an analogue for szego polynomials of the clenshaw algorithm abstract linear combinations of polynomials that are orthogonal with respect to an inner product defined on part of the real axis are commonly evaluated by the clenshaw algorithm we present an analogous algorithm for the evaluation of a linear combination n j 0 j j of polynomials j that are orthogonal with respect to an inner product defined on part of the unit circle the j are known as szeg polynomials and find applications e g in signal processing we also discuss how to express n j 0 j j as a linear combination of monomials

a novel asymptotic extraction technique for the efficient evaluation of a class of double sommerfeld integrals when the galerkin method of moments mom is used in the numerical analysis of microwave microstrip networks elements of a network impedance matrix are required two dimensional 2d sommerfeld spectral integrals must be evaluated in such cases when semi infinite circuit feed lines are used certain specific types of sommerfeld integrals arise whose integrands oscillate rapidly and present serious numerical difficulties for their evaluation in this paper a novel accurate and much more efficient asymptotic extraction technique aet is developed that involves extracting an inner asymptotic angular integral which is then evaluated analytically before evaluating the outer integral numerically

uniform approximation to fractional derivatives of functions of algebraic singularity fractional derivative d qf x 0 1 with g x being a well behaved function we propose a quadrature method for uniformly approximating d q x ag x the present method consists of interpolating g x at abscissae in 0 1 by a finite sum of chebyshev polynomials it is shown that the use of the lower endpoint x 0 as an abscissa is essential for the uniform approximation namely to bound the approximation errors independently of x 0 1 numerical examples demonstrate the performance of the present method

the abc of hyper recursions each member of the family of gauss hypergeometric functions fn 2f1 a e1n b e2n c e3n z where a b c and z do not depend on n and ej 0 1 not all ej equal to zero satisfies a second order linear difference equation of the form anfn 1 bnfn cnfn 1 0 because of symmetry relations and functional relations for the gauss functions the set of 26 cases for different ej values can be reduced to a set of 5 basic forms of difference equations in this paper the coefficients an bn and cn of these basic forms are given in addition domains in the complex z plane are given where a pair of minimal and dominant solutions of the difference equation have to be identified the determination of such a pair asks for a detailed study of the asymptotic properties of the gauss functions fn for large values of n and of other gauss functions outside this group this will be done in a later paper

time dependent lagrangian systems a geometric approach to the theory of systems with constraints a geometric approach to the theory of time dependent regular lagrangian systems with constraints is presented using the framework of the exact contact manifold tq r l the main subject of the article concerns the properties of the time dependent nonholonomic constraints and the geometric approach to the method of the lagrange multipliers it is shown that every constraint determines a contact one form a vertical vector field and a nonvertical vector field the explicit form of the vector field representing the constrained dynamics is obtained and finally the properties of all these one forms and vector fields are discussed

computation of jacobi functions of the second kind for use in nearside farside scattering theory abstract the nearside farside decomposition of a partial wave series is currently being used to understand the angular scattering of atom diatom collision systems in this theory it is necessary to compute jacobi functions of the second kind on the cut these functions are denoted by q n cos where n may be large positive integers the q n cos can be computed from a three term linear recurrence relation provided the initial values corresponding to n 0 and 1 are known we derive explicit formulas for q 0 cos q 1 cos in terms of elementary transcendental functions a new generating function for jacobi functions of nonintegral degree off the cut is obtained a special case of which yields a generating function for q n cos this is used to check the numerical results as is a casoratian relation we show that the recurrence for q n cos is stable in the forward direction with errors growing like o n we also present some numerics demonstrating the success of the method

fast algorithms for discrete polynomial transforms consider the vandermonde like matrix p p k cos j n j k 0 n where the polynomials p k satisfy a three term recurrence relation if p k are the chebyshev polynomials t k then p coincides with c n 1 cos jk n j k 0 n this paper presents a new fast algorithm for the computation of the matrix vector product pa in o n log 2 n arithmetical operations the algorithm divides into a fast transform which replaces pa with c n 1 and a subsequent fast cosine transform the first and central part of the algorithm is realized by a straightforward cascade summation based on properties of associated polynomials and by fast polynomial multiplications numerical tests demonstrate that our fast polynomial transform realizes pa with almost the same precision as the clenshaw algorithm but is much faster for n 128

unified formalism for nonautonomous mechanical systems we present a unified geometric framework for describing both the lagrangian and hamiltonian formalisms of regular and non regular time dependent mechanical systems which is based on the approach of skinner and rusk 18 the dynamical equations of motion and their compatibility and consistency are carefully studied making clear that all the characteristics of the lagrangian and the hamiltonian formalisms are recovered in this formulation as an example it is studied a semidiscretization of the nonlinear wave equation proving the applicability of the proposed formalism

time dependent lagrangian systems a geometric approach using semibasic forms a new presentation of the geometric formalism in r tq of the time dependent lagrangian systems is given the formalism is used to prove some properties of the equations determining the euler lagrange vector field xl for relating the theory with the time independent case and for studying the time dependent lagrangian inverse problem

arnoldi and jacobi davidson methods for generalized eigenvalue problems ax λbx with singular b in many physical situations a few specific eigenvalues of a large sparse generalized eigenvalue problem ax abx are needed if exact linear solves with a b are available implicitly restarted arnoldi with purification is a common approach for problems where b is positive semidefinite in this paper a new approach based on implicitly restarted arnoldi will be presented that avoids most of the problems due to the singularity of b secondly if exact solves are not available jacobi davidson qz will be presented as a robust method to compute a few specific eigenvalues results are illustrated by numerical experiments

computation of multiple eigenvalues of infinite tridiagonal matrices in this paper it is first given as a necessary and sufficient condition that infinite matrices of a certain type have double eigenvalues the computation of such double eigenvalues is enabled by the newton method of two variables the three term recurrence relations obtained from its eigenvalue problem evp subsume the well known relations of a the zeros of j v z b the zeros of zj f v z hj v z c the evp of the mathieu differential equation and d the evp of the spheroidal wave equation the results of experiments are shown for the three cases a c for the computation of their double pairs

preconditioning constrained eigenvalue problems abstract the purpose of our paper is to introduce a robust preconditioning scheme for the numerical solution of the leftmost eigenvalues and corresponding eigenvectors of a constrained eigenvalue problem this constrained eigenvalue problem is congruent to a nonsymmetric eigenvalue problem with nontrivial jordan blocks associated with infinite eigenvalues the proposed preconditioning scheme is relevant to the application of krylov subspace methods and preconditioned eigensolvers the two key results are a semi orthogonal decomposition and a transformation process that implicitly combines a preconditioning step followed by abstract projection onto the subspace associated with the finite eigenvalues numerical results demonstrate the effectiveness of the preconditioning scheme

asymptotic expansions for second order linear difference equations abstract formal series solutions are obtained for the difference equation y n 2 a n y n 1 b n y n 0 where a n and b n have asymptotic expansions of the form a n s 0 a s n s and b n s 0 b s n s for large values of n and b0 0 these solutions are characterized by the roots of the characteristic equation 2 a0 b0 0 our discussion is divided into three cases according to whether the roots are distinct or equal and do not satisfy the auxiliary equation a1 b1 0 or equal and do satisfy the auxiliary equation the last case is further divided into three subcases according to whether the roots of the indicial equation 1 2 a1 a2 b2 0 do not differ by a nonnegative integer or differ by a positive integer or are equal in all cases the formal series solutions will be shown to be asymptotic our approach is based on the method of successive approximations

an accurate and efficient analysis for transient plane waves obliquely incident on a conductive half space te case we develop a method for the analytical evaluation of the inverse laplace transform representations for transient transverse magnetic tm plane wave obliquely incident on a conductive half space we assume that the permittivity and conductivity of the dispersive half space are independent of frequency the time domain expressions for the reflected and transmitted waves are first represented as inverse laplace transforms the transient fields are then shown to consist of two canonical integrals the canonical integrals in turn are solved analytically thereby yielding close form solutions involving incomplete lipschitz hankel integrals ilhis the ilhis are computed numerically using efficient convergent and asymptotic series expansions thus enabling the efficient computation of the transient fields the exact closed form expressions are verified by comparing with previously published results and with results obtained using standard numerical integration and fast fourier transform fft algorithms an asymptotic series representation for the ilhis is also employed to obtain a relatively simple late time approximation for the transient fields this approximate late time expression is shown to accurately model the fields over a large portion of its time history

evaluation of bessel function integrals with algebraic singularities abstract a new method for the numerical evaluation of 1 0 1 x x j v ax dx is proposed where a are given constants and j is the bessel function of the first kind and of order it is assumed that f is a smooth function the method is based on approximation of the function f by a polynomial given in the form of a linear combination of the shifted jacobi polynomials r k and on the use of chebyshev series expansion for the function j ax

convergence rates for inverse free rational approximation of matrix functions abstract this article deduces geometric convergence rates for approximating matrix functions via inverse free rational krylov methods in applications one frequently encounters matrix functions such as the matrix exponential or matrix logarithm often the matrix under consideration is too large to compute the matrix function directly and krylov subspace methods are used to determine a reduced problem if many evaluations of a matrix function of the form f a v with a large matrix a are required then it may be advantageous to determine a reduced problem using rational krylov subspaces these methods may give more accurate approximations of f a v with subspaces of smaller dimension than standard krylov subspace methods unfortunately the system solves required to construct an orthogonal basis for a rational krylov subspace may create numerical difficulties and or require excessive computing time this paper investigates a novel approach to determine an orthogonal basis of an approximation of a rational krylov subspace of small dimension from a standard orthogonal krylov subspace basis of larger dimension the approximation error will depend on properties of the matrix a and on the dimension of the original standard krylov subspace we show that our inverse free method for approximating the rational krylov subspace converges geometrically for increasing dimension of the standard krylov subspace to a rational krylov subspace the convergence rate may be used to predict the dimension of the standard krylov subspace necessary to obtain a certain accuracy in the approximation computed examples illustrate the theory developed

restrictions on implicit filtering techniques for orthogonal projection methods abstract we consider the class of the orthogonal projection methods opm to solve iteratively large eigenvalue problems an opm is a method that projects a large eigenvalue problem on a smaller subspace in this subspace an approximation of the eigenvalue spectrum can be computed from a small eigenvalue problem using a direct method examples of opms are the arnoldi and the davidson method we show how an opm can be restarted implicitly and explicitly this restart can be used to remove a specific subset of vectors from the approximation subspace this is called explicit filtering an implicit restart can also be combined with an implicit filtering step i e the application of a polynomial or rational function on the subspace even if inaccurate arithmetic is assumed however the condition for the implicit application of a filter is that the rank of the residual matrix must be small

numerical computation of 2d sommerfeld integrals decomposition of the angular integral abstract spectral domain techniques are frequently used in conjunction with galerkin s method to obtain the current distribution on planar structures when this technique is employed a large percentage of the computation time is spent filling the impedance matrix therefore it is important to develop accurate and efficient numerical techniques for the computation of the impedance elements which can be written as two dimensional 2d sommerfeld integrals once the current distribution has been found then the near zone electric field distribution can be obtained by computing another set of 2d sommerfeld integrals the computational efficiency of the 2d sommerfeld integrals can be improved in two ways the first method which is discussed in this paper involves finding a new way to compute the inner angular integral in the polar representation of these integrals it turns out that the angular integral can be decomposed into a finite number of incomplete lipschitz hankel integrals which in turn can be calculated using series expansions therefore the angular integral can be computed by summing a series instead of applying a standard numerical integration algorithm this new technique is found to be more accurate and efficient when piecewise sinusoidal basis functions are used to analyze a printed strip dipole antenna in a layered medium the incomplete lipschitz hankel integral representation for the angular integral is then used in another paper to develop a novel asymptotic extraction technique for the outer semi infinite integral

halley maps for a trigonometric and rational function among the methods available for the characterization of complicated mathematical and physical phenomena computers with graphics are emerging as an important tool in this article i present computational and graphical results on halley s method for one parameter functions of the form 1 0 and sin 0 in order to gain insight as to where the method can be relied upon and where it behaves strangely the resulting plots reveal a visually striking and intricate class of patterns indicating behavior ranging from stable attractive and repulsive points to chaos iterative approximation methods such a halley s method occur frequently in science and engineering

a complete method for the computations of mathieu characteristic numbers of integer orders a complete method and expressions for computing mathieu characteristic numbers mcns of integer orders are given the paper reviews available algorithms and their shortcomings and then suitable methods are selected on the basis of simplicity and accuracy for small orders n 4 new approximate formulas for mcns are developed these have much larger range than available expressions a new normalization technique for the mcns is proposed with the aid of this normalization a new approximate recurrence relation for mcns is obtained using the same normalization general formulas are obtained for regional limits where the approximate and asymptotic expressions can be used to compute mcns with these expressions the algorithms for computing mcns can handle very high orders up to n 1000 and beyond

uniform approximation to cauchy principal value integrals with logarithmic singularity abstract an approximation of clenshaw curtis type is given for cauchy principal value integrals of logarithmically singular functions i f c 1 1 f x log x c x c d x c 1 1 with a given function f using a polynomial p n of degree n interpolating f at the chebyshev nodes we obtain an approximation i p n c i f c we expand p n in terms of chebyshev polynomials with o n log n computations by using the fast fourier transform our method is efficient for smooth functions f for which p n converges to f fast as n grows and so simple to implement this is achieved by exploiting three term inhomogeneous recurrence relations in three stages to evaluate i p n c for f z analytic on the interval 1 1 in the complex plane z the error of the approximation i p n c is shown to be bounded uniformly using numerical examples we demonstrate the performance of the present method

current trends in asymptotics some problems and some solutions abstract i discuss five topics of current interest in asymptotic analysis the use of probabilistic methods to estimate the growth of combinatorial sequences asymptotic methods in the theory of random walks the asymptotic estimation of solutions of difference equations asymptotic expansions in generalized scales and the computation by asymptotic methods of distributions whose moments are known i give a number of examples and mention some outstanding unsolved problems

numerical analysis of arbitrarily shaped probe excited single arm printed wire antennas a general integral equation technique is described for analysis of an arbitrarily shaped single arm printed wire antenna excited through a vertical probe a unified current integral equation is formulated on the basis of dyadic green s functions and the reciprocity theorem the current distribution is obtained by using a parametric moment method in which parameter segments are adopted for the printed wire instead of the commonly employed wire length segments the radiation field solution involving both the printed antenna and vertical probe is also presented the validity of the formulation is verified by comparing the numerically obtained input impedance and radiation patterns for a linear antenna and a meander antenna with measured data a circular open loop and an archimedian spiral are investigated to illustrate the applicability of the present technique

asymptotic extraction approach for antennas in a multilayered spherical media an efficient algorithm is introduced to enhance the convergence of dyadic green s functions dgf in a layered spherical media where asymptotic expressions have been developed the formulated expressions involve an infinite series of spherical eigenmodes that can be reduced to the simple homogenous media green s function using the addition theorem of spherical hankel functions substantial improvements in the convergence speed have been attained by subtracting the asymptotic series representation from the original dgf the subtracted components are then added to the solution using the homogenous media green s function format

on the hill determinant method the conditions of applicability of the hill determinant method to the potentials v r v1r v2r2 and v x x2 lambda x2 1 gx2 are investigated it is shown that the procedure leads to the actual eigenvalues in the former case provided the ansatz is properly chosen but it completely fails to yield the right answer in the latter one

an implicit filter for rational krylov using core transformations abstract the rational krylov method is a powerful tool for computing a selected subset of eigenvalues in large scale eigenvalue problems in this paper we study a method to implicitly apply a filter in a rational krylov iteration by directly acting on a qr factorized representation of the hessenberg pair from the rational krylov method this filter is used to restart the iteration which is generally required to limit the orthogonalization and storage costs the contribution in this paper is threefold we reformulate existing procedures in terms of operations on core transformations this has the advantage of improved convergence monitoring secondly we demonstrate that the extended qz method is a special case of this more general method finally numerical experiments show the validity and the increased accuracy of the new approach compared with existing methods

analytical technique to evaluate the asymptotic part of the impedance matrix of sommerfeld type integrals the propagation of electromagnetic waves in a grounded dielectric slab has numerous applications in printed antenna technology and in the analysis of microwave and millimeter wave integrated circuits for the accurate analysis of microstrip dipoles and circuits based on the moment of method mom a crucial step is the precise evaluation of the impedance matrix elements which contain the integration of sommerfeld type integrals the integral transform method with the asymptotic extraction technique is formulated for calculating a sommerfeld type integral problem this formulation allows the infinite double integral of the asymptotic part of the impedance matrix to be transformed into a finite one dimensional 1 d integral this finite 1 d integral contains a spherical legendre function and can be easily evaluated numerically after the singular part of the integral is performed analytically it is shown that the proposed method dramatically reduces the computation time and improves the accuracy over the conventional method to evaluate the asymptotic part of impedance matrix

geometric theory of the equivalence of lagrangians for constrained systems the lagrangian description of a system is analysed from a geometric viewpoint in order to find a concept for equivalence of singular lagrangians generalising that of the regular case geometric and gauge equivalence of singular lagrangians are studied and the authors also give some conditions in which second order differential equations exist satisfying the dynamical equation on the final constraint submanifold

network oriented dyadic time domain green s function for a sequentially excited infinite planar array of dipoles in free space during the past few years we have performed an effective wideband analysis for characterizing the electrodynamic behavior of phased array antennas infinite periodic structures frequency selective surfaces and related applications with emphasis on gaining physical insight into the phenomenology of short pulse radiation the present contribution shows the current status of our network oriented dyadic time domain green s function td gf for a planar array of sequentially excited dipoles that constitutes a prototype study of sequentially short pulsed radiation by infinite periodic arrays the dispersive effects of the td floquet waves fw and the consequences of a te and tm field decomposition are discussed in details in this network formulation the equivalent td transmission line tl voltages and currents excited by equivalent td current generators are determined an expression for the tm voltage tl gf is provided for the first time in terms of incomplete lipschitz hankel integrals and bessel functions together with some remarks for its evaluation as shown in previous publications for the scalar potentials real td physical observables are determined by pairing td floquet waves with positive and negative indices it is shown that the vectorial td radiated field is reconstructed at any time with the superposition of a few td fws

a reflection on the implicitly restarted arnoldi method for computing eigenvalues near a vertical line abstract in this article we will study the link between a method for computing eigenvalues closest to the imaginary axis and the implicitly restarted arnoldi method the extension to computing eigenvalues closest to a vertical line is straightforward by incorporating a shift without loss of generality we will restrict ourselves here to computing eigenvalues closest to the imaginary axis in a recent publication meerbergen and spence discussed a new approach for detecting purely imaginary eigenvalues corresponding to hopf bifurcations which is of interest for the stability of dynamical systems the novel method is based on inverse iteration inverse power method applied on a lyapunov like eigenvalue problem to reduce the computational overhead significantly a projection was added this method can also be used for computing eigenvalues of a matrix pencil near a vertical line in the complex plane we will prove in this paper that the combination of inverse iteration with the projection step is equivalent to sorensen s implicitly restarted arnoldi method utilizing well chosen shifts

coulomb and bessel functions of complex arguments and order abstract the coulomb wavefunctions originally constructed for real 0 real and integer 0 are defined for and all complex we examine the complex continuation of a variety of series and continued fraction expansions for the coulomb functions and their logarithmic derivatives and then see how these expansions may be selectively combined to calculate both the regular and irregular functions and their derivatives the resulting algorithm 46 is a complex generalisation of steed s method 6 7 as it appears in the real procedure coulfg 10 complex whittaker confluent hypergeometric and bessel functions can also be calculated

adjoint symmetries for time dependent second order equations the authors extend part of their previous work on autonomous second order systems sarlet et al 1987 to time dependent differential equations the main subject of the paper concerns the notion of adjoint symmetries they are introduced as a particular type of 1 form whose leading coefficients satisfy the adjoint equations of the equations determining symmetry vector fields it is shown that all interesting properties of adjoint symmetries known from the autonomous theory have their counterparts in the present framework of particular interest is a result establishing that lagrangian systems seem to be the only ones for which there is a natural duality between symmetries and adjoint symmetries a number of examples illustrate how the construction of adjoint symmetries of a given system can be explored in a systematic way

the chebyshev fast gauss and nonuniform fast fourier transforms and their application to the evaluation of distributed heat potentials we present a method for the fast and accurate computation of distributed heat potentials in two dimensions the distributed source is assumed to be given in terms of piecewise space time chebyshev polynomials we discretize uniformly in time whereas in space the polynomials are defined on the leaf nodes of a quadtree the quadtree can vary at each time step we combine a product integration rule with fast algorithms fast heat potentials nonuniform fft fast gauss transform to obtain a high order accurate method with optimal complexity if n is the number of time steps m is the maximum number of leaf nodes over all the time steps and the input contains a qth order polynomial representation of f then our method requires o q 3mnlogm work to evaluate the heat potential at arbitrary mn space time target locations the overall convergence rate of the method is of order q we present numerical experiments for q 4 8 and 16 and we verify the theoretical convergence rate of the method when the solution is sufficiently smooth the 16th order variant results in significant computational savings even in the case in which we require only a few digits of accuracy

extended legendre transformation approach to the time dependent hamiltonian formalism the hamiltonian formalism for time dependent systems is developed using vector fields defined in the extended cotangent bundle t q r the extended hamiltonian h is obtained using the extended legendre transformation associated to an extended lagrangian l previously defined in t q r and then the techniques of symplectic mechanics are employed in t q r to study the properties of the extended hamiltonian system finally this approach is related with the multisymplectic formalism and other extended approaches

how to implement the spectral transformation the general linear eigenvalue equations h x m z 0 where h and m are real symmetric matrices with m positive semidefinite must be transformed if the lanczos algorithm is to be used to compute eigenpairs x z when the matrices are large and sparse but not diagonal some factorization must be performed as part of the transformation step if we are interested in only a few eigenvalues x near a specified shift then the spectral transformation of ericsson and ruhe 1 proved itself much superior to traditional methods of reduction the purpose of this note is to show that a small variant of the spectral transformation is preferable in all respects perhaps the lack of symmetry in our formulation deterred previous investigators from choosing it it arises in the use of inverse iteration a second goal is to introduce a systematic modification of the computed ritz vectors which improves the accuracy when m is ill conditioned or singular we confine our attention to the simple lanczos algorithm although the first two sections apply directly to the block algorithms as well 1 overview this contribution is an addendum to the paper by ericsson and ruhe 1 and also 7 the value of the spectral transformation is reiterated in a later section here we outline our implementation of this transformation the equation to be solved for an eigenvalue a and eigenvector z is 1 h xm z 0 h and m are real symmetric n x n matrices and m is positive semidefinite a practical instance of 1 occurs in dynamic analysis of structures where h and m are the stiffness and mass matrices respectively we assume that a linear combination of h and m is positive definite it then follows that all eigenvalues x are real in addition one has a real scalar a distinct from any eigenvalue and we seek a few eigenvalues a close to a together with their eigenvectors z ericsson and ruhe replace 1 by a standard eigenvalue equation 2 c h am ct v y 0 where c is the choleski factor of m m ctc and y cz if m is singular then so is c but fortunately the eigenvector z can be recovered from y via z h am icty of course there is no intention to invert h a m explicitly the received may 14 1984 revised december 20 1985 1980 mathematics subject classification primary 65f15 this research was supported in part by the afosr contract f49620 84 c 0090 the third author was also supported in part by the swedish natural science research council the paper was written while this author was visiting the center for pure and applied mathematics university of california berkeley california 94720 1987 american mathematical society 0025 5718 87 1 00 25 per page

a user friendly method for computing indefinite integrals of oscillatory functions for indefinite integrals q f x 1 x f t e i t d t x 1 1 torii and the first author hasegawa and torii 1987 developed a quadrature method of clenshaw curtis c c type its improvement was made and combined with sidi s m w transformation by sidi and the first author hasegawa and sidi 1996 to compute infinite oscillatory integrals the improved method per se however has not been elucidated in its attractive features which here we reveal with new results and its detailed algorithm a comparison with a method of c c type for definite integrals q f 1 due to dominguez et al 2011 suggests that a smaller number of computations is required in our method this is achieved by exploiting recurrence and normalization relations and their associated linear system we show their convergence and stability properties and give a verified truncation error bound for a result computed from the linear system with finite dimension for f z analytic on and inside an ellipse in the complex plane z the error of the approximation to q f x of the improved method is shown to be bounded uniformly numerical examples illustrate the stability and performance of the method

completely monotonic functions associated with the gamma function and its q analogues several functions involving the gamma function x and the q gamma function q x are proved to be completely monotonic some of these results are used to establish the infinite divisibility of a number of probability distributions defined via their moment generating functions

an asymptotic extraction technique for evaluating sommerfeld type integrals an asymptotic extraction process is developed which leads to an efficient method for the evaluation of the electric field from a hertzian dipole in a layered geometry this technique allows for a rapidly converging expression for the electric field which remains well behaved as the source and observation heights z z coincide a useful application of this method is in the calculation of mutual impedance between printed dipoles

moment method analysis of printed wire spirals using curved piecewise sinusoidal subdomain basis and testing functions archimedian and logarithmic printed wire spirals are analyzed using a moment method incorporating curved piecewise sinusoidal subdomain basis and testing functions results for spiral parameters including input impedance and current distribution are presented and are shown to have good agreement with published results for the same antennas obtained using linear segmentation since the curved basis functions exactly follow the spiral contour significantly fewer curved segments are therefore required for accurate analysis compared with linear segmentation

on the differential geometry of the euler lagrange equations and the inverse problem of lagrangian dynamics the conditions for a system of second order differential equations to be derivable from a lagrangian the conditions of self adjointness in the terminology of santilli 1978 and others are related in the time independent case to the differential geometry of the tangent bundle of configuration space these conditions are simply expressed in terms of the horizontal distribution which is associated with any vector field representing a system of second order differential equations necessary and sufficient conditions for such a vector field to be derivable from a lagrangian may be stated as the existence of a two form with certain properties it is interesting that it is a deduction not an assumption that this two form is closed and thus defines a symplectic structure some other differential geometric properties of euler lagrange second order differential equations are described

linear stability analysis in fluid structure interaction with transpiration part ii numerical analysis and applications this paper constitutes the numerical counterpart of the mathematical framework introduced in part i we address the problem of flutter analysis of a coupled fluid structure system involving an incompressible newtonian fluid and a reduced structure we use the linearization principle approach developed in part i particularly suited for fluid structure problems involving moving boundaries thus the stability analysis is reduced to the computation of the leftmost eigenvalues of a coupled eigenproblem of minimal complexity this eigenproblem involves the linearized incompressible navier stokes equations and those of a reduced linear structure the coupling is realized through specific transpiration interface conditions the eigenproblem is discretized using a finite element approximation and its smallest real part eigenvalues are computed by combining a generalized cayley transform and an implicit restarted arnoldi method finally we report three numerical experiments a structure immersed in a fluid at rest a cantilever pipe conveying a fluid flow and a rectangular bridge deck profile under wind effects the numerical results are compared to former approaches and experimental data the quality of these numerical results is very satisfactory and promising

recurrence relations for the connection coefficients of orthogonal polynomials of a discrete variable on the lattice x s q 2 abstract we give explicitly recurrence relations satisfied by the connection coefficients between two families of the classical orthogonal polynomials of a discrete variable on a non uniform lattice x s q2s i e the q analogues of charlier meixner krawtchouk and hahn polynomials in terms of the coefficients and of the pearson equation satisfied by the weight function and the coefficients of the three term recurrence relation and of two structure relations obeyed by these polynomials

a high order solver for the heat equation in 1d domains with moving boundaries we describe a fast high order accurate method for the solution of the heat equation in domains with moving dirichlet or neumann boundaries and distributed forces we assume that the motion of the boundary is prescribed our method extends the work of greengard and strain comm pure appl math xliii 1990 pp 949 963 our scheme is based on a time space chebyshev pseudo spectral collocation discretization which is combined with a recursive product quadrature rule to accurately and efficiently approximate convolutions with green s function for the heat equation we present numerical results that exhibit up to eighth order convergence rates assuming n time steps and m spatial discretization points the evaluation of the solution of the heat equation at the same number of points in space time requires mathcal o n m log m work thus our scheme can be characterized as fast that is it is work optimal up to a logarithmic factor

extended tangent bundle formalism for time dependent lagrangian systems the lagrangian formalism for time dependent systems is developed using vector fields defined in the extended tangent bundle t q r the definition of the time independent extended lagrangian function l associated to a time dependent lagrangian l is given and then the techniques of symplectic mechanics are used to prove some properties of the equations determining the euler lagrange vector field the relations between the time independent and the time dependent helmholtz conditions are analyzed from a geometric perspective and finally the noether theorem for the extended lagrangian l is considered

on the computation of fourier transforms of singular functions abstract in this paper we consider a method for the computation of finite fourier transforms of functions having endpoint singularities of algebraic or logarithmic type this method may be considered as a modification of clenshaw curtis quadrature the nonoscillatory and nonsingular part of the integrand is replaced by a truncated chebyshev series approximation and the integral is then evaluated using recurrence relations the numerical stability of these recurrence relations is investigated

generalized canonical transformations for time dependent systems we introduce the concept of generalized canonical transformations as symplectomorphisms of the extended phase space we prove that any such transformation factorizes in a standard canonical transformation times another one that changes only the time variable the theory of generating functions as well as that of hamilton jacobi is developed some further applications are developed

reduced cancellation in the evaluation of entire functions and applications to the error function a general concept for the reduction of cancellation problems in the evaluation of taylor sections of certain entire functions is proposed the resulting method is applied to and tested in the case of the complementary error function

difference equations in statistical mechanics i cluster statistics models a review and some new results are presented for several cluster statistics models solutions of which can be reduced to difference equations mathematical techniques suitable for solving these equations are surveyed

geometry of multisymplectic hamiltonian first order field theories in the jet bundle description of field theories multisymplectic models in particular there are several choices for the multimomentum bundle where the covariant hamiltonian formalism takes place as a consequence several proposals for this formalism can be stated and on each one of them the differentiable structures needed for setting the formalism are obtained in different ways in this work we make an accurate study of some of these hamiltonian formalisms showing their equivalence in particular the geometrical structures canonical or not needed for the hamiltonian formalism are introduced and compared and the derivation of hamiltonian field equations from the corresponding variational principle is shown in detail furthermore the hamiltonian formalism of systems described by lagrangians is performed both for the hyper regular and almost regular cases finally the role of connections in the construction of hamiltonian field theories is clarified

polynomial optimization and a jacobi davidson type method for commuting matrices in this paper we introduce a new jacobi davidson type eigenvalue solver for a set of commuting matrices called jdcomm used for the global optimization of so called minkowski norm dominated polynomials in several variables the stetter moller matrix method yields such a set of real non symmetric commuting matrices since it reformulates the optimization problem as an eigenvalue problem a drawback of this approach is that the matrix most relevant for computing the global optimum of the polynomial under investigation is usually large and only moderately sparse however the other matrices are generally much sparser and have the same eigenvectors because of the commutativity this fact is used to design the jdcomm method for this problem the most relevant matrix is used only in the outer loop and the sparser matrices are exploited in the solution of the correction equation in the inner loop to greatly improve the efficiency of the method some numerical examples demonstrate that the method proposed in this paper is more efficient than approaches that work on the main matrix standard jacobi davidson and implicitly restarted arnoldi as well as conventional solvers for computing the global optimum i e sostools gloptipoly and phcpack

an analytic method for convergence acceleration of certain hypergeometric series a method is presented for convergence acceleration of the generalized hypergeometric series 3 f 2 with the argument 1 using analytic properties of their terms iterated transformation of the series is performed analytically which results in obtaining new fast converging expansions for some special functions and mathematical constants

numerical computation of the incomplete lipschitz hankel integral je o a z abstract two factorial neumann series expansions are derived for the incomplete lipschitz hankel integral je 0 a z these expansions are used together with the neumann series expansion given by agrest in an algorithm which efficiently computes je 0 a z to a user defined number of significant digits other expansions for je 0 a z which are found in the literature are also discussed but these expansions are found to offer no significant computational advantages when compared with the expansions used in the algorithm

the associated askey wilson polynomials we derive some contiguous relations for very well poised 8 7 series and use them to construct two linearly independent solutions of the three term recurrence relation of the associated askey wilson polynomials we then use these solutions to find explicit representations of two families of associated askey wilson polynomials we identify the corresponding continued fractions as quotients of two very well poised 807 series and find the weight functions

the constraint algorithm for time dependent lagrangians the aim of this paper is to develop a constraint algorithm for time dependent lagrangian systems which permit us to solve the motion equations this algorithm extends the gotay and nester algorithm for autonomous lagrangians which is in fact a particular case to do this the almost stable tangent geometry of the evolution space and the notion of cosymplectic structure are used

resurrecting the asymptotics of linear recurrences abstract once on the forefront of mathematical research in america the asymptotics of the solutions of linear recurrence equations is now almost forgotten especially by the people who need it most namely combinatorists and computer scientists here we present this theory in a concise form and give a number of examples that should enable the practicing combinatorist and computer scientist to include this important technique in her or his asymptotics tool kit

a new algorithm for computing orthogonal polynomials the aim of this paper is to present an algorithm for computing orthogonal polynomials the fast fourier transform completes the connection given by draux between the three term recurrence relationship and the euclidian algorithm applications to hankel systems and numerical examples illustrate our purpose

poisson maps and canonoid transformations for time dependent hamiltonian systems after a new presentation of the geometric theory of time dependent systems in the hamiltonian formulation using poisson structures a characterization of canonoid transformations with respect to a dynamical vector field is given the associated constants of motion and the generating functions of canonoid transformations are also studied the theory is illustrated with several examples

solution of the eigenvalue problems resulting from global non parallel flow stability analysis abstract the eigensolution method more flexible and efficient than previously used for global non parallel flow stability analysis is presented the differential eigenvalue problem resulting from linear stability equations of non parallel flow is discretized with the penalty fem the large dimensional algebraic eigenvalue problem is solved with the use of subspace iteration method separation of the eigenvalues interesting for the flow stability analysis is performed with the inverse cayley transformation it is shown that certain barriers resulting from very large dimensions of the eigenvalue problem and limiting the non parallel flow stability method can be overcome with this approach to demonstrate the algorithm the stability of the flow around circular cylinder is analyzed

extended hamiltonian systems in multisymplectic field theories we consider hamiltonian systems in first order multisymplectic field theories we review the properties of hamiltonian systems in the so called restricted multimomentum bundle including the variational principle which leads to the hamiltonian field equations in an analogous way to how these systems are defined in the so called extended symplectic formulation of nonautonomous mechanics we introduce hamiltonian systems in the extended multimomentum bundle the geometric properties of these systems are studied the hamiltonian equations are analyzed using integrable multivector fields the corresponding variational principle is also stated and the relation between the extended and the restricted hamiltonian systems is established all these properties are also adapted to certain kinds of submanifolds of the multimomentum bundles in order to cover the case of almost regular field theories

a generalized computational approach to stability of static equilibria of nonlinearly elastic rods in the presence of constraints abstract we present a generalized approach to stability of static equilibria of nonlinearly elastic rods subjected to general loading boundary conditions and constraints of both point wise and integral type based upon the linearized dynamics stability criterion discretization of the governing equations leads to a non standard singular generalized eigenvalue problem a new efficient sparse matrix friendly algorithm is presented to determine its few left most eigenvalues which in turn yield stability instability information for conservative problems the eigenvalue problem arising from the linearized dynamics stability criterion is also shown to be equivalent to that arising in the determination of constrained local minima of the potential energy we illustrate the method with several examples a novel variational formulation for extensible and unshearable rods is also proposed within the context of one of the example problems

exact closed form representations for the time domain surface impedances of a homogeneous lossy half space in this paper we derive exact closed form representations for the transverse electric and transverse magnetic time domain surface impedances associated with a plane wave that is obliquely incident on a homogeneous lossy half space the time domain surface impedances are expressed in terms of modified bessel functions and incomplete lipschitz hankel integrals the limitations on the validity of a previously published approximate expression are also discussed

numerical computation of 2d sommerfeld integrals a novel asymptotic extraction technique abstract the accurate and efficient computation of the elements in the impedance matrix is a crucial step in the application of galerkin s method to the analysis of planar structures as was demonstrated in a previous paper it is possible to decompose the angular integral in the polar representation for the 2d sommerfeld integrals in terms of incomplete lipschitz hankel integrals ilhis when piecewise sinusoidal basis functions are employed since bessel series expansions can be used to compute these ilhis a numerical integration of the inner angular integral is not required this technique provides an efficient method for the computation of the inner angular integral however the outer semi infinite integral still converges very slowly when a real axis integration is applied therefore it is very difficult to compute the impedance elements accurately and efficiently in this paper it is shown that this problem can be overcome by using the ilhi representation for the angular integral to develop a novel asymptotic extraction technique for the outer semi infinite integral the usefulness of this asymptotic extraction technique is demonstrated by applying it to the analysis of a printed strip dipole antenna in a layered medium

an asymptotic closed form microstrip surface green s function for the efficient moment method analysis of mutual coupling in microstrip antennas a relatively simple closed form asymptotic representation for the single layer microstrip dyadic surface green s function is developed the large parameter in this asymptotic development is proportional to the lateral separation between the source and field points along the air dielectric interface this asymptotic solution remains surprisingly accurate even for very small a few tenths of a free space wavelength lateral separation of the source and field points thus using the present asymptotic approximation of the green s function can lead to a very efficient moment method mm solution for the currents on an array of microstrip antenna patches and feed lines numerical results based on the efficient mm analysis using the present closed form asymptotic approximation to the microstrip surface green s function are given for the mutual coupling between a pair of printed dipoles on a single layer grounded dielectric slab the accuracy of the latter calculation is confirmed by comparison with numerical results based on a mm analysis which employs an exact integral representation for the microstrip green s function

uniform approximation to finite hilbert transform of oscillatory functions and its algorithm abstract for the finite hilbert transform of oscillatory functions q f c 1 1 f x e i x x c d t with a smooth function f and real 0 for c 1 1 in the sense of cauchy principal value or for c 1 of hadamard finite part we present an approximation method of clenshaw curtis type and its algorithm interpolating f by a polynomial p n of degree n and expanding in terms of the chebyshev polynomials with o n log n operations by the fft we obtain an approximation q p n c q f c we write q p n c as a sum of the sine and cosine integrals and an oscillatory integral of a polynomial of degree n 1 we efficiently evaluate the oscillatory integral with a combination of authors previous method and keller s method for f z analytic on the interval 1 1 in the complex plane z the error of q p n c is bounded uniformly with respect to c and numerical examples illustrate the performance of our method

solutions of linear recurrence equations solutions are given to general homogeneous and non homogeneous recurrence equations defined on the set of integers solutions to homogeneous recurrence equations are given as i an infinite series of terms involving the partial ordinary bell polynomial ii an infinite series of terms involving the complete ordinary bell polynomial iii a weighted finite sum of terms involving powers solutions to non homogeneous recurrence equations are given as i a finite series of terms involving the complete ordinary bell polynomial ii a weighted infinite sum of terms involving powers computational issues of these solutions are also discussed

using implicitly filtered rks for generalised eigenvalue problems the rational krylov sequence rks method can be seen as a generalisation of arnoldi s method it projects a matrix pencil onto a smaller subspace this projection results in a small upper hessenberg pencil as for the arnoldi method rks can be restarted implicitly using the qr decomposition of a hessenberg matrix this restart comes with a projection of the subspace using a rational function in this paper it is shown how the restart can be worked out in practice in a second part it is shown when the filtering of the subspace basis can fail and how this failure can be handled by deflating a converged eigenvector from the subspace using a schur decomposition

a generalized asymptotic extraction solution for antennas in multilayered spherical media an efficient model is developed to accelerate the convergence of the dyadic green s function s dgf infinite summation when the source and observation points are placed in different layers of a dielectric sphere thereby expediting computational analysis the proposed procedure is based on asymptotic extraction principles in which the quasi static images are extracted from the spectral domain dgf the effectiveness of the approach is demonstrated in a method of moment model where a microstrip antenna as well as a conformal dipole array have been studied

symmetry theory and lagrangian inverse problem for time dependent second order differential equations a set xgamma of vector fields in the evolution space e playing the role of newtonian vector fields with respect to a second order equation field gamma is introduced and endowed with a cinfinity e module structure a dual set m gamma is used for giving an answer to the lagrangian inverse problem the symmetry theory is also developed in this framework and in particular the characterisation of symmetries of gamma in terms of the transformation properties of the lagrangian l is also given

tangent bundle geometry lagrangian dynamics various aspects of the differential geometry of the tangent bundle of a differentiable manifold are examined and the results applied to time independent lagrangian dynamics it is shown that a certain type 1 1 tensor field which is part of the intrinsic geometry of a tangent bundle being a tensorial equivalent of the projection map of tangent vectors plays a role in lagrangian theory scarcely less important than that of the canonical one form on a cotangent bundle in hamiltonian theory recent results in lagrangian theory are interpreted from this new viewpoint

implicitly restarted arnoldi with purification for the shift invert transformation the need to determine a few eigenvalues of a large sparse generalised eigenvalue problem az bx with positive semidefinite b arises in many physical situations for example in a stability analysis of the discretised navier stokes equation a common technique is to apply arnoldi s method to the shift invert transformation but this can suffer from numerical instabilities as is illustrated by a numerical example in this paper a new method that avoids instabilities is presented which is based on applying the implicitly restarted arnoldi method with the b semi inner product and a purification step the paper contains a rounding error analysis and ends with brief comments on some extensions

exploiting structure in large scale electrical circuit and power system problems abstract the rapid increase in complexity of systems such as electrical circuits and power systems calls for the development of efficient numerical methods in many cases direct application of standardized methods for numerical problems is computationally not feasible or inefficient however the performance of such methods can be improved considerably by taking into account the structure of the underlying problem in this paper we describe when and how this mathematical and or physical structure can be exploited to arrive at efficient algorithms that also suffer less from other numerical issues such as round off errors eigenvalue and stability problems are considered in particular but applications to other problems are shown as well

stripline fed arbitrarily shaped printed aperture antennas a full wave analysis method is presented for modeling the radiation properties of a stripline fed planar printed aperture antenna element in this formulation both the finite length of stripline and the finite aperture may be of any arbitrary shape since their equivalent electric and magnetic currents are modeled with triangular patch basis functions galerkin s method is applied to numerically solve the coupled mixed potential integral equation mpie exact spatial domain green s functions are used to account for all radiation surface wave and mutual coupling effects interactions between the stripline feed and the radiating aperture are rigorously included numerical analysis is presented for the following nonrectangular shapes an exponentially tapered slot an annular slot an annular slot with opposing stubs and a monofilar archimedean spiral slot element results are shown for input return loss radiation patterns and axial ratio for the circularly polarized elements

application of the fast fourier transform to the computation of the sommerfeld integral for a vertical electric dipole above a half space the fast fourier transform fft is used in conjunction with analytical techniques to obtain an approximate expression for the sommerfeld integral associated with a vertical electric dipole above a half space the resulting expression which is an analytic function of the source and observation locations can be used in lieu of a numerical integration of the sommerfeld integral it can be used in either the near zone or the far zone and it can even be applied for observation angles close to grazing the accuracy of the expansion directly depends on the number of terms used in the fft results obtained using the expansion are compared with those obtained both from a direct numerical integration of the sommerfeld integral and the reflection coefficient method

a new representation for the green s function of multilayer media based on plane wave expansion in this paper a new representation for the space domain green s function of general multilayer media is presented this approach is based on an efficient plane wave expansion of the source incident field using the transmission line model for the multilayer medium the effect of the layered medium on the incident plane waves is determined by a transmission coefficient and the amplitude and phase of each plane wave at the field point are obtained the total field is evaluated by summing the resultant plane waves at the field point the main advantage of the proposed method is that the spatial domain green s function of multilayer media can be obtained easily as a summation of simple exponential functions without the need for sommerfeld integration or complex image approximation the plane wave approximation is independent of the parameters of the medium and is valid over a wide frequency range furthermore the exponential form of the plane wave solution makes it possible to compute the method of moments matrix elements analytically for most important types of basis functions in multilayer problems

an algorithm for nondominant solutions of linear second order inhomogeneous difference equations an algorithm is given for computing a weighted sum of a nondominant solution of a linear second order inhomogeneous difference equation to a prescribed accuracy by estimating the truncation error the present method is an extension of both the stable numerical method due to olver and sookne and a summation technique due to deuflhard for computing minimal solutions of a homogeneous difference equation the method is illustrated by numerical examples

skinner rusk unified formalism for optimal control systems and applications a geometric approach to time dependent optimal control problems is proposed this formulation is based on the skinner and rusk formalism for lagrangian and hamiltonian systems the corresponding unified formalism developed for optimal control systems allows us to formulate geometrically the necessary conditions given by a weak form of pontryagin s maximum principle provided that the differentiability with respect to controls is assumed and the space of controls is open furthermore our method is also valid for implicit optimal control systems and in particular for the so called descriptor systems optimal control problems including both differential and algebraic equations

jet fields connections and second order differential equations jet fields are defined on fibred manifolds and represented as type 1 1 tensor fields corresponding to cartan ehresmann connections when defined on the first jet bundle they may be used to generalise the second order differential equation fields used in theoretical mechanics and to extend several existing results to field theories