a high order numerical method for the helmholtz equation with nonstandard boundary conditions we describe a high order accurate methodology for the numerical simulation of time harmonic waves governed by the helmholtz equation our approach combines compact finite difference schemes that provide an inexpensive venue toward high order accuracy with the method of difference potentials developed by ryaben kii the latter can be interpreted as a generalized discrete version of the method of calderon s operators in the theory of partial differential equations the method of difference potentials can accommodate nonconforming boundaries on regular structured grids with no loss of accuracy due to staircasing it introduces a universal framework for treating boundary conditions of any type a significant advantage of this method is that changing the boundary condition within a fairly broad variety does not require any major changes to the algorithm and is computationally inexpensive in this paper we address various types of boundary conditions using the method of difference potentials we demonstrate th

a new eighth order implicit finite difference method to solve the three dimensional helmholtz equation abstract this paper introduces a novel method for deducing high order finite difference schemes for the three dimensional helmholtz equation p 2 p f the new method can reach an eighth order of accuracy it is based on an implicit formulation by computing simultaneously the unknown variable and its corresponding derivatives the scheme is obtained from taylor series expansion and plane wave theory analysis and it is constructed from a few modifications to the standard centered finite differences a classical dispersion analysis and the error between the numerical wavenumber and the exact wavenumber is given numerical experiments are presented to verify the feasibility and accuracy of the proposed methods for a broad range of values moreover the numerical method is capable to solve with great precision the monotone and oscillatory helmholtz problem with variable values thus the proposed formulation results in an attractive method easy to implement with high order of accuracy but nearly the same computational cost as those of an explicit formulation

fast and high accuracy multigrid solution of the three dimensional poisson equation we employ a fourth order compact finite difference scheme fos with the multigrid algorithm to solve the three dimensional poisson equation we test the influence of different orderings of the grid space and different grid transfer operators on the convergence and efficiency of our high accuracy algorithm fourier smoothing analysis is conducted to show that fos has a smaller smoothing factor than the traditional second order central difference scheme cds a new method of fourier smoothing analysis is proposed for the partially decoupled red black gauss seidel relaxation with fos numerical results are given to compare the computed accuracy and the computational efficiency of fos with multigrid against cds with multigrid

a dispersion minimizing compact finite difference scheme for the 2d helmholtz equation in this paper we present a dispersion minimizing compact finite difference scheme for solving the 2d helmholtz equation which is a fourth order scheme the error between the numerical wavenumber and the exact wavenumber is analyzed and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters numerical results are presented to demonstrate the efficiency and accuracy of the compact finite difference scheme with refined parameters

component averaged row projections a robust block parallel scheme for sparse linear systems a new method for the parallel solution of large sparse linear systems is introduced it proceeds by dividing the equations into blocks and operating in block parallel iterative mode i e all the blocks are processed in parallel and the partial results are merged to form the next iterate the new scheme performs kaczmarz row projections within the blocks and merges the results by certain component averaging operations hence it is called component averaged row projections or carp the system matrix can be general nonsymmetric and ill conditioned and the division into blocks is unrestricted for partial differential equations pdes if the blocks are domain based then only variables at the boundaries between domains are averaged thereby minimizing data transfer between processors carp is very robust its application to test cases of linear systems derived from pdes shows that it converges in difficult cases where state of the art methods fail it is also very memory efficient and exhibits an almost linear speedup ratio with efficiency greater than unity in some cases a formal proof of convergence is presented it is shown that the component averaging operations are equivalent to row projections in a certain superspace so the convergence properties of carp are identical to those of kaczmarz s algorithm in the superspace carp and its convergence proof also apply to the consistent convex feasibility problem

the alternating greedy expansion and applications to computing digit expansions from left to right in cryptography the central topic of this paper is the alternating greedy expansion of integers which is defined to be a binary expansion with digits 0 1 with the property that the nonzero digits have alternating signs we collect known results about this alternating greedy expansion and complement it with other useful properties and algorithms in the second part we apply it to give an algorithm for computing a joint expansion of d integers of minimal joint hamming weight from left to right i e from the column with the most significant bits towards the column with the least significant bits furthermore we also compute an expansion equivalent to the so caled w naf from left to right using the alternating greedy expansion

a new 9 point sixth order accurate compact finite difference method for the helmholtz equation a new 9 point sixth order accurate compact finite difference method for solving the helmholtz equation in one and two dimensions is developed and analyzed this scheme is based on sixth order approximation to the derivative calculated from the helmholtz equation a sixth order accurate symmetrical representation for the neumann boundary condition was also developed the efficiency and accuracy of the scheme is validated by its application to two test problems which have exact solutions numerical results show that this sixth order scheme has the expected accuracy and behaves robustly with respect to the wave number

a transformation free hoc scheme and multigrid method for solving the 3d poisson equation on nonuniform grids a high order compact hoc difference scheme is proposed to solve the three dimensional 3d poisson equation on nonuniform orthogonal cartesian grids involving no coordinate transformation from the physical space to the computational space theoretically the proposed scheme has third to fourth order accuracy its fourth order accuracy is achieved under uniform grid settings then a multigrid method is developed to solve the linear system arising from this hoc difference scheme and the corresponding multigrid restriction and interpolation operators are constructed using the volume law numerical experiments are conducted to show the computed accuracy of the hoc scheme and the computational efficiency of the multigrid method

compact finite difference schemes of sixth order for the helmholtz equation new compact finite difference schemes of sixth order are derived for the three dimensional helmholtz equation du k 2u f convergence characteristics and accuracy are compared and a truncation error analysis is presented for a broad range of k values

a fast spectral solver for a 3d helmholtz equation we present a fast solver for the helmholtz equation delta u pm lambda 2 u f in a 3d rectangular box the method is based on the application of the discrete fourier transform accompanied by a subtraction technique which allows us to reduce the errors associated with the gibbs phenomenon and achieve any prescribed rate of convergence the algorithm requires o n3 log n operations where n is the number of grid points in each direction we solve a dirichlet boundary problem for the helmholtz equation we also extend the method to the solution of mixed problems where dirichlet boundary conditions are specified on some faces and neumann boundary conditions are specified on other faces high order accuracy is achieved by a comparatively small number of points for example for the accuracy of 10 8 the resolution of only 16 32 points in each direction is necessary

inequalities for series in q shifted factorials and q gamma functions abstract the paper studies logarithmic convexity and concavity of power series with coefficients involving q gamma functions or q shifted factorials with respect to a parameter contained in their arguments the principal motivating examples of such series are basic hypergeometric functions we consider four types of series for each type we establish conditions sufficient for the power series coefficients of the generalized turanian formed by these series to have constant sign finally we furnish seven examples of basic hypergeometric functions satisfying our general theorems this investigation extends our previous results on power series with coefficient involving the ordinary gamma functions and the shifted factorials

compact 2d and 3d sixth order schemes for the helmholtz equation with variable wave number several studies have presented compact fourth order accurate finite difference approximation for the helmholtz equation in two or three dimensions several of these formulae allow for the wave number k to be variable other papers have extended this further to include variable coefficients within the laplacian which models non homogeneous materials in electromagnetism later papers considered more accurate compact sixth order methods but these were restricted to constant k in this paper we extend these compact sixth order schemes to variable k in both two and three dimensions results on 2d and 3d problems with known analytic solutions verify the sixth order accuracy we demonstrate that for large wave numbers the second order scheme cannot produce comparable results with reasonable grid sizes

a high order fast direct solver for singular poisson equation we present a fourth order numerical solution method for the singular neumann boundary problem of poisson equations such problems arise in the solution process of incompressible navier stokes equations and in the time harmonic wave propagation in the frequence space with the zero wavenumber the equation is first discretized with a fourth order modified collatz difference scheme producing a singular discrete equation then an efficient singular value decomposition svd method modified from a fast poisson solver is employed to project the discrete singular equation into the orthogonal complement of the null space of the singular matrix in the complement of the null space the projected equation is uniquely solvable and its solution is proven to be a solution of the original singular discrete equation when the original equation has a solution analytical and experimental results show that this newly proposed singular equation solver is efficient while retaining the accuracy of the high order discretization c 2001 academic press

algorithm of asynchronous binary signed digit recoding on fast multiexponentiation multiexponentiation for the 2 dimension one especially is a very important but time consuming operation in many elgamal like public key cryptosystems an efficient multiexponentiation method is firstly proposed by shamir in elgamal s paper shamir s method requires 1 75n modular multiplications in average where n is the bit length of the exponents using minimal weight binary signed digit bsd representations in shamir s method can reduce the number of multiplications to 1 556n due to there are many minimal weight bsd representations for an integer dimitrov et al proposed a new 2 dimension multiexponentiation algorithm to recode the canonical binary signed digit representations the average number of the number 1 556n is reduced to 1 534n after the publish of the dimitrov jullien miller algorithm there are many research on receding bsd representation of a pair of integers the most interesting one is the joint sparse form the number of multiplication can be reduced to 1 500n in joint sparse form when the exponent length is near infinite in this paper we propose a new asynchronous recoding algorithm to reduce the factor to 1 509n for comparison to all the previous algorithm the property of asynchronous receding is especially suitable for multi dimension multiexponentiation

asymptotics of basic bessel functions and q laguerre polynomials abstract we establish a large n complete asymptotic expansion for q laguerre polynomials and a complete asymptotic expansion for a q bessel function of large argument these expansions are needed in our study of an exactly solvable random transfer matrix model for disordered electronic systems we also give a new derivation of an asymptotic formula due to littlewood 1907

on asymptotics of the q exponential and q gamma functions abstract in this work we present a derivation for the complete asymptotic expansions of euler s q exponential function and jackson s q gamma function via mellin transform these formulas are valid everywhere uniformly on any compact subset of the complex plane

high order implicit finite difference schemes for the two dimensional poisson equation abstract in this paper a new family of high order finite difference schemes is proposed to solve the two dimensional poisson equation by implicit finite difference formulas of 2 m 1 operator points the implicit formulation is obtained from taylor series expansion and wave plane theory analysis and it is constructed from a few modifications to the standard finite difference schemes the approximations achieve 2 m 4 order accuracy for the inner grid points and up to eighth order accuracy for the boundary grid points using a successive over relaxation method the high order implicit schemes have faster convergence as m is increased compensating the additional computation of more operator points thus the proposed solver results in an attractive method easy to implement with higher order accuracy but nearly the same computation cost as those of explicit or compact formulation in addition particular case m 1 yields a new compact finite difference schemes of sixth order of accuracy numerical experiments are presented to verify the feasibility of the proposed method and the high accuracy of these difference schemes

single cell high order difference methods for the helmholtz equation abstract a procedure for deriving certain high order difference formulas for the helmholtz equation is given families of formulas of order 2 4 and 6 are derived the distinction between atomic and nonatomic formulas is made and a nonatomic formula of order 4 is given for a similar equation with variable coefficients numerical results using these formulas are given

a dispersion minimizing finite difference scheme and preconditioned solver for the 3d helmholtz equation in this paper a new 27 point finite difference method is presented for solving the 3d helmholtz equation with perfectly matched layer pml which is a second order scheme and pointwise consistent with the equation an error analysis is made between the numerical wavenumber and the exact wavenumber and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters a full coarsening multigrid based preconditioned bi cgstab method is developed for solving the linear system stemming from the helmholtz equation with pml by the finite difference scheme the shifted laplacian is extended to precondition the 3d helmholtz equation and a spectral analysis is given the discrete preconditioned system is solved by the bi cgstab method with a multigrid method used to invert the preconditioner approximately full coarsening multigrid is employed and a new matrix based prolongation operator is constructed accordingly numerical results are presented to demonstrate the efficiency of both the new 27 point finite difference scheme with refined parameters and the preconditioned bi cgstab method with the 3d full coarsening multigrid

discrete dispersion relation for hp version finite element approximation at high wave number the dispersive properties of high order finite element schemes are analyzed in the setting of the helmholtz equation and an explicit form of the discrete dispersion relation is obtained for elements of arbitrary order it is shown that the numerical dispersion displays three different types of behavior depending on the size of the order of the method relative to the mesh size and the wave number quantitative estimates are obtained for the behavior and rates of decay of the dispersion error in the differing regimes all estimates are fully explicit and are shown to be sharp limits are obtained where transitions between the different regimes occur and used to provide guidelines for the selection of the mesh size and the polynomial order in terms of the wave number so that the dispersion error is controlled

high order finite difference methods for the helmholtz equation high order finite difference methods for solving the helmholtz equation are developed and analyzed in one and two dimensions on uniform grids the standard pointwise representation has a second order accurate local truncation error we also study two schemes which have a fourth order accurate local truncation error one of the high order schemes is based on generalizations of the pade approximation the second scheme is based on high order approximation to the derivative calculated from the helmholtz equation itself a symmetric high order representation is developed for a neumann boundary condition numerical results are presented on model problems approximated with the developed schemes

expansion and asymptotic in terms of basic bessel functions abstract this work aims to study the expansion and asymptotic for solutions of q difference equations in terms of the basic bessel functions namely j 2 x q for this purpose we will show that the constructive method introduced by olver f w j olver asymptotics and special functions academic press inc 1974 and exploited early by fitouhi et al h chebli a fitouhi m m hamza expansion in series of bessel functions and transmutations for perturbed bessel operator j math anal appl 181 3 1994 a fitouhi m m hamza uniform expansion for eigenfunction of singular second order differential operator siam j math anal 21 6 1990 a fitouhi m m hamza expansion in series of laguerre functions for solution of perturbed laguerre equations contemp math 183 1995 a fitouhi j el kamel expansion in series of gegenbauer polynomials int trans sp funct 5 3 4 1997 213 226 can be extended in this context as application new expansions of some basic special functions are established in particular these given by ismail y chen m e h ismail k a muttalib asymptotics of basic bessel functions and q laguerre polynomilas j comput appl math 54 1994 263 272 m e h ismail classical and quantum orthogonal polynomials in one variable cambridge university press 2005

on the derivation of highest order compact finite difference schemes for the one and two dimensional poisson equation with dirichlet boundary conditions the primary aim of this paper is to answer the question what are the highest order five or nine point compact finite difference schemes to answer this question we present several simple derivations of finite difference schemes for the one and two dimensional poisson equation on uniform quasi uniform and nonuniform face to face hyperrectangular grids and directly prove the existence or nonexistence of their highest order local accuracies our derivations are unique in that we do not make any initial assumptions on stencil symmetries or weights for the one dimensional problem the derivation using the three point stencil on both uniform and nonuniform grids yields a scheme with arbitrarily high order local accuracy however for the two dimensional problem the derivation using the corresponding five point stencil on uniform and quasi uniform grids yields a scheme with at most second order local accuracy and on nonuniform grids yields at most first order local accuracy when expanding the five poin

accurate finite difference methods for time harmonic wave propagation finite difference methods for solving problems of time harmonic acoustics are developed and analyzed multi dimensional inhomogeneous problems with variable possibly discontinuous coefficients are considered accounting for the effects of employing non uniform grids a weighted average representation is less sensitive to transition in wave resolution due to variable wave numbers or non uniform grids than the standard pointwise representation further enhancement in method performance is obtained by basing the stencils on generalizations of pade approximation or generalized definitions of the derivative reducing spurious dispersion anisotropy and reflection and by improving the representation of source terms the resulting schemes have fourth order accurate local truncation error on uniform grids and third order in the non uniform case guidelines for discretization pertaining to grid orientation and resolution are presented

high order compact solvers for the three dimensional poisson equation new compact approximation schemes for the laplace operator of fourth and sixth order are proposed the schemes are based on a pade approximation of the taylor expansion for the discretized laplace operator the new schemes are compared with other finite difference approximations in several benchmark problems it is found that the new schemes exhibit a very good performance and are highly accurate especially on large grids they outperform noncompact schemes

an infinite class of completely monotonic functions involving the q gamma function abstract in this paper the q analogue of the stirling formula for the q gamma function is used to prove the complete monotonicity property for an infinite class of functions which are all closely related to the q gamma function and its logarithmic derivatives q polygamma functions as an application of this result the two sided inequalities for the q gamma and the q polygamma functions are established

distribution results for low weight binary representations for pairs of integers we discuss an optimal method for the computation of linear combinations of elements of abelian groups which uses signed digit expansions this has applications in elliptic curve cryptography we compute the expected number of operations asymptotically including a periodically oscillating second order term and prove a central limit theorem apart from the usual right to left i e least significant digit first approach we also discuss a left to right computation of the expansions this exhibits fractal structures that are studied in some detail

oscillatory matrix model in chern simons theory and jacobi theta determinantal point process the partition function of the chern simons theory on the three sphere with the unitary group u n provides a one matrix model the corresponding n particle system can be mapped to the determinantal point process whose correlation kernel is expressed by using the stieltjes wigert orthogonal polynomials the matrix model and the point process are regarded as q extensions of the random matrix model in the gaussian unitary ensemble and its eigenvalue point process respectively we prove the convergence of the n particle system to an infinite dimensional determinantal point process in n in which the correlation kernel is expressed by jacobi s theta functions we show that the matrix model obtained by this limit realizes the oscillatory matrix model in chern simons theory discussed by de haro and tierz

parallel solution of high frequency helmholtz equations using high order finite difference schemes abstract high frequency helmholtz equations pose a challenging computational task especially at high frequencies this work examines the parallel solution of such problems using 2nd 4th and 6th order finite difference schemes the examples include problems with known analytic solutions enabling error evaluation of the different schemes on various grids including problems on l shaped domains the resulting linear systems are solved with the block parallel carp cg algorithm which succeeded in lowering the relative residual indicating that it is a robust and reliable parallel solver of the resulting linear systems however lowering the error of the solution to reasonable levels with practical mesh sizes is obtained only with the higher order schemes these results corroborate the known limitations of the 2nd order scheme at modeling the helmholtz equation at high frequencies and they indicate that carp cg can also be used effectively with high order finite difference schemes furthermore the parallel scalability of carp cg improves when the wave number is increased on a fixed grid or when for a fixed wave number the grid size is decreased

a perfectly matched layer for the helmholtz equation in a semi infinite strip the perfectly matched layer pml has become a widespread technique for preventing reflections from far field boundaries for wave propagation problems in both the time dependent and frequency domains we develop a discretization to solve the helmholtz equation in an infinite two dimensional strip we solve the interior equation using high order finite differences schemes the combined helmholtz pml problem is then analyzed for the parameters that give the best performance we show that the use of local high order methods in the physical domain coupled with a specific second order approximation in the pml yields global high order accuracy in the physical domain we discuss the impact of the parameters on the effectiveness of the pml numerical results are presented to support the analysis

redundancy of minimal weight expansions in pisot bases motivated by multiplication algorithms based on redundant number representations we study representations of an integer n as a sum n k kuk where the digits k are taken from a finite alphabet and uk k is a linear recurrent sequence of pisot type with u0 1 the most prominent example of a base sequence uk k is the sequence of fibonacci numbers we prove that the representations of minimal weight k k are recognised by a finite automaton and obtain an asymptotic formula for the average number of representations of minimal weight furthermore we relate the maximal number of representations of a given integer to the joint spectral radius of a certain set of matrices

on a family of integrals that extend the askey wilson integral we study a family of integrals parameterised by n 2 3 generalising the askey wilson integral n 2 which has arisen in the theory of q analogs of monodromy preserving deformations of linear dierential systems and in theory of the baxter q operator for the xxz open quantum spin chain these integrals are particular examples of moments dened by weights generalising the askey wilson weight and we show the integrals are characterised by various n 1 th order linearq dierence equations which we construct in addition we demonstrate that these integrals can be evaluated as a nite sum of n 1 bc1 type jackson integrals or 2n 2 2n 1 basic hypergeometric functions

asymptotic expansions for q gamma q exponential and q bessel functions abstract new asymptotic expansions are given for the q gamma function the q exponential functions and for the hahn exton q bessel function for the theta functions four expansions are given and for the hahn exton q bessel difference equation a new solution is given which forms with the hahn exton q bessel function a numerically satisfactory pair of solutions

on accuracy conditions for the numerical computation of waves abstract the helmholtz equation k 2 n 2 u f with a variable index of refraction n and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems such problems can be solved numerically by first truncating the given unbounded domain imposing a suitable outgoing radiation condition on an artificial boundary and then solving the resulting problem on the bounded domain by direct discretization for example using a finite element method in practical applications the mesh size h and the wave number k are not independent but are constrained by the accuracy of the desired computation it will be shown that the number of points per wavelength measured by kh 1 is not sufficient to determine the accuracy of a given discretization for example the quantity k 3 h 2 is shown to determine the accuracy in the l 2 norm for a second order discretization method applied to several propagation models

a general meshsize fourth order compact difference discretization scheme for 3d poisson equation a fourth order compact difference scheme with unrestricted general meshsizes in different coordinate directions is derived to discretize three dimensional poisson equation on a regular cubic domain the difference scheme derivation procedure makes use of the symbolic representation of the finite difference schemes and is easier to understand in such complex three dimensional manipulations we use a preconditioned conjugate gradient method to solve the resulting sparse linear systems and verify the formal order of convergence of the derived fourth order finite difference scheme

an optimal compact sixth order finite difference scheme for the helmholtz equation abstract in this paper we present an optimal compact finite difference scheme for solving the 2d helmholtz equation a convergence analysis is given to show that the scheme is sixth order in accuracy based on minimizing the numerical dispersion a refined optimization rule for choosing the scheme s weight parameters is proposed numerical results are presented to demonstrate the efficiency and accuracy of the compact finite difference scheme with refined parameters

asymptotics for some fundamental q functions abstract in this work we derive asymptotic formulas for the q exponential function e q z q gamma function q z and q shifted factorial z q n as q approaches 1 without using theta functions or summation formulas these asymptotics are valid uniformly on compact subsets of the complex plane