pseudospectral method based on prolate spheroidal wave functions for frequency domain electromagnetic simulations we apply prolate spheroidal wave functions of order zero as basis functions in the pseudospectral method for frequency domain electromagnetic simulation problems like the traditional pseudospectral frequency domain psfd methods based on chebyshev and legendre polynomial series the prolate psfd method yields exponential order of accuracy in terms of the number of samples utilized per wavelength the prolate expansion is superior to the chebyshev and legendre polynomial series by a factor of spl pi 2 in addition the prolate psfd method employs a more uniform spatial grid achieving better resolution near the center of the domain

on hp convergence of prolate spheroidal wave functions and a new well conditioned prolate collocation scheme the first purpose of this paper is to provide further illustrations from both theoretical and numerical perspectives for the nonconvergence of h refinement in hp approximation by the prolate spheroidal wave functions pswfs a surprising convergence property that was first discovered by boyd et al 2013 3 the second purpose is to offer a new basis that leads to prolate collocation systems with condition numbers independent of c n the intrinsic bandwidth parameter and the number of collocation points we highlight that the collocation scheme together with a very practical rule for pairing up c n significantly outperforms the legendre polynomial based method and likewise other jacobi polynomial based methods in approximating highly oscillatory bandlimited functions

extrapolation algorithms for discrete signals with application in spectral estimation in this paper we present some new algorithms useful for extrapolation and spectral estimation of band limited sequences in one and two dimensions first we show that many of the existing extrapolation algorithms for noiseless observations are unified under the criterion of minimum norm least squares mnls extrapolation for example the iterative algorithms proposed in 2 and 8 10 are shown to be special cases of a one step gradient algorithm which has linear convergence convergence and other numerical properties are improved by going to a conjugate gradient algorithm for noisy observations these algorithms could be extended by considering a mean square extrapolation criterion which gives rise to a mean square extrapolation filter and also to a recursive extrapolation filter examples and application of these methods are given extension of these algorithms is made for problems where the signal is known to be periodic a new set of functions called the periodic discrete prolate spheroidal sequences p dpss analogous to dpss 21 22 are introduced and their properties are studied finally several of these algorithms are generalized to two dimensions and the relevant equations are given

development of noncentered wavenumber based optimized interpolation schemes with amplification control for overlapping grids a noncentered interpolation technique has been constructed to perform simulations using overlapping grids for complex geometries high order centered lagrange polynomial interpolations and interpolations optimized in the fourier space are first generalized to the noncentered case these noncentered interpolations either generate significant dispersion errors or strongly amplify high wavenumber components accordingly a noncentered high order wavenumber based optimized interpolation method is developed with the addition of a nonlinear constraint for the control of the amplitude amplification induced by decentering high order piecewise polynomial regressions of the obtained interpolation coefficients are performed the time stability of the method is investigated in the 1 d case when the interpolation method is used in conjunction with explicit high order differencing filtering schemes as well as a 6 step runge kutta time integration algorithm a criterion is formulated to predict its stability as a function of the filtering strength and the courant friedrichs lewy constant finally 1 d convection simulations are presented to illustrate the stability and the accuracy of the developed noncentered interpolations

fast computation of the spectral differentiation by the fast multipole method abstract this paper is concerned with the application of the fast multipole method fmm to the pseudospectral methods the spectral differentiation in pseudospectral methods involves dense matrix vector products which if evaluated directly entail o n 2 operations the core of this work lies in the fact that the fmm leads to a significant speed up in cpu time with a major reduction in the amount of computer memory needed when performing matrix vector products the fmm is exploited to estimate the barycentric weights and the pseudospectral differentiation which reduces the complexity to o n furthermore we show the convergence analysis and present the applications to the second order boundary value problems numerical examples confirm the advantages of the method

the use of tricubic interpolation with spectral derivatives to integrate particle trajectories in complicated electromagnetic fields codes that calculate the trajectories of particles in complicated electromagnetic fields often include spectral methods which take advantage of the speed of ffts to rapidly solve for the fields in this case hermite tricubic interpolation can be used to calculate the fields between grid points with spectral derivates used to determine the interpolation coefficients this method is extremely accurate and in the case of electrostatic fields produces an energy conserving force when incorporated into a particle follower code

some comments on fourier analysis uncertainty and modeling investigation of the problem of simultaneously concentrating a function and its fourier transform has led to some interesting special functions that have widespread applications in engineering they provide a means of proving a rigorous version of an engineering folk theorem called the 2wt theorem many generalizations of these ideas seem to possess a similar elegant mathematical structure a brief descriptive review is given of these developments

a performance comparison of associated legendre projections the associated legendre projection provides a means for accelerating the dynamical core of a global weather or climate model therefore the goal is to determine the fastest possible projection algorithm of which this paper compares six the standard method which computes the projection using a forward and backward associated legendre transform the direct method which uses a single projection matrix when this approach results in fewer operations the fast multipole method the weighted orthogonal complement method the seminaive method and a projection based on transforms proposed by mohlenkamp timing results indicate that all the projections behave like o n3 algorithms up to at least n 200 spectral truncation for this range of resolutions the weighted orthogonal complement has the lowest operation count best cache utilization and best overall timings

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

direct projection decoding algorithm for sigma delta modulated signals usually sigma delta modulators spl sigma spl delta m are modeled by replacing the quantizer with an additive white noise source based on this linear model linear decoder structures can be designed however the spl sigma spl delta m is a nonlinear system and corresponding nonlinear decoder structures may be able to achieve a better signal to quantization noise ratio sqnr performance in this paper a new nonlinear decoding algorithm is presented it is based on the projection onto complex sets pocs algorithm developed by hein and zakhor our algorithm reduces the decoding problem to the solution of a single quadratic programming problem based on the state equations of the modulator and the condition that the modulator input signal is bandlimited the bandlimitation constraint is directly applied to the state equations thus the resulting quadratic programming problem needs to be solved only once

an o 1 integration scheme for three dimensional surface scattering problems we present an accurate method of o 1 complexity with respect to frequency i e a method that to achieve a prescribed error tolerance requires a bounded computational cost for arbitrarily high frequencies for the computation of singular oscillatory integrals arising in the boundary integral formulation of problems of acoustic scattering by surfaces in three dimensional space like the two dimensional counterpart of this algorithm which we introduced recently and which is applicable to scattering by curves in the plane the present method is based on a combination of two main elements 1 a high frequency ansatz for the unknown density in a boundary integral formulation of the problem and 2 an extension of the ideas of the method of stationary phase to allow for o 1 high order accurate integration of oscillatory functions the techniques we introduce to implement an efficient o 1 integrator in the present three dimensional context differ significantly from those used in the earlier two dimensional algorithm in particular in the present text we introduce an efficient canonical hybrid analytic numerical algorithm which in addition to allowing for integration of oscillatory functions around both singular points and points of stationary phase can handle the significant difficulty that arises as singular points and one or more stationary points approach each other within the two dimensional scattering surface we include numerical results illustrating the behavior of the integration algorithm on sound soft spheres with diameters of up to 5000 wavelengths in such cases for a single integral the algorithm yields accuracies of the order of three digits in computational times of less than two seconds in a preliminary full scattering simulation we present a solution with two digits of accuracy in the surface density was obtained in about three hours running time in a single 1 5ghz amd athlon processor for a sphere of 500 wavelengths in diameter

a fast algorithm for chebyshev fourier and sinc interpolation onto an irregular grid a chebyshev or fourier series may be evaluated on the standard collocation grid by the fast fourier transform fft unfortunately the fft does not apply when one needs to sum a spectral series at n points which are spaced irregularly the cost becomes o n2 operations instead of the ffts o n log n this sort of off grid interpolation is needed by codes which dynamically readjust the grid every few time steps to resolve a shock wave or other narrow features it is even more crucial to semi lagrangian spectral algorithms for solving convection diffusion and navier stokes problems because off grid interpolation must be performed several times per time step in this work we describe an alternative algorithm the first step is to pad the set of spectral coefficients an with zeros and then take an fft of length 3n to interpolate the chebyshev series to a very fine grid the second step is to apply either the mth order euler sum acceleration or 2m 1 point lagrangian interpolation to approximate the sum of the series on the irregular grid we show that both methods yield full precision with m n allowing an order of magnitude reduction in cost with no loss of accuracy

on the use of singular value decomposition and decimation in discrete time band limited signal extrapolation the problem of extrapolating a band limited signal in discrete time is viewed as one of solving an underdetermined system of linear equations choosing the minimum norm least squares mnls solution is one criterion for singling out an extrapolation from all the possible solutions to the linear system use of the moore penrose inverse yields the mnls solution and singular value decomposition svd provides a means for implementing the moore penrose inverse an expression for the mean square error incurred in solving a linear system via svd is derived this can be used to estimate the number of singular values needed to form the inverse the error expression also indicates that decimation can be applied in the extrapolation problem to reduce the high computational cost of svd without degrading the extrapolation the results developed for the one dimensional case are extended to higher dimensions examples of the svd approach to extrapolation are given along with examples using other extrapolation techniques for comparison the svd approach compares favorably with known mnls extrapolation methods

an instability phenomenon in spectral methods the eigenvalues of chebyshev and legendre spectral differentiation matrices which determine the allowable time step in an explicit time integration are extraordinarily sensitive to rounding errors and other perturbations on a grid of n points per space dimension machine rounding leads to errors in the eigenvalues of size o n 2 this phenomenon may lead to inconsistency between predicted and observed time step restrictions one consequence of it is that spectral differentiation by interpolation in legendre points which has a favorable o n 1 time step restriction for the model problem u t u x in theory is subject to an o n 2 restriction in practice the same effect occurs with chebyshev points for the model problem u t xu x another consequence is that a spectral calculation with a fixed time step may be stable in double precision but unstable in single precision we know of no other examples in numerical computation of this kind of precision dependent stability

spectral integration and two point boundary value problems a numerical method for two point boundary value problems with constant coefficients is developed which is based on integral equations and the spectral integration matrix for chebyshev nodes the method is stable achieves superalgebraic convergence and requires o n log n operations where n is the number of nodes in the discretization although stable spectral methods have been constructed in the past they have generally been based on reformulating the recurrence relations obtained through spectral differentiation in an attempt to avoid the ill conditioning introduced by that process

accuracy enhancement for higher derivatives using chebyshev collocation and a mapping technique a new method is investigated to reduce the roundoff error in computing derivatives using chebyshev collocation methods by using a grid mapping derived by kosloff and tal ezer and the proper choice of the parameter alpha the roundoff error of the k th derivative can be reduced from o n 2k to o n lge k where epsilon is the machine precision and n is the number of collocation points this drastic reduction of roundoff error makes mapped chebyshev methods competitive with any other algorithm in computing second or higher derivatives with large n several other aspects of the mapped chebyshev differentiation matrix are also studied revealing that the mapped chebyshev methods require much less than pi points to resolve a wave the eigenvalues are less sensitive to perturbation by roundoff error and larger time steps can be used for solving pdes all these advantages of the mapped chebyshev methods can be achieved while maintaining spectral accuracy

spectral methods based on prolate spheroidal wave functions for hyperbolic pdes we examine the merits of using prolate spheroidal wave functions pswfs as basis functions when solving hyperbolic pdes using pseudospectral methods the relevant approximation theory is reviewed and some new approximation results in sobolev spaces are established an optimal choice of the band limit parameter for pswfs is derived for single mode functions our conclusion is that one might gain from using the pswfs over the traditional chebyshev or legendre methods in terms of accuracy and efficiency for marginally resolved broadband solutions

spectral methods on arbitrary grids stable and spectrally accurate numerical methods are constructed on arbitrary grids for partial differential equations these new methods are equivalent to conventional spectral methods but do not rely on specific grid distributions specifically we show how to implement legendre galerkin legendre collocation and laguerre galerkin methodology on arbitrary grids

high accuracy approximation techniques for analytic functions a generalization of the familiar mesh point technique for numerical approximation of functions is presented high accuracy and very rapid convergence may be obtained by thoughtful choice of the reference function chosen for interpolation between the mesh points in particular derivative operators are represented by highly nonlocal matrices but this is no drawback when one has computing machines to perform the algebraic manipulations some examples are given from familiar quantum mechanical problems

another preprocessing algorithm for generalized one dimensional fast multipole method the fast multipole method fmm which is originally an algorithm for fast evaluation of particle interactions is also effective for accelerating several numerical computations yarvin and rokhlin proposed generalized fmm using the singular value decomposition svd which gives the optimum low rank approximation their algorithm reduces the computational costs of the fmm evaluation and frees the fmm from analytical approximation formulae however the computational complexity of the preprocessing for an n n matrix is o n3 because of the svd and it requires orthogonal matrices of the low rank approximations in this paper we propose another preprocessing algorithm for the generalized fmm our algorithm runs in time o n2 even with the svd and releases the low rank approximations from orthogonal matrices the triangularization by the qr decomposition with sparsification which reduces the costs of the fmm more than the diagonalization is enabled although the algorithm by yarvin and rokhlin can be accelerated to o n2 using the qr decomposition our preprocessing algorithm outperforms it in fast spherical filter fast polynomial interpolation and fast legendre transform

non linear reconstruction of delta sigma modulated signals randomized surrogate constraint decoding algorithm delta sigma a d conversion is a popular technique used to achieve high resolution data conversion for low to moderate bandwidth applications it is commonly accepted that the digital circuitry of the a d converters should perform a linear time invariant lti filtering on the output of the modulator in order to reconstruct the input signal however it has been shown that when higher order a d converters are operated at higher oversampling ratios osrs non linear reconstruction algorithms extract more information about the input signal than lti methods still use of non linear algorithms has been limited due to their complexity and stability issues two practical non linear methods presented in the literature are fast projection onto convex sets pocs and direct projection dp in this work we show that the non linear reconstruction process for modulated sequences can be treated as a linear feasibility problem lfp we then show that if the reconstruction process is defined as an lfp its implementation is less complex than in the case of the dp method we further describe how the lfp can be well conditioned and demonstrate the reconstruction process by using the surrogate constraint algorithm with random constraint selection the described algorithm considers a small number of constraints at a time which makes it suitable for serial implementation

collocation on uniform grids in this paper we derive four sets of sinc like functions defined on a finite interval and obeying different boundary conditions the functions in each set are orthogonal and their nodes are uniformly distributed on the interval we have applied each set to solve a large class of eigenvalue equations with different boundary conditions both on finite intervals and on the real line showing that precise numerical results can be obtained efficiently and rapidly a comparison with results available in the literature is also performed

generalized discrete spherical harmonic transforms two generalizations of the spherical harmonic transforms are provided first they are generalized to an arbitrary distribution of latitudinal points i this unifies transforms for gaussian and equally spaced distributions and provides transforms for other distributions commonly used to model geophysical phenomena the discrete associated legendre functions pnm i are shown to be orthogonal to within roundoff error with respect to a weighted inner product thus providing the forward transform to spectral space second the representation of the transforms is also generalized to rotations of the discrete basis set pnm i a discrete function basis is defined that provides an alternative to pnm i on a grid with n latitudes the new basis requires o n2 memory compared to the usual o n3 the resulting transforms differ in spectral space but provide identical results for certain applications for example a forward transform followed immediately by a backward transform projects the original discrete function in a manner identical to the existing transforms namely they both project the original function onto the same smooth least squares approximation without the high frequencies induced by the closeness of the points in the neighborhood of the poles finally a faster projection is developed based on the new transforms

multipole expansions and pseudospectral cardinal functions a new generalization of the fast fourier transform the polynomial or trigonometric interpolant of an arbitrary function f x may be presented as a critical function series whose coefficients are the values of f x at the interpolation points we show that the cardinal series is identical to the sum of the forces due to a set of n point charges with appropriate force laws it follows that the cardinal series can be summed via the fast multipole method fmm in o nlog sub 2 n operations which is much cheaper than the o n sup 2 cost of direct summation the fmm is slower than the fast fourier transform fft so the latter should always be used where applicable however the multipole expansion succeeds where the fft fails in particular the fmm can be used to evaluate fourier and chebyshev series on an irregular grid as is needed when adaptively regridding in a time integration also the multipole expansion can be applied to basis sets for which the fft is inapplicable even on the canonical grid including legendre polynomials hermite and laguerre functions spherical harmonics and zinc functions 12 refs

a spectral method of characteristics for hyperbolic problems a new numerical technique is introduced for first order hyperbolic equations based on combining the spectral method with explicit timestepping along the characteristics which permits the use of large timesteps at no cost in accuracy the method is unconditionally stable and spectrally accurate the theoretical results are illustrated by numerical experiments

a nonuniform fast fourier transform based on low rank approximation by viewing the nonuniform discrete fourier transform nudft as a perturbed version of a uniform discrete fourier transform we propose a fast and quasi optimal algorithm for computing the nudft based on the fast fourier transform fft our key observation is that an nudft and dft matrix divided entry by entry is often well approximated by a low rank matrix allowing us to express a nudft matrix as a sum of diagonally scaled dft matrices our algorithm is simple to implement automatically adapts to any working precision and is competitive with state of the art algorithms in the fully uniform case our algorithm is essentially the fft we also describe quasi optimal algorithms for the inverse nudft and two dimensional nudfts

a fast block based nonlinear decoding algorithm for spl sigma spl delta modulators previous work has established that the digital output of a spl sigma spl delta modulator as an a d converter contains more information about the analog input than is extracted with conventional linear filtering under reasonable mathematical assumptions optimal nonlinear decoding of the digital output can achieve significantly larger signal to noise ratios than linear filtering however the hitherto proposed decoding algorithms only demonstrate conceptual feasibility and are impractical from a computational point of view we present a new block based decoding algorithm that like previous work employs projections onto convex sets the algorithm owes its speed to a change of projection norm an accelerated convergence scheme and a decimation like subsampling it is on the order of 10 sup 4 10 sup 5 times faster than one previously published algorithm for typical parameter values and about 2 10 times slower than linear decoding the new algorithm is applicable to all currently popular spl sigma spl delta architectures

direct operatorial tau method for pantograph type equations abstract the lanczos tau method is applied to find chebyshev polynomial approximations for the solutions of pantograph differential equations the results are accompanied by an error analysis numerical examples calculated by our matlab package chebpack confirm the theory and prove the importance for practice of this approach

a new compact spectral scheme for turbulence simulations we propose a new kind of compact difference scheme for the computation of the first and second derivatives in the simulation of high reynolds number turbulent flows the scheme combines and truncates the pseudospectral representation of derivative for convergence acceleration comparison of the wave resolution property with available optimized compact schemes minimizing the prescribed wave resolution error reveals our scheme s superiority for the same size of stencils without introducing optimization parameters an accompanying boundary scheme is also proposed with the stability analysis the proposed scheme is tested for the evaluation of derivatives of a function that decays very slowly in the wavenumber space and for the simulation of three dimensional isotropic turbulence

deterministic analysis of oversampled a d conversion and decoding improvement based on consistent estimates discusses the deterministic analysis of oversampled a d conversion adc the properties derivable from such an analysis and the consequences on reconstruction using nonlinear decoding given a band limited input x producing a quantized version c the authors consider the set of all input signals that are band limited and produce c they call any element of this set a consistent estimate of x regardless of the type of encoder simple predictive or noise shaping they show that this set is convex and as a consequence any nonconsistent estimate can be improved they also show that the classical linear decoding estimates are not necessarily consistent numerical tests performed on simple adc single loop and multiloop spl sigma spl delta modulation show that consistent estimates yield a mean square error mse that decreases asymptotically with the oversampling ratio faster than the linear decoding mse by approximately 3 db octave this implies an asymptotic mse of the order of spl oscr r sup spl minus 2n 2 instead of spl oscr r sup spl minus 2n 1 in linear decoding where r is the oversampling ratio and n the order of the modulator methods of improvement of nonconsistent estimates based on the deterministic knowledge of the quantized signal are proposed for simple adc predictive adc single loop and multiloop spl sigma spl delta modulation

fast approximate fourier transforms for irregularly spaced data several algorithms for efficiently evaluating trigonometric polynomials at irregularly spaced points are presented and analyzed the algorithms can be viewed as approximate generalizations of the fast fourier transform fft and they are compared with regard to their accuracy and their computational efficiency

a pseudospectral method of solution of fisher s equation in this paper we develop an accurate and efficient pseudospectral solution of fisher s equation a prototypical reaction diffusion equation the solutions of fisher s equation are characterized by propagating fronts that can be very steep for large values of the reaction rate coefficient there is an ongoing effort to better adapt pseudospectral methods to the solution of differential equations with solutions that resemble shock waves or fronts typical of hyperbolic partial differential equations the collocation method employed is based on chebyshev gauss lobatto quadrature points we compare results for a single domain as well as for a subdivision of the main domain into subintervals instabilities that occur in the numerical solution for a single domain analogous to those found by others are attributed to round off errors arising from numerical features of the discrete second derivative matrix operator however accurate stable solutions of fisher s equation are obtained with a multidomain pseudospectral method a detailed comparison of the present approach with the use of the sine interpolation is also carried out

accurate function sinc interpolation and derivative estimations over finite intervals function expansion in terms of finite sinc bases is considered for finite intervals we considered proper transformation and weighted barycentric methods we adjusted the transformation function to comply with the truncation of the expansion this removes the unbounded behavior of the terminal derivatives and yields accurate results efficient weighted barycentric expansions are presented to improve the accuracies of function and derivative estimations a new expression for the estimation of the second derivative is derived the derived expressions are successfully applied to accurately locate the minimum of the solution and solve a singularly perturbed problem

a hybrid fast multipole pseudo spectral time domain method the major computation cost of pseudo spectral method comes from the evaluation of differentiation matrix multiplication in the past uniform or chebyshev collocation points are used for sampling the differentiation matrix multiplication was evaluated by fast fourier transform fft or fast cosine transform fct in order to reduce the computation complexity from o n2 to o n log n however the intrinsic properties of fft or fct may cause the wraparound effect and gibbs phenomenon moreover fft or fct is not applicable to other collocation points such as legendre and hermite in order to improve the accuracy and applicability of the pseudo spectral method the fast multipole method fmm is exploited to substitute the fft or fct by making use of the similarity of the n body problem and the collocation problem a new fmm based pseudo spectral time domain method is developed in this paper

the period of the bell numbers modulo a prime we discuss the numbers in the title and in particular whether the minimum period of the bell numbers modulo a prime p can be a proper divisor of np p p 1 p 1 it is known that the period always divides np the period is shown to equal np for most primes p below 180 the investigation leads to interesting new results about the possible prime factors of np for example we show that if p is an odd positive integer and m is a positive integer and q 4m 2 p 1 is prime then q divides p m2p 1 then we explain how this theorem influences the probability that q divides np

rapid prolate pseudospectral differentiation and interpolation with the fast multipole method pseudospectral methods utilizing prolate spheroidal wave functions as basis functions have been shown to possess advantages over the conventional pseudospectral methods based on trigonometric and orthogonal polynomials however the spectral differentiation and interpolation steps of the prolate pseudospectral method involve matrix vector products which if evaluated directly entail o n2 memory requirement and computational complexity where n is the number of unknowns utilized for discretization and interpolation in this work we show that the fast multipole method fmm can be used to reduce the memory requirement and computational complexity of the prolate pseudospectral method to o n example simulation results demonstrate the speed and accuracy of the resulting fast prolate pseudospectral solver

construction of lighthill s unitary functions the imbricate series of unity the poisson sum theorem implies that every spatially periodic function f x can be written as a so called imbricate series that is as the superposition of an infinite number of evenly space copies of a pattern function a x the pattern is the fourier transform of a function a n that specifies the usual fourier coefficients because there are many a n which coincide with a given set of fourier coefficients aj when n is an integer the pattern function a x is not unique in this work we show how to construct pattern functions of the form a x f x w x where f x is the periodic function itself the imbrication of a x and where w x is what lighthill called a unitary function we derive an explicit formula to construct all possible unitary functions and tabulate eight examples we also prove that all unitary functions are pattern functions of unity that is for the imbricate series that sums to f x 1 x

variational collocation on finite intervals in this paper we study a set of functions defined on an interval of finite width which are orthogonal and which reduce to the sinc functions when the appropriate limit is taken we show that these functions can be used within a variational approach to obtain accurate results for a variety of problems we have applied them to the interpolation of functions on finite domains and to the solution of the schrodinger equation and we have compared the performance of the present approach with others

stability of the method of lines it is well known that a necessary condition for the lax stability of the method of lines is that the eigenvalues of the spatial discretization operator scaled by the time stepk lie within a distanceo k of the stability region of the time integration formula ask 0 in this paper we show that a necessary and sufficient condition for stability except for an algebraic factor is that the pseudo eigenvalues of the same operator lie within a distanceo o k of the stability region ask 0 our results generalize those of an earlier paper by considering a runge kutta and other one step formulas b implicit as well as explicit linear multistep formulas c weighted norms d algebraic stability e finite and infinite time intervals and f stability regions with cusps in summary the theory presented in this paper amounts to a transplantation of the kreiss matrix theorem from the unit disk for simple power iterations to an arbitrary stability region for method of lines calculations

fast algorithms for polynomial interpolation integration and differentiation for functions tabulated at chebyshev nodes on an interval spectral interpolation and indefinite integration can be performed stably and efficiently via the fast fourier transform for many other sets of nodes such as the gaussian nodes on an interval the classical interpolation and indefinite integration schemes are stable but slow requiring o n 2 arithmetic operations with n the number of nodes in the discretization of the interval in this paper a group of algorithms is presented for the efficient evaluation of lagrange polynomial interpolants at multiple points on the line and for the rapid indefinite integration and differentiation of functions tabulated at nodes other than chebyshev the interpolation scheme requires o n cdot log 1 varepsilon arithmetic operations and o n cdot log n n cdot log 1 varepsilon operations are required for the integration and differentiation schemes where varepsilon is the precision of computations and n is the number of nodes t

equivalence of some quadratic programming algorithms we formulate a general algorithm for the solution of a convex but not strictly convex quadratic programming problem conditions are given under which the iterates of the algorithm are uniquely determined the quadratic programming algorithms of fletcher gill and murray best and ritter and van de panne and whinston dantzig are shown to be special cases and consequently are equivalent in the sense that they construct identical sequences of points the various methods are shown to differ only in the manner in which they solve the linear equations expressing the kuhn tucker system for the associated equality constrained subproblems equivalence results have been established by goldfarb and djang for the positive definite hessian case our analysis extends these results to the positive semi definite case

analysis of spectral approximations using prolate spheroidal wave functions in this paper the approximation properties of the prolate spheroidal wave functions of order zero pswfs are studied and a set of optimal error estimates are derived for the pswf approximation of non periodic functions in sobolev spaces these results serve as an indispensable tool for the analysis of pswf spectral methods a pswf spectral galerkin method is proposed and analyzed for elliptic type equations illustrative numerical results consistent with the theoretical analysis are also presented

a preconditioned implementation of pseudospectral methods on arbitrary grids in this article a preconditioner for pseudospectral methods in terms of expansion coefficients on arbitrary grids is presented the preconditioner is first developed for application in ordinary differential equations and then is generalized for partial differential equations a fully discrete pseudospectral approximation is then used to solve bivariate problems this preconditioner in effect makes the pseudospectral methods insensitive to specific grid points the numerical results show the efficiency of the method

divisors of mersenne numbers we add to the heuristic and empirical evidence for a conjecture of gillies about the distribution of the prime divisors of mersenne numbers we list some large prime divisors of mersenne numbers mp in the range 17000 p 105

rapid convergence of some seismic processing algorithms the rate of convergence of many numerical algorithms can be greatly improved by repeated application of the method of summation by parts cases of interest in seismology arise when we need to resample the spectrum of a function at unevenly spaced frequency values related examples include the time domain evaluation of the hilbert transform and the extrapolation to the real axis of spectra evaluated in the complex domain a formula of this type was first presented by lanczos 1956 the validity of the algorithms depends upon the fact that summation by parts of the cardinal series representation is justified as long as a function is somewhat oversampled

rayleigh schrodinger perturbation theory with a strong perturbation anharmonic oscillators the bound state solutions of schrodinger s equation for the anharmonic oscillator potentials v x2 lambda x2k k 2 3 have been investigated using elementary techniques of low order variational perturbation theory for the quartic oscillator k 2 a scaled harmonic potential provides a remarkably accurate model for all lambda although this model is slightly less satisfactory for higher order anharmonicities k or 3 the perturbation procedures remain effective and can be applied successfully provided that higher order terms are calculated

image restoration by the method of convex projections part 1ߞtheory a projection operator onto a closed convex set in hilbert space is one of the few examples of a nonlinear map that can be defined in simple abstract terms moreover it minimizes distance and is nonexpansive and therefore shares two of the more important properties of ordinary linear orthogonal projections onto closed linear manifolds in this paper we exploit the properties of these operators to develop several iterative algorithms for image restoration from partial data which permit any number of nonlinear constraints of a certain type to be subsumed automatically their common conceptual basis is as follows every known property of an original image f is envisaged as restricting it to lie in a well defined closed convex set thus m such properties place f in the intersection e0 ei of the corresponding closed convex sets e1 e2 em given only the projection operators pi onto the individual ei s i 1 m we restore f by recursive means clearly in this approach the realization of the pi s in a hilbert space setting is one of the major synthesis problems section i describes the geometrical significance of the three main theorems in considerable detail and most of the underlying ideas are illustrated with the aid of simple diagrams section ii presents rules for the numerical implementation of 11 specific projection operators which are found to occur frequently in many signal processing applications and the appendix contains proofs of all the major results

aurifeuillian factorizations and the period of the bell numbers modulo a prime we show that the minimum period modulo p of the bell exponential integers is p p 1 p 1 for all primes p 102 and several larger p our proof of this result requires the prime factorization of these periods for some primes p the factoring is aided by an algebraic formula called an aurifeuillian factorization we explain how the coefficients of the factors in these formulas may be computed

a fast spherical harmonics transform algorithm the spectral method with discrete spherical harmonics transform plays an important role in many applications in spite of its advantages the spherical harmonics transform has a drawback of high computational complexity which is determined by that of the associated legendre transform and the direct computation requires time of o n3 for cut off frequency n in this paper we propose a fast approximate algorithm for the associated legendre transform our algorithm evaluates the transform by means of polynomial interpolation accelerated by the fast multipole method fmm the divide and conquer approach with split legendre functions gives computational complexity o n2 log n experimental results show that our algorithm is stable and is faster than the direct computation for n 511

anharmonic oscillator and the optimized basis expansion we introduce various optimization schemes for highly accurate calculation of the eigenvalues and the eigenfunctions of the one dimensional anharmonic oscillators we present several methods of analytically fixing the nonlinear variational parameter specified by the domain of the trigonometric basis functions we show that the optimized parameter enables us to determine the energy spectrum to an arbitrary accuracy also using the harmonic oscillator basis functions we indicate that the resulting optimal frequency agrees with the one obtained by the principle of the minimal sensitivity

prolate spheroidal wavefunctions as an alternative to chebyshev and legendre polynomials for spectral element and pseudospectral algorithms prolate spheroidal functions of order zero are generalizations of legendre polynomials which when the bandwidth parameter c 0 oscillate more uniformly on x 1 1 than either chebyshev or legendre polynomials this suggests that compared to these polynomials prolate functions give more uniform spatial resolution further when used as the spatial discretization for time dependent partial differential equations in combination with explicit time marching prolate functions allow a longer stable timestep than legendre polynomials we show that these advantages are real and further that it is almost trivial to modify existing pseudospectral and spectral element codes to use the prolate basis however improvements in spatial resolution are at most a factor of 2 approached slowly as n the timestep can be lengthened by several times but not by a factor that grows rapidly with n the prolate basis is not likely to radically expand the range of problems that can be done on a workstation however for production runs on the bleeding edge edge of arithmurgy such as numerical weather prediction the rewards for switching to a prolate basis may be considerable

sixth order factorization of the evolution operator for time dependent potentials the evolution operator of a quantum system in a time dependent potential is factorized in unitary exponential operators at order 6 this expression is derived with the time ordering method it is compared with lower order factorizations on several simple one dimensional examples better accuracies are reached at sixth order for a given time step than at lower orders due to a significant increase of computation duration per time step the sixth order approximation is mainly useful when high accuracies are required

image restoration by the method of convex projections part 2 applications and numerical results the image restoration theory discussed in a previous paper by youla and webb 1 is applied to a simulated image and the results compared with the well known method known as the gerchberg papoulis algorithm the results show that the method of image restoration by projection onto convex sets by providing a convenient technique for utilizing a priori information performs significantly better than the gerchberg papoulis method

must the multilevel sinc transform a fast multilevel algorithm must for evaluating an n sample sinc interpolant at mn points is presented for uniform grids its complexity is 25mn log 1 delta flops for the sinc kernel and 75mn log 1 delta for the sincd kernel where delta is the target evaluation accuracy must is faster than fft and fmm based evaluations for large n and or for large delta it is also applicable to nonuniform grids and to other kernels numerical experiments demonstrating the algorithm s practicality are presented

hybrid chebyshev function bases for sparse spectral methods in parity mixed pdes on an infinite domain abstract we present a numerical spectral method to solve systems of differential equations on an infinite interval y in presence of linear differential operators of the form q y y b where q y is a rational fraction and b a positive integer even when these operators are not parity preserving we demonstrate how a mixed expansion in interleaved chebyshev rational functions t b n y and s b n y preserves the sparsity of their discretization this paves the way for fast o n ln n and spectrally accurate mixed implicit explicit time marching of sets of linear and nonlinear equations in unbounded geometries

spectral differencing with a twist spectral collocation methods have become very useful in providing highly accurate solutions to differential equations a straightforward implementation of these methods involves the use of spectral differentiation matrices to obtain optimal accuracy these matrices must be computed carefully we demonstrate that naive algorithms for computing these matrices suffer from severe loss of accuracy due to roundoff errors several improvements are analyzed and compared a number of numerical examples are provided demonstrating significant differences between the sensitivity of the forward problem and the inverse problem

the eps method a new method for constructing pseudospectral derivative operators we develop a new type of derivative matrix for pseudospectral methods the norm of these matrices grows at the optimal rate o n 2 for n by n matrices in contrast to standard pseudospectral constructions that result in o n 4 growth of the norm the smaller norm has a big advantage when using the derivative matrix for solving time dependent problems such as wave propagation the construction is based on representing the derivative operator as an integral kernel and does not rely on the interpolating polynomials in particular we construct second derivative matrices that incorporate dirichlet or neumann boundary conditions on an interval and on the disk but the method can be used to construct a wide variety of commonly used operators for solving pdes and integral equations the construction can be used with any quadrature including traditional gauss legendre quadratures but we have found that by using quadratures based on prolate spheroidal wave functions we can achieve a near optimal sampling rate close to two points per wavelength even for non periodic problems we provide numerical results for the new construction and demonstrate that the construction achieves similar or better accuracy than traditional pseudospectral derivative matrices while resulting in a norm that is orders of magnitude smaller than the standard construction to demonstrate the advantage of the new construction we apply the method for solving the wave equation in constant and discontinuous media and for solving pdes on the unit disk we also present two compression algorithms for applying the derivative matrices in o nlogn operations

an adaptive pseudo spectral method for reaction diffusion problems the spectral interpolation error was considered for both the chebyshev pseudo spectral and galerkin approximations a family of functionals i sub r u with the property that the maximum norm of the error is bounded by i sub r u j sub r where r is an integer and j is the degree of the polynomial approximation was developed these functionals are used in the adaptive procedure whereby the problem is dynamically transformed to minimize i sub r u the number of collocation points is then chosen to maintain a prescribed error bound the method is illustrated by various examples from combustion problems in one and two dimensions

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

reconstruction of oversampled band limited signals from spl sigma spl delta encoded binary sequences considers the application of spl sigma spl delta modulators to analog to digital conversion the authors have previously shown that for constant input signals optimal nonlinear decoding can achieve large gains in signal to noise ratio snr over linear decoding the present paper shows a similar result for band limited input signals the new nonlinear decoding algorithm is based on projections onto convex sets pocs and alternates between a time domain operation and a band limitation to find a signal invariant under both the time domain operation results in a quadratic programming problem the band limitation can be based on singular value decomposition of a certain matrix the authors show simulation results for the snr performance of a pocs based decoder and a linear decoder for the single loop double loop and two stage spl sigma spl delta modulators and for a specific fourth order interpolative modulator depending on the modulator and the oversampling ratio improvements in snr of up to 10 20 db can be achieved

a fast spherical filter with uniform resolution this paper introduces a fast algorithm for obtaining a uniform resolution representation of a function known at a latitude longitude grid on the surface of a sphere equivalent to a triangular isotropic truncation of the spherical harmonic coefficients for the function the proposedspectral truncation method which is based on the fast multipole method and the fast fourier transform projects the function to a space with uniform resolution while avoiding surface harmonic transformations the method requireso n2logn operations foro n2 grid points as opposed too n3 operations for the standard spectral transform method providing a reduced complexity spectral method obviating the pole problem in the integration of time dependent partial differential equations on the sphere the filter s performance is demonstrated with numerical examples

a fast algorithm for spectral differentiation a simple algorithm is presented for matrix multiplication in just over half the number of operations entailed by the conventional algorithm in cases where the matrix possesses the degree of symmetry widely exhibited by derivative matrices the algorithm is used to multiply the chebyshev derivative matrix by a vector for the larger values of n the ratio of execution times approached the expected value of 2