauss and clenshaw curtis quadrature for functions of limited regularity we study the optimal general rate of convergence of the n point quadrature rules of gauss and clenshaw curtis when applied to functions of limited regularity if the chebyshev coefficients decay at a rate o n s 1 for some s 0 clenshaw curtis and gauss quadrature inherit exactly this rate the proof for gauss if 0 s 2 there is numerical evidence only is based on work of curtis johnson riess and rabinowitz from the early 1970s and on a refined estimate for gauss quadrature applied to chebyshev polynomials due to petras 1995 the convergence rate of both quadrature rules is up to one power of n better than polynomial best approximation hence the classical proof strategy that bounds the error of a quadrature rule with positive weights by polynomial best approximation is doomed to fail in establishing the optimal rate

optimizing talbot s contours for the inversion of the laplace transform talbot s method for the numerical inversion of the laplace transform consists of numerically integrating the bromwich integral on a special contour by means of the trapezoidal or midpoint rules in this paper we address the issue of parameter selection in the method for the particular situation when parabolic pdes are solved in the process the well known subgeometric convergence rate o exp c sqrt n of this method is improved to the geometric rate o exp c n with n the number of nodes in the integration rule the value of the maximum decay rate c is explicitly determined numerical results for two versions of the heat equation are presented with the choice of parameters derived here the rule of thumb is that to achieve an accuracy of 10 ell at any given time the associated elliptic problem has to be solved no more than ell times

function classes for double exponential integration formulas the double exponential de formulas for numerical integration are known to be highly efficient more efficient than the single exponential se formulas in many cases function classes suited to the se formulas have already been investigated in the literature through rigorous mathematical analysis whereas this is not the case with the de formulas this paper identifies function classes suited to the de formulas in a way compatible with the existing theoretical results for the se formulas the de formulas are good for more restricted classes of functions but more efficient for such functions two concrete examples demonstrate the subtlety in the behavior of the de formulas that is revealed by our theoretical analysis

euler maclaurin expansions for integrals with endpoint singularities a new perspective in this note we provide a new perspective on euler maclaurin expansions of offset trapezoidal rule approximations of the finite range integrals i f baf x dx where f c a b but can have general algebraic logarithmic singularities at one or both endpoints these integrals may exist either as ordinary integrals or as hadamard finite part integrals we assume that f x has asymptotic expansions of the general forms where ps y and qs y are some polynomials in y here the s and s are complex in general and different from 1 2 the results we obtain in this work generalize and include as special cases those pertaining to the known special cases in which f x x a log x a pga x b x log b x qgb x where p and q are nonnegative integers and ga c a b and gb c a b in addition they have the pleasant feature that they are expressed in very simple terms based only on the asymptotic expansions of f x as x a and x b with h b a n where n is a positive integer and with one of these results reads as h 0 where z is the riemann zeta function

a 2 5d finite difference algorithm for elastic wave modeling using near optimal quadratures abstractwe have developed a 2 5d finite difference algorithm to model the elastic wave propagation in heterogeneous media in 2 5d problems it is assumed that the elastic properties of models are invariant along a certain direction therefore we can convert the 3d problem into a set of 2d problems in the spectral domain the 3d solutions are then obtained by applying a numerical integration in the spectral domain usually the quadrature points used in the numerical integration scheme are sampled from the real axis in the spectral domain the convergence of this quadrature can be very slow especially at high frequencies we have applied the optimal quadrature scheme for the spectral integration this is equivalent to transforming the contour of integration from the real axis into a path in the complex plane our numerical studies have indicated more than 10 times of reduction in the number of quadrature points compared with sampling along the real axis in the spectral domain this scheme alleviates the

new exponential variable transform methods for functions with endpoint singularities the focus of this paper is the approximation of functions which are analytic on a compact interval except at the endpoints typical numerical methods for approximating such functions depend upon the use of particular conformal maps from the original interval to either a semi infinite or an infinite interval followed by an appropriate approximation procedure on the new region we first analyze the convergence of these existing methods and show that in a precisely defined sense they are suboptimal specifically they exhibit poor resolution properties by which we mean that many more degrees of freedom are required to resolve oscillatory functions than standard approximation schemes for analytic functions such as chebyshev interpolation to remedy this situation we introduce two new transforms one for each of the above settings we provide full convergence results for these new approximations and then demonstrate that for a particular choice of parameters these methods lead to substantially better re

whispering bloch modes we investigate eigenvalue problems for the planar helmholtz equation in open systems with a high order of rotational symmetry the resulting solutions have similarities with the whispering gallery modes exploited in photonic micro resonators and elsewhere but unlike these do not necessarily require a surrounding material boundary with confinement instead resulting from the geometry of a series of inclusions arranged in a ring the corresponding fields exhibit angular quasi periodicity reminiscent of bloch waves and hence we refer to them as whispering bloch modes wbms we show that if the geometry of the system is slightly perturbed such that the rotational symmetry is broken modes with asymmetric field patterns can be observed resulting in field enhancement and other potentially desirable effects we investigate the wbms of two specific geometries first using expansion methods and then by applying a two scale asymptotic scheme

likelihood and bayesian prediction of chaotic systems abstract there has recently been considerable interest in both applied disciplines and in mathematics as well as in the popular science literature in the areas of nonlinear dynamical systems and chaotic processes by a nonlinear deterministic dynamical system we mean a time series in which starting at some initial condition the values of the series are some fixed nonlinear function of the previous states one of the more intriguing aspects of these models is their propensity for displaying very complex apparently random behavior even when simple models are analyzed a consequence of such chaotic behavior is that it is difficult to predict the exact behavior of a chaotic system the difficulty in prediction stems from the fact that even the tiniest of errors including computer roundoff in either the specification of the function or the initial condition can lead to huge errors in prediction after a brief review of dynamical systems and the role of probability in dealing with uncertainty a com

o 1 computation of legendre polynomials and gauss legendre nodes and weights for parallel computing a self contained set of algorithms is proposed for the fast evaluation of legendre polynomials of arbitrary degree and argument in 1 1 more specifically the time required to evaluate any legendre polynomial regardless of argument and degree is bounded by a constant i e the complexity is mathcal o 1 the proposed algorithm also immediately yields an mathcal o 1 algorithm for computing an arbitrary gauss legendre quadrature node such a capability is crucial for efficiently performing certain parallel computations with high order legendre polynomials such as computing an integral in parallel by means of gauss legendre quadrature and the parallel evaluation of legendre series in order to achieve the mathcal o 1 complexity novel efficient asymptotic expansions are derived and used alongside known results a c implementation is available from the authors that includes the evaluation routines of the legendre polynomials and gauss legendre quadrature rules

using complex variables to estimate derivatives of real functions a method to approximate derivatives of real functions using complex variables which avoids the subtractive cancellation errors inherent in the classical derivative approximations is described numerical examples illustrating the power of the approximation are presented

computing a alpha log a and related matrix functions by contour integrals new methods are proposed for the numerical evaluation of f mathbf a or f mathbf a b where f mathbf a is a function such as mathbf a 1 2 or log mathbf a with singularities in infty 0 and mathbf a is a matrix with eigenvalues on or near 0 infty the methods are based on combining contour integrals evaluated by the periodic trapezoid rule with conformal maps involving jacobi elliptic functions the convergence is geometric so that the computation of f mathbf a b is typically reduced to one or two dozen linear system solves which can be carried out in parallel

explicit nearly optimal linear rational approximation with preassigned poles this paper gives explicit rational functions for interpolating and approximating functions on the intervals 1 1 0 oc and oc oc the rational functions are linear in the functions to be approximated and they have preassigned poles the error of approximation of these rationals is nearly as small as the error of best rational approximation with numerator and denominator polynomials of the same degrees regions of analyticity are described which make it possible to tell a priori the accuracy which we can expect from this type of rational approximation 1 introduction and summary in this paper we attempt to give a constructive affirmative answer to each of the following questions

on two types of convergence in the method of auxiliary sources the method of auxiliary sources mas is often applied to problems involving an externally illuminated smooth perfect electric conductor pec one seeks to approximately satisfy the boundary condition on the pec surface using auxiliary sources located inside the pec surface usually the underlying auxiliary surface is also smooth and closed the currents on the auxiliary sources mas currents are the initial unknowns once they have been found one can easily determine the field due to them mas field at all points external to the auxiliary surface and in particular at all points external to the pec scatterer we show that in the limit n it is possible to have a convergent mas field together with divergent mas currents and that this phenomenon is accompanied by an abrupt behavior of the limiting value of the mas field we show this possibility through an analytical study of a two dimensional scattering problem involving a circular cylinder in which mas fields and currents can be determined explicitly the analytical study proceeds from first principles it involves fundamental electromagnetics and relatively simple mathematical manipulations numerical results supplement the analytical study and demonstrate the nature of the aforementioned divergence our study sheds light on other aspects of mas in particular it establishes interesting similarities and differences between mas and its continuous version and reveals many similarities between mas currents and numerical solutions of hallen s integral equation with the approximate kernel

inversion formulas for the discretized hilbert transform on the unit circle a discrete version of the hilbert transform on the unit circle is considered its moore penrose inverse with respect to suitable scalar products is derived for different side conditions furthermore stability of the pseudo inverse is studied these results allow the efficient computation of approximate solutions of singular integral equations with hilbert kernel furthermore the stability analysis of such methods becomes much easier even for graded meshes which are useful for weakly singular solutions

the double exponential transformation in numerical analysis the double exponential transformation was first proposed by takahasi and mori in 1974 for the efficient evaluation of integrals of an analytic function with end point singularity afterwards this transformation was improved for the evaluation of oscillatory functions like fourier integrals recently it turned out that the double exponential transformation is useful not only for numerical integration but also for various kinds of sinc numerical methods the purpose of the present paper is to review the double exponential transformation in numerical integration and in a variety of sinc numerical methods

fast fourier transforms for nonequispaced data a group of algorithms is presented generalizing the fast fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval pi pi the schemes of this paper are based on a combination of certain analytical considerations with the classical fast fourier transform and generalize both the forward and backward ffts each of the algorithms requires o n cdot log n n cdot log 1 varepsilon arithmetic operations where varepsilon is the precision of computations and n is the number of nodes the efficiency of the approach is illustrated by several numerical examples

on acoustic and electric faraday cages two dimensional problems involving many identical small circles are considered the circles are the cross sections of parallel wires modelling a cage or a grating both electrostatic and acoustic fields are considered the main emphasis is on periodic configurations of n circles distributed evenly around a large circle a ring foldy s theory is used for acoustic problems and then adapted for electrostatic problems in both cases circulant matrices are encountered the fields can be calculated explicitly then the limit n is studied a connection between the n circle problem and the limiting problem fields exterior to the ring is established using known results on the convergence of a defective form of the trapezoidal rule defective in that endpoint contributions are ignored because the integrand has logarithmic singularities at those points this shows that the solution of the limiting problem is approached very slowly as n 1 log n as n

zolotarev quadrature rules and load balancing for the feast eigensolver the feast method for solving large sparse eigenproblems is equivalent to subspace iteration with an approximate spectral projector and implicit orthogonalization this relation allows to characterize the convergence of this method in terms of the error of a certain rational approximant to an indicator function we propose improved rational approximants leading to feast variants with faster convergence in particular when using rational approximants based on the work of zolotarev numerical experiments demonstrate the possible computational savings especially for pencils whose eigenvalues are not well separated and when the dimension of the search space is only slightly larger than the number of wanted eigenvalues the new approach improves both convergence robustness and load balancing when feast runs on multiple search intervals in parallel

finite size corrections to the spectrum of regular random graphs an analytical solution we develop a thorough analytical study of the o 1 n correction to the spectrum of regular random graphs with n nodes the finite size fluctuations of the resolvent are given in terms of a weighted series over the contributions coming from loops of all possible lengths from which we obtain the isolated eigenvalue as well as an analytical expression for the o 1 n correction to the continuous part of the spectrum the comparison between this analytical formula and direct diagonalization results exhibits an excellent agreement confirming the correctness of our expression

a trapezoidal rule error bound unifying the euler maclaurin formula and geometric convergence for periodic functions the error in the trapezoidal rule quadrature formula can be attributed to discretization in the interior and non periodicity at the boundary using a contour integral we derive a unified bound for the combined error from both sources for analytic integrands the bound gives the euler maclaurin formula in one limit and the geometric convergence of the trapezoidal rule for periodic analytic functions in another similar results are also given for the midpoint rule

computing the gamma function using contour integrals and rational approximations some of the best methods for computing the gamma function are based on numerical evaluation of hankel s contour integral for example temme evaluates this integral based on steepest descent contours by the trapezoid rule here we investigate a different approach to the integral the application of the trapezoid rule on talbot type contours using optimal parameters recently derived by weideman for computing inverse laplace transforms relatedly we also investigate quadrature formulas derived from best approximations to exp z on the negative real axis following cody meinardus and varga the two methods are closely related and both converge geometrically we find that the new methods are competitive with existing ones even though they are based on generic tools rather than on specific analysis of the gamma function

on a certain quadrature formula traduction anglaise d un article en japonais publie en aout 1970 dans la serie kokyuroku vol 91 pp 82 118

semi infinite arrays of isotropic point scatterers a unified approach we solve the two dimensional problem of acoustic scattering by a semi infinite pe riodic array of identical isotropic point scatterers i e objects whose size is negligible compared to the incident wavelength and which are assumed to scatter incident waves uniformly in all directions this model is appropriate for scatterers on which dirichlet boundary conditions are applied in the limit as the ratio of wavelength to body size tends to infinity the problem is also relevant to the scattering of an e polarized electromagnetic wave by an array of highly conducting wires the actual geometry of each scatterer is characterized by a single parameter in the equations related to the single body scattering problem and determined from a harmonic boundary value problem using a mixture of analytical and numerical techniques we confirm that a number of phenomena reported for specific geometries are in fact present in the general case such as the presence of shadow boundaries in the far field and the vanishing of the circular wave scattered by the end of the array in certain specific directions we show that the semi infinite array problem is equivalent to that of inverting an infinite toeplitz matrix which in turn can be formulated as a discrete wiener hopf problem numerical results are presented which compare the amplitude of the wave diffracted by the end of the array for scatterers having different shapes

rapidly convergent representations for green functions for laplace equation representations for green s functions for laplace equation in domains with infinite boundaries are obtained by integrating solutions to appropriate heat conduction problems with respect to time by using different representations for these heat equation solutions for small and large times the changeover being determined by an arbitrary positive parameter a a one parameter family of formulae for the required green function is derived and by varying a the convergence characteristics of this new representation can be altered letting a zero results in known eigenfunction expansions and in those situations in which they exist letting a recovers known image series representations the method which is essentially equivalent to ewald summation is applied to two types of problem first it is applied to potential flow between parallel planes and in a rectangular channel and second to two and three dimensional water wave problems in which the depth is constant in all cases the results of computations