frequency domain analysis of exponentially damped linear systems abstract in this paper frequency domain analysis of linear exponentially damped systems subjected to arbitrary initial conditions is studied the standard discrete fourier transform dft procedure combined with fast fourier transform fft algorithms is applied to compute dynamic responses after transforming the original motion equations subjected to non zero initial conditions into new ones with completely null initial conditions by using the pseudo force approach a multi degree of freedom mdof system and a continuum system with exponential damping are considered as the illustrative problems the accuracy of the proposed method is evaluated by comparing the results with those obtained from simulink based on the state space method

linear damping models for structural vibration abstract linear damping models for structural vibration are examined first the familiar dissipation matrix model then the general linear model in both cases an approximation of small damping is used to obtain simple expressions for damped natural frequencies complex mode shapes and transfer functions results for transfer functions can be expressed in the form of very direct extensions of the familiar expression for the undamped case this allows a detailed discussion of the implications of the various damping models for the interpretation of measured transfer functions especially in the context of experimental modal analysis in the case of a dissipation matrix model it would be possible in principle to determine all the model parameters from measurements in the case of the general model however there is a fundamental ambiguity which prevents full determination of the model from measurements on a single structure

initial conditions in frequency domain analysis the fem applied to the scalar wave equation the present paper describes a procedure to consider initial conditions in the frequency domain analysis of continuous media discretized by the fem the frequency domain formulation presented here is based on a standard dft procedure the fft algorithm being employed to transform from time to the frequency domain and vice versa the standard galerkin finite element method displacement model is used to replace the original differential governing equation by an integral equation amenable to numerical solution the scalar wave equation one and two dimensional is used to illustrate the proposed approach at the end of the paper examples of wave propagation for a taut string a one dimensional rod and a membrane are presented to illustrate the robustness of the formulation presented here

electrical characteristics of a class of n order triangular network abstract this paper investigates the electrical properties of a class of n order triangular resistor network model the authors establish by constructing equivalent model the second order nonlinear difference equation model we propose by using equivalent transformation method the general and specific methods of solution for the nonlinear difference equation and at once obtain an accurate equivalent resistance formula of n order triangular resistor network and a series of special conclusions are analyzed and compared finally we study the oscillation and resonance characteristics of the triangular lc network under different conditions where the method used is variable substitution by using the equivalent resistance results obtained this study reveals the electrical characteristics of triangular lc network and establishes a new theoretical foundation for related disciplines

explicit preconditioned domain decomposition schemes for solving nonlinear boundary value problems abstract a new class of inner outer iterative procedures in conjunction with picard newton methods based on explicit preconditioning iterative methods for solving nonlinear systems is presented explicit preconditioned iterative schemes based on the explicit computation of a class of domain decomposition generalized approximate inverse matrix techniques are presented for the efficient solution of nonlinear boundary value problems on multiprocessor systems applications of the new composite scheme on characteristic nonlinear boundary value problems are discussed and numerical results are given

dynamic analysis of the linear discrete dynamic systems subjected to the initial conditions by using an fft based spectral analysis method abstract in this paper a fast fourier transforms fft based spectral analysis method is introduced for the dynamic analysis of the linear discrete dynamic systems subjected to the non zero initial conditions the fft based spectral analysis method is developed first for the one degree of freedom dof system and then extended to the multi dof systems to evaluate the accuracy and convergence of the fft based spectral analysis method the forced vibration of a viscously damped 3 dof system is considered as an illustrative problem the accuracy of the proposed fft based spectral analysis method is evaluated by comparing the forced vibration responses obtained by using the fft based spectral analysis method with the exact analytical solutions as well as with the numerical results obtained by using the runge kutta method

iterative solution of linear systems in the 20th century this paper sketches the main research developments in the area of iterative methods for solving linear systems during the 20th century although iterative methods for solving linear systems nd their origin in the early 19th century work by gauss the eld has seen an explosion of activity spurred by demand due to extraordinary technological advances in engineering and sciences the past ve decades have been particularly rich in new developments ending with the availability of large toolbox of specialized algorithms for solving the very large problems which arise in scientic and industrial computational models as in any other scientic area research in iterative methods has been a journey characterized by a chain of contributions building on each other it is the aim of this paper not only to sketch the most signicant of these contributions during the past century but also to relate them to one another c 2000 elsevier science b v all rights reserved

identification of linear non viscous damping with different kernel functions in the time domain abstract the non viscous damping model considers the complete velocity history via a convolution over the kernel functions in this paper a procedure is proposed for identifying a damping model involving different kernel functions in the time domain the method uses responses calculated by the generic time domain integration with the known mass matrix and the stiffness matrix the relaxation parameter in the kernel function is identified through a summation of the error function and the damping coefficient matrix is identified via the linear least squares method two examples show that reasonable estimates could be obtained by the proposed method

the iterative solver risolv with application to the exterior helmholtz problem the innermost computational kernel of many large scale scientific applications is often a large set of linear equations of the form ax b which typically consumes a significant portion of the overall computational time required by the simulation the traditional approach for solving this problem is to use direct methods this approach is often preferred in industry because direct solvers are robust and effective for moderate size problems however direct methods can consume a huge amount of memory and cpu time in large scale cases in these cases iterative techniques are the only viable alternative unfortunately iterative methods lack the robustness of direct methods the situation is especially difficult when the matrix is nonsymmetric a lot of research has been devoted to trying to develop a robust iterative algorithm for nonsymmetric systems the present paper describes a new robust and efficient algorithm aimed at solving iteratively nonsymmetric linear systems it is based on looking for an approximation to the optimal polynomial p m z which satisfies p m z infty min q in pi m q z infty z in d where pi m is the set of all polynomials of degree m which satisfies q m 0 1 and d is a domain in the complex plane which includes all the eigenvalues of a the resulting algorithm is an efficient one especially in the case where we have a set of linear systems which share the same matrix a we present several applications including the exterior helmholtz problem which leads to a large indefinite nonsymmetric and complex system

fft based spectral element analysis for the linear continuum dynamic systems subjected to arbitrary initial conditions by using the pseudo force method in this paper a fast fourier transform fft based spectral element method sem is developed for the linear continuum dynamic systems subjected to arbitrary non null initial conditions in the fft based sem the original equations of motion subjected to arbitrary initial conditions are transformed into a new set of equations of motion subjected to completely null initial conditions by using the pseudo force method so that the conventional spectral element analysis can be applied to obtain desired dynamic responses a simply supported beam and a cantilevered beam are considered as the illustrative problems to evaluate the fft based sem the dynamic responses obtained by using the fft based sem are shown to be in good agreement with the analytical solutions obtained by using the mode superposition method copyright 2007 john wiley sons ltd

anderson acceleration for fixed point iterations this paper concerns an acceleration method for fixed point iterations that originated in work of d g anderson j assoc comput mach 12 1965 pp 547 560 which we accordingly call anderson acceleration here this method has enjoyed considerable success and wide usage in electronic structure computations where it is known as anderson mixing however it seems to have been untried or underexploited in many other important applications moreover while other acceleration methods have been extensively studied by the mathematics and numerical analysis communities this method has received relatively little attention from these communities over the years a recent paper by h fang and y saad numer linear algebra appl 16 2009 pp 197 221 has clarified a remarkable relationship of anderson acceleration to quasi newton secant updating methods and extended it to define a broader anderson family of acceleration methods in this paper our goals are to shed additional light on anderson acceleration and to draw further attention to its usefulness as a general tool we first show that on linear problems anderson acceleration without truncation is essentially equivalent in a certain sense to the generalized minimal residual gmres method we also show that the type 1 variant in the fang saad anderson family is similarly essentially equivalent to the arnoldi full orthogonalization method we then discuss practical considerations for implementing anderson acceleration and illustrate its performance through numerical experiments involving a variety of applications

the quasi newton least squares method a new and fast secant method analyzed for linear systems we present a new quasi newton method that can solve systems of equations of which no information is known explicitly and which requires no special structure of the system matrix like positive definiteness or sparseness the method builds an approximate jacobian based on input output combinations of a black box system uses a rank one update of this jacobian after each iteration and satisfies the secant equation while it has originally been developed for nonlinear equations we analyze its properties and performance when applied to linear systems analytically the method is shown to be convergent in n 1 iterations n being the number of unknowns irrespective of the nature of the system matrix the performance of this method is greatly superior to other quasi newton methods and comparable with gmres when tested on a number of standardized test cases

on investigating gmres convergence using unitary matrices for a given matrix a and right hand side b this paper investigates unitary matrices generating with some right hand sides c the same gmres residual norms as the pair a b we give characterizations of this class of unitary matrices and point out the relationship with krylov subspaces and krylov residual subspaces for the pair a b we investigate the eigenvalues of these unitary matrices in relation to the convergence behavior of gmres for the pair a b and describe the indispensable role of the eigenvector information we conclude with a formula for the gmres residual norms generated by a normal matrix b in terms of its eigenvalues and components of the right hand side c in the eigenvector basis

initial conditions contribution in a bem formulation based on the convolution quadrature method this work presents a two dimensional boundary element method bem formulation for the analysis of scalar wave propagation problems the formulation is based on the so called convolution quadrature method cqm by means of which the convolution integral presented in time domain bem formulations is numerically substituted by a quadrature formula whose weights are computed using the laplace transform of the fundamental solution and a linear multistep method this bem formulation was initially developed for scalar wave propagation problems with null initial conditions in order to overcome this limitation this work presents a general procedure that enables one to take into account non homogeneous initial conditions after replacing the initial conditions by equivalent pseudo forces the numerical results included in this work show the accuracy of the proposed bem formulation and its applicability to such kind of analysis copyright 2006 john wiley sons ltd

exact values for the cubic lattice green functions it is shown that the value of the cubic lattice green functions g z l m n at the upper critical point z 3 for the simple cubic and face centred cubic lattices or z 1 for the body centred cubic lattice is expressible rationally in terms of the known value of g 3 0 0 0 or g 1 0 0 0 and

a generalized conjugate gradient least square method a generalizeds term truncated conjugate gradient method of least square type proposed in 1a b is extended to a form more suitable for proving when the truncated version is identical to the full term version advantages with keeping a control term in the truncated version is pointed out a computationally efficient new algorithm based on a special inner product with a small demand of storage is also presented we also give simplified and slightly extended proofs of termination of the iterative sequence and of existence of ans term recursion identical to the full term version important earlier results on this latter topic are found in 15 16 8 and 11

least squares residuals and minimal residual methods we study krylov subspace methods for solving unsymmetric linear algebraic systems that minimize the norm of the residual at each step minimal residual mr methods mr methods are often formulated in terms of a sequence of least squares ls problems of increasing dimension we present several basic identities and bounds for the ls residual these results are interesting in the general context of solving ls problems when applied to mr methods they show that the size of the mr residual is strongly related to the conditioning of different bases of the same krylov subspace using different bases is useful in theory because relating convergence to the characteristics of different bases offers new insight into the behavior of mr methods different bases also lead to different implementations which are mathematically equivalent but can differ numerically our theoretical results are used for a finite precision analysis of implementations of the gmres method y saad and m h schultz siam j sci statist comput 7 1986 pp 856 869 we explain that the choice of the basis is fundamental for the numerical stability of the implementation as demonstrated in the case of simpler gmres h f walker and l zhou numer linear algebra appl 1 1994 pp 571 581 the best orthogonalization technique used for computing the basis does not compensate for the loss of accuracy due to an inappropriate choice of the basis in particular we prove that simpler gmres is inherently less numerically stable than the classical gmres implementation due to saad and schultz siam j sci statist comput 7 1986 pp 856 869

scaled total least squares fundamentals the standard approaches to solving overdetermined linear systems bx approx c construct minimal corrections to the vector c and or the matrix b such that the corrected system is compatible in ordinary least squares ls the correction is restricted to c while in data least squares dls it is restricted to b in scaled total least squares stls 22 corrections to both c and b are allowed and their relative sizes depend on a real positive parameter gamma stls unifies several formulations since it becomes total least squares tls when gamma 1 and in the limit corresponds to ls when gamma rightarrow 0 and dls when gamma rightarrow infty this paper analyzes a particularly useful formulation of the stls problem the analysis is based on a new assumption that guarantees existence and uniqueness of meaningful stls solutions for all parameters gamma 0 it makes the whole stls theory consistent our theory reveals the necessary and sufficient condition for preserving the smallest singular value of a matrix while appending or deleting a column this condition represents a basic matrix theory result for updating the singular value decomposition as well as the rank one modification of the hermitian eigenproblem the paper allows complex data and the equivalences in the limit of stls with dls and ls are proven for such data it is shown how any linear system bx approx c can be reduced to a minimally dimensioned core system satisfying our assumption consequently our theory and algorithms can be applied to fully general systems the basics of practical algorithms for both the stls and dls problems are indicated for either dense or large sparse systems our assumption and its consequences are compared with earlier approaches

improved approximate methods for calculating frequency response function matrix and response of mdof systems with viscoelastic hereditary terms as we know it is difficult and unnecessary to obtain all the eigenpairs of a large scaled viscoelastic nonviscous or hysteretic damping systems which means that the mode truncation scheme is generally used and the mode truncated error is therefore introduced this study is aimed at eliminating the influence of the unavailable modes on the dynamic response of mdof systems with viscoelastic hereditary terms the energy dissipation terms of the system depend on the past history of motion via convolution integrals over some kernel functions therefore the system is a nonviscously damped system which has been considered as the most generalized damping model within the scope of a linear mechanical analysis to approximate frequency response function frf matrix and response without using the unavailable modes we suggest two methods which attempt to approximate the influence of the unavailable modes in terms of the lower modes and system matrices by using the first one or two terms of neumann expansion of the contribution of the unavailable modes in contrast with the frf matrix approximated in terms of the first two terms of neumann expansion these procedures cannot be extended to further high order terms since all of them will be affected by the frequency dependent variation of damping matrix from previous terms finally an example is shown that the two presented methods can make the mode truncated error reduce and may be used to approximate the influence of nonviscous modes contributed to frf matrix due to the fact that the nonviscous modes are difficult to be obtained accurately even if a small scaled model is used for some eigensolution methods

sharpness in rates of convergence for the symmetric lanczos method the lanczos method is often used to solve a large and sparse symmetric matrix eigenvalue problem there is a well established convergence theory that produces bounds to predict the rates of convergence good for a few extreme eigenpairs these bounds suggest at least linear convergence in terms of the number of lanczos steps assuming there are gaps between individual eigenvalues in practice often superlinear convergence is observed the question is do the existing bounds tell the correct convergence rate in general an affirmative answer is given here for the two extreme eigenvalues by examples whose lanczos approximations have errors comparable to the error bounds for all lanczos steps

exact evaluation of the resistance in an infinite face centered cubic network the equivalent resistance between the origin and the lattice site 2n 0 0 in an infinite face centered cubic fcc network consisting from identical resistors each of resistance r has been evaluated analytically and numerically the asymptotic behavior of the equivalent resistance has been also investigated finally some numerical values for the equivalent resistance are presented

computational methods for complex eigenproblems in finite element analysis of structural systems with viscoelastic damping treatments in this paper efficient numerical methods to approximate the complex eigenvalues and eigenvectors in non proportional and non viscous systems are presented these methods are specially conceived for practical engineering applications making use of the finite element analysis to determinate the effect that potential damping treatments have on the natural frequencies and mode shapes of structural systems considering the solution of the undamped problem the complex eigenpair is estimated by finite increments using the eigenvector derivatives for non proportional viscous systems with low and medium damping a simple single step technique is presented whose rapidity and accuracy is verified by means of numerical applications for higher damped systems an incremental approach is proposed that keeps the accuracy without significantly increasing the computational time for non viscously damped systems a fast iterative modality is suggested which allows to approximate in an efficient and simple way the complex eigenpair as numerical applications the study of a metallic beam with free layer damping treatment is completed using finite element procedures where the damping material is modelized by an exponential model whose parameters are obtained from curve fitting to experimental data

identification of damping part 1 viscous damping abstract characterization of damping forces in a vibrating structure has long been an active area of research in structural dynamics the most common approach is to use viscous damping where the instantaneous generalized velocities are the only relevant state variables that affect damping forces however viscous damping is by no means the only damping model within the scope of linear analysis any model which makes the energy dissipation functional non negative is a possible candidate for a valid damping model this paper and its companion see pp 63 88 of this issue are devoted to developing methodologies for identification of such general damping models responsible for energy dissipation in a vibrating structure this paper considers identification of viscous damping under circumstances when the actual damping model in the structure is non viscous a method is presented to obtain a full non proportional viscous damping matrix from complex modes and complex natural frequencies it is assumed that the damping is small so that a first order perturbation method is applicable the proposed method and several related issues are discussed by considering numerical examples based on a linear array of damped spring mass oscillators it is shown that the method can predict the spatial location of damping with good accuracy and also give some indication of the correct mechanism of damping

vandermonde matrices with chebyshev nodes abstract for an n n vandermonde matrix v n j i 1 1 ij n with translated chebyshev zero nodes it is discovered that v n t admits an explicit qr decomposition with the r factor consisting of the coefficients of the translated chebyshev polynomials this decomposition then leads to an exact expression for the frobenius condition number of its submatrix v k n j i 1 1 i k 1 j n so called a rectangular vandermonde matrix bounds on individual singular value and more it is explained how these results can be used to establish asymptotically optimal lower bounds on condition numbers of real rectangular vandermonde matrices and nearly optimally conditioned real rectangular vandermonde matrices on a given interval extensions are also made for v n with nodes being zeros of any translated orthogonal polynomials other than chebyshev ones similar results hold for v n with translated chebyshev extreme nodes too owing to that v n t admits an explicit qr like decomposition close formulas of or tight bounds on the residuals are also presented for the conjugate gradient method the minimal residual method and the generalized minimal residual method on certain linear systems ax b with a having eigenvalues the same as the nodes mentioned above as a by product they yield positive definite linear systems for which the residuals by the conjugate gradient method are always comparable to the existing error bounds for all iteration steps

the condition number of equivalence transformations that block diagonalize matrix pencils how ill conditioned must a matrix s be if its columns are constrained to span certain subspaces we answer this question in order to find nearly best conditioned matrices sr and sl that block diagonalize a given matrix pencil t a b i e s l 1 tsr is bloc diagonal we show that the best conditioned sr has a condition number approximately equal to the cosecant of the smallest angle between right subspaces belonging to different diagonal blocks of thus the more nearly the right subspaces overlap the more ill conditioned sr must be the same is true of sl and the left subspaces for the standard eigenproblem t a i sl sr and the cosecant of the angle between subspaces turns out equal to an earlier estimate of the smallest condition number namely the norm of the projection matrix associated with one of the subspaces we apply this result to bound the error in an algorithm to compute analytic functions of matrices for instance exp t

symmetric state space method for a class of nonviscously damped systems multiple degree of freedom linear nonviscously damped systems are considered it is assumed that the nonviscousdampingforcesdependonthepasthistoryofvelocitiesviaconvolutionintegralsoverexponentiallydecaying kernel functions the traditional state space approach well known for viscously damped systems is extended to such nonviscously damped systems using a set of internal variables suitable numerical examples are provided to illustrate the proposed approach i introduction v iscousdampingisthemostcommonmodelforthemodeling of vibration damping in linear systems this model e rst introduced by rayleigh 1 assumes that the instantaneous generalized velocities are the only relevant variables that determine damping viscous damping models are used widely for their simplicity and mathematical convenience even though the behavior of real structural materials is at best poorly mimicked by simple viscous models forthisreason itiswellrecognizedthat ingeneral aphysically realistic model of damping will not be viscous damping models in whichthedissipativeforcesdependonanyquantityotherthantheinstantaneous generalized velocities are nonviscous damping models mathematically any causal model that makes the energy dissipation functional nonnegative is a possible candidate for a nonviscous damping model clearly a wide range of choice is possible either based on the physics of the problem or by selecting a model a priori and e tting its parameters from experiments here we will use a particular type of damping model that is not viscous and throughout the paper the terminology nonviscous damping will refer to this specie c model only possiblythemostgeneralwaytomodeldampingwithinthelinear range is to use nonviscous damping models that depend on the past history of motion via convolution integrals over kernel functions 2 the equations of motion of an n degree of freedom linear system with such damping can be expressed by

implementation of sparse matrix algorithms in an advection diffusion chemistry module a two dimensional advection diffusion chemistry module of a large scale environmental model is taken the module is described mathematically by a system of partial differential equations sequential splitting is used in the numerical treatment the non linear chemistry is most time consuming part and it is handled by six implicit algorithms for solving ordinary differential equations this leads to the solution of very long sequences of systems of linear algebraic equations it is crucial to solve these systems efficiently this is achieved by applying four different algorithms the numerical results indicate that the algorithms based on a preconditioned sparse matrix technique and on a specially designed algorithm for the particular problem under consideration perform best

a comparison of some gmres like methods abstract gmres and cgs are well known iterative methods for the solution of certain sparse linear systems with a nonsymmetric matrix these methods have been compared experimentally in many studies and specific observations on their convergence behavior have been reported a new iterative method to solve a nonsymmetric system has been proposed by eirola and nevanlinna the purpose of this paper is to investigate this method and to compare it with gmres we have seen problems for which this method is more efficient than gmres the original method has as drawbacks that it is not scaling invariant and that it may suffer from numerical instability but it is shown that these deficiencies can be repaired a method proposed by broyden 1969 seems related to the new method and is therefore included in the comparison

residual and backward error bounds in minimum residual krylov subspace methods minimum residual norm iterative methods for solving linear systems ax b can be viewed as and are often implemented as sequences of least squares problems involving krylov subspaces of increasing dimensions the minimum residual method minres c paige and m saunders siam j numer anal 12 1975 pp 617 629 and generalized minimum residual method gmres y saad and m schultz siam j sci statist comput 7 1986 pp 856 869 represent typical examples in c paige and z strakos bounds for the least squares distance using scaled total least squares numer math to appear revealing upper and lower bounds on the residual norm of any linear least squares ls problem were derived in terms of the total least squares tls correction of the corresponding scaled tls problem in this paper theoretical results of c paige and z strakos bounds for the least squares distance using scaled total least squares numer math to appear are extended to the gmres context the bounds that are developed are important in theory but they also have fundamental practical implications for the finite precision behavior of the modified gram schmidt implementation of gmres and perhaps for other minimum norm methods

on the use of pseudo forces to consider initial conditions in 3d time and frequency domain acoustic analysis this work is mainly concerned with a general strategy based on well known concepts of classical mechanics for taking into account initial conditions in frequency domain fd and time domain td analyses a general approach extended here to three dimensional applications is presented special problems associated with analyses through discrete fourier transform dft algorithms as those occurring in consequence of a non correct choice of extended period or those connected with aliasing phenomenon are also discussed furthermore an alternative starting procedure for time marching schemes in td analyses is proposed at the end of the paper to validate the proposed techniques and to demonstrate their generality two and three dimensional problems with non homogeneous initial conditions are solved through frequency and time domain approaches by employing the finite element method fem numerical results are compared with existing analytical solutions

on commuting matrices in max algebra and in classical nonnegative algebra abstract this paper studies commuting matrices in max algebra and nonnegative linear algebra our starting point is the existence of a common eigenvector which directly leads to max analogs and nonnegative analogs of some classical results for complex matrices we also investigate frobenius normal forms of commuting matrices particularly when the perron roots of the components are distinct for the case of max algebra we show how the intersection of eigencones of commuting matrices can be described and we consider connections with boolean algebra which enables us to prove that two commuting irreducible matrices in max algebra have a common eigennode

accelerating with rank one updates abstract consider the iteration xk 1 xk h b axk for solving ax b a is n x n nonsingular we discuss rank one updates to improve h as an approximation to a 1 during the iteration the update kills and reduces singular values of i ah and thus speeds up the convergence the algorithm terminates after at most n sweeps and if all n sweeps are needed then a 1 has been computed

a new implementation of en method en method proposed by eirola and nevalinna can be more efficient than gmres in some cases but be not the case in general which has been studied by vuik and van der vorst comparison of the efficiency of the two methods in practice shows that gmres is preferable a new implementation of en method with significant saving both in the amount of work and memory requirements is introduced in the present paper numerical experiments indicate that this new version is competitive

on the exact evaluation of the face centred cubic lattice green function the mathematical properties of the lattice green function open image in new window are investigated where w w 1 iw 2 lies in a complex plane which is cut from w 1 to w 3 and l 1 l 2 l 3 is a set of integers with l 1 l 2 l 3 equal to an even integer in particular it is proved that g 2n 0 0 w where n 0 1 2 is a solution of a fourth order linear differential equation of the fuchsian type with four regular singular points at w 1 0 3 and it is also shown that g 2n 0 0 w satisfies a five term recurrence relation with respect to the integer variable n the limiting function g 2n 0 0 w 1 equiv lim epsilon rightarrow0 g 2n 0 0 w 1 mathrm i epsilon g mathrm r 2n 0 0 w 1 mathrm i g mathrm i 2n 0 0 w 1 nonumber where w 1 1 3 is evaluated exactly in terms of 2 f 1 hypergeometric functions and the special cases g 2n 0 0 0 g 2n 0 0 1 and g 2n 0 0 3 are analysed using singular value theory more generally it is demonstrated that g l 1 l 2 l 3 w can be written in the form open image in new window where open image in new window are rational functions of the variable k k and e k are complete elliptic integrals of the first and second kind respectively with k 2 equiv k 2 w 1 over2 2 over w biggl 1 1 over w biggr 3 over2 1 over2 biggl 1 1 over w biggr biggl 1 1 over w biggr 3 over2 biggl 1 3 over w biggr 1 over2 nonumber and the parameter is defined as xi equiv xi w biggl 1 sqrt 1 3 over w biggr 1 biggl 1 sqrt 1 1 over w biggr nonumber this result is valid for all values of w which lie in the cut plane the asymptotic behaviour of g 2n 0 0 w 1 and g 2n 0 0 w 1 as n is also determined in the final section of the paper a new 2 f 1 product form for the anisotropic face centred cubic lattice green function is given