calculation of cfie impedance matrix elements with rwg and n spl times rwg functions the method of moments mom solution of combined field integral equation cfie for electromagnetic scattering problems requires calculation of singular double surface integrals when galerkin s method with triangular vector basis functions rao wilton glisson functions and the cfie are applied to solve electromagnetic scattering by a dielectric object both rwg and n spl times rwg functions n is normal unit vector should be considered as testing functions robust and accurate methods based on the singularity extraction technique are presented to evaluate the impedance matrix elements of the cfie with these basis and test functions in computing the impedance matrix elements including the gradient of the green s function we can avoid the logarithmic singularity on the outer testing integral by modifying the integrand in the developed method all singularities are extracted and calculated in closed form and numerical integration is applied only for regular functions in addition we present compact iterative formulas for computing the extracted terms in closed form by these formulas we can extract any number of terms from the singular kernels of cfie formulations with rwg and n spl times rwg functions

fast and accurate evaluation of double surface integrals in time domain integral equation formulations a novel integration approach is developed for efficiently evaluating double surface integrals in time domain integral equation formulations the approach mainly consists of an angular integration scheme for inner 2 d integrals and a modified outer integration scheme the approach proposed can handle the singularity and near hypersingularity of the 4 d integrals and is sufficiently general for arbitrary types of basis functions over planar domains as several numerical results will demonstrate the proposed approach has high accuracy and rapid convergence rate for a variety of integral kernels

on the convergence of local expansions of layer potentials in a recently developed quadrature method quadrature by expansion or qbx it was demonstrated that weakly singular or singular layer potentials can be evaluated rapidly and accurately on surface by making use of local expansions about carefully chosen off surface points in this paper we derive estimates for the rate of convergence of these local expansions providing the analytic foundation for the qbx method the estimates may also be of mathematical interest particularly for microlocal or asymptotic analysis in potential theory

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

a mesh adapted closed form regular kernel for 3d singular integral equations the green s functions employed in the method of moments mom diverge when observation and source points coincide this is at the origin of the difficulties in computing the mom matrix entries and in handling the near field interactions in fast fourier transform fft based fast methods and other sampling based methods in this paper we show that this singularity can be avoided and a modified regular green s function can be used instead this latter is obtained from the spectral representation of the usual green s function via windowing of its spectrum the width of the spectral window depends on the size of the mesh employed for discretizing the problem so that the proposed regular green s function is a mesh adapted regular kernel we address a general 3d problem we relate the mom reaction integrals to the 2d fourier spectrum of the green s function that allows to discuss the necessary spectral bandwidth for the windowed green s function we employ a tapered window and present a closed form expression for the spatial green s function numerical results are presented for 3d antenna and scattering problems discretized with rao wilton glisson rwg functions and for uniform and nonuniform meshing they show that the proposed method yields accurate solutions also for the antenna input impedance the meaning of the regularized green s function is also discussed and put in perspective

principal angles between subspaces in an a based scalar product algorithms and perturbation estimates computation of principal angles between subspaces is important in many applications e g in statistics and information retrieval in statistics the angles are closely related to measures of dependency and covariance of random variables when applied to column spaces of matrices the principal angles describe canonical correlations of a matrix pair we highlight that all popular software codes for canonical correlations compute only cosine of principal angles thus making impossible because of round off errors finding small angles accurately we review a combination of sine and cosine based algorithms that provide accurate results for all angles we generalize the method to the computation of principal angles in an a based scalar product for a symmetric and positive definite matrix a we provide a comprehensive overview of interesting properties of principal angles we prove basic perturbation theorems for absolute errors for sine and cosine of principal angles with improved constants numerical examples and a detailed description of our code are given

computing thecs decomposition of a partitioned orthonormal matrix this paper describes an algorithm for simultaneously diagonalizing by orthogonal transformations the blocks of a partitioned matrix having orthonormal columns

tangent bundle maps on the grassmann manifold application to empirical arithmetic averaging the present paper elaborates on tangent bundle maps on the grassmann manifold with application to subspace arithmetic averaging in particular the present contribution elaborates on the work about retraction lifting maps devised for the stiefel manifold in the recently published paper t kaneko s fiori and t tanaka empirical arithmetic averaging over the compact stiefel manifold ieee trans signal process vol 61 no 4 pp 883 894 february 2013 and discusses the extension of such maps to the grassmann manifold tangent bundle maps are devised on the basis of the thin qr matrix decomposition the polar matrix decomposition and the exponential map also tangent bundle pseudo maps based on the matrix cayley transform are devised theoretical and numerical comparisons about the devised tangent bundle maps are performed in order to get an insight into their relative merits and demerits with special emphasis to their computational burden the averaging algorithm based on the thin qr decomposition maps stands out as it exhibits the best trade off between numerical precision and computational burden such algorithm is further compared with two grassmann averaging algorithms drawn from the scientific literature on an handwritten digits recognition data set the thin qr tangent bundle maps based algorithm exhibits again numerical features that make it preferable over such algorithms

extrapolation techniques for ill conditioned linear systems in this paper the regularized solutions of an ill conditioned system of linear equations are computed for several values of the regularization parameter lambda then these solutions are extrapolated at lambda 0 by various vector rational extrapolations techniques built for that purpose these techniques are justified by an analysis of the regularized solutions based on the singular value decomposition and the generalized singular value decomposition numerical results illustrate the effectiveness of the procedures

a numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions we present a numerical algorithm for the construction of efficient high order quadratures in two and higher dimensions quadrature rules constructed via this algorithm possess positive weights and interior nodes resembling the gaussian quadratures in one dimension in addition rules can be generated with varying degrees of symmetry adaptable to individual domains we illustrate the performance of our method with numerical examples and report quadrature rules for polynomials on triangles squares and cubes up to degree 50 these formulae are near optimal in the number of nodes used and many of them appear to be new

analytical best approximate hermitian and generalized skew hamiltonian solution of matrix equation axah cych f abstract in this paper we consider the matrix equation a x a h c y c h f where a c and f are given matrices with appropriate sizes and x y is an unknown hermitian and generalized skew hamiltonian matrix pair based on matrix differential calculus and projection theorem in inner product spaces we exploit the best approximate solution x y in the set s to a given matrix pair x y where s signifies the least squares hermitian and generalized skew hamiltonian solution set of the matrix equation a x a h c y c h f the analytical expression of the best approximate solution is presented by applying the canonical correlation decomposition and the generalized singular value decomposition finally a numerical algorithm and an illustrated example are given

high order divergence conforming constrained bases for triangular cells this paper presents an algebraic technique for generating arbitrary order divergence conforming bases for curvilinear triangular cells the bases are constructed by enforcing appropriate constraints on a linear combination of general functions and then extracting the desired bases using singular value decompositions koornwinder dubiner polynomials are chosen as the general function set basic constraints are presented to obtain divergence conforming bases and additional constraints are presented to further enforce the bases to be nedelec results from a variety of problems are given to show that the bases exhibit high order convergence and also produce systems that are relatively well conditioned compared to other basis sets

a new matrix decomposition for signal processing we extend the generalized singular value decomposition to a new decomposition that can be updated at a low cost in addition we show how a forgetting factor can be incorporated in our decomposition

numerical evaluation via singularity cancellation schemes of near singular integrals involving the gradient of helmholtz type potentials in this paper we present a purely numerical procedure to evaluate strongly near singular integrals involving the gradient of helmholtz type potentials for observation points at finite arbitrarily small distances from the source domain in the proposed approach the source domain is subdivided into a disc plus truncated subtriangles and proper variable transformations are applied in each integration domain to exactly cancel the kernel singularity a novel feature of the proposed angular transform is that required discrete values of the inverse transform which is transcendental are determined via a root finding procedure the same idea can also be applied to other transforms that arise in singularity cancellation methods the resulting integral may then evaluated via a low order gauss legendre quadrature scheme

high degree efficient symmetrical gaussian quadrature rules for the triangle gaussian quadrature is required for the computation of matrices based on the isoparametric formulztion of the finite element method a brief review of existing quadrature rules for the triangle is given and the method for the determination of high degree efficient symmetrical rules for the triangle is discussed new quadrature rules of degree 12 20 are presented and a short fortran program is included

quadrature over a pyramid or cube of integrands with a singularity at a vertex a simple transformation is introduced which facilitates the evaluation of integrals over certain square based pyramids or cubes in cases where the integrand has a singularity at a vertex

individual based predator prey model for biological coevolution fluctuations stability and community structure we study an individual based predator prey model of biological coevolution using linear stability analysis and large scale kinetic monte carlo simulations the model exhibits approximate 1 f noise in diversity and population size fluctuations and it generates a sequence of quasisteady communities in the form of simple food webs these communities are quite resilient toward the loss of one or a few species which is reflected in different power law exponents for the durations of communities and the lifetimes of species the exponent for the former is near ensuremath 1 while the latter is close to ensuremath 2 statistical characteristics of the evolving communities including degree predator and prey distributions and proportions of basal intermediate and top species compare reasonably with data for real food webs

on a variational formulation of the qsvd and the rsvd abstract recently m t chu r f funderlic and g h golub siam j matrix anal appl 18 1997 1082 1092 presented a variational formulation for the quotient singular value decomposition qsvd of two matrices a r n m c r p m which is a generalization to two matrices of the ordinary singular value decomposition svd and characterizes the role of two orthogonal matrices in qsvd in this paper we give an alternative derivation of this variational formulation and extend it to establish an analogous variational formulation for the restricted singular value decomposition rsvd of a matrix triplet a r n m b r n l c r p m which provides a new understanding of the orthogonal matrices appearing in this decomposition

singularity subtraction technique for high order polynomial vector basis functions on planar triangles in this paper a singularity subtraction technique is developed for computing the impedance matrix elements of various electromagnetic surface integral equation formulations with the galerkin method and high order basis functions analytical closed form formulas for computing surface integrals with r r sup n n spl ges 3 singularities times polynomial nodal shape functions of arbitrary order on a planar triangle are presented

a tangent algorithm for computing the generalized singular value decomposition we present two new algorithms for floating point computation of the generalized singular values of a real pair a b of full column rank matrices and for floating point solution of the generalized eigenvalue problem hx lambda mx with symmetric positive definite matrices h and m the pair a b is replaced with an equivalent pair a b and the generalized singular values are computed as the singular values of the explicitly computed matrix f a b 1 the singular values of f are computed using the jacobi method the relative accuracy of the computed singular value approximations does not depend on column scalings of a and b that is the accuracy is nearly the same for all pairs ad 1 bd 2 with d 1 d 2 arbitrary diagonal nonsingular matrices similarly the pencil h lambda m is replaced with an equivalent pencil h lambda m and the eigenvalues of h lambda m are computed as the squares of the singular values of g l h l m 1 where l h l m are the cholesky factors of h m respectively and the matrix g is explicitly computed as the solution of a linear system of equations for the computed approximation lambda delta lambda of any exact eigenvalue lambda the relative error delta lambda lambda is of order p n mbox boldmath varepsilon max min delta in cal d kappa 2 delta h delta min delta in cal d kappa 2 delta m delta where p n is a modestly growing polynomial of the dimension of the problem mbox boldmath varepsilon is the round off unit of floating point arithmetic cal d denotes the set of diagonal nonsingular matrices and kappa 2 cdot is the spectral condition number furthermore floating point computation corresponds to an exact computation with h delta h m delta m where for all i j delta h ij sqrt h ii h jj and delta m ij sqrt m ii m jj are of order of mbox boldmath varepsilon times a modest function of n

a high order locally corrected nystrom scheme for charge current integral equations a high order nystrom scheme is developed for electromagnetic em scattering problems of smooth objects the implementation is based on different well known charge current formulations this study aims to develop a novel quadrature scheme to compute different singular integrals that appear on integral equation methods in ems the proposed quadrature scheme is used on a locally corrected nystrom method and applied to charge current integral equations to obtain a fast accurate and a low frequency stable method to solve em scattering problems the proposed method allows obtaining accuracy in the range 4 10 digits in the near electric and magnetic scattered field for certain simple geometries and a wide range of frequencies we also do a preliminary research on nonsmooth geometries obtaining a certain degree of accuracy depending on the particular charge current integral equation used

asymmetric dynamics and critical behavior in the bak sneppen model abstract we investigate using mean field theory and simulation the effect of asymmetry on the critical behavior and probability density of bak sneppen models two kinds of anisotropy are investigated i different numbers of sites to the left and right of the central minimum site are updated and ii sites to the left and right of the central site are renewed in different ways of particular interest is the crossover from symmetric to asymmetric scaling for weakly asymmetric dynamics and the collapse of data with different numbers of updated sites but the same degree of asymmetry all non symmetric rules studied fall independent of the degree of asymmetry in the same universality class conversely symmetric variants reproduce the exponents of the original model our results confirm the existence of two symmetry based universality classes for extremal dynamics

numerical evaluation of singular and near singular potential integrals a simple and efficient numerical procedure using a singularity cancellation scheme is presented for evaluating singular and near singular potential integrals with 1 r singularities the procedure not only has several advantages over singularity subtraction methods but also improves on some aspects of other singularity cancellation methods such as polar and duffy transformations a theoretical analysis is presented for triangles quadrilaterals tetrahedrons bricks and prisms and numerical results are presented for triangles and prisms

computing the cs and the generalized singular value decompositions if the columns of a matrix are orthonormal and it is partitioned into a 2 by 1 block matrix then the singular value decompositions of the blocks are related this is the essence of the cs decomposition the computation of these related svd s requires some care stewart has given an algorithm that uses the linpack svd algorithm together with a jacobitype clean up operation on a cross product matrix our technique is equally stable and fast but avoids the cross product matrix the simplicity of our technique makes it more amenable to parallel computation on systolic type computer architectures these developments are of interest because a good way to compute the generalized singular value decomposition of a matrix pair a b is to compute the cs decomposition of a certain orthogonal column matrix related toa andb

on singular probability densities generated by extremal dynamics abstract extremal dynamics is the mechanism that drives the bak sneppen model into a self organized critical state marked by a singular stationary probability density p x with the aim of understanding this phenomenon we study the bs model and several variants via mean field theory and simulation in all cases we find that p x is singular at one or more points as a consequence of extremal dynamics furthermore we show that the extremal barrier x i always belongs to the prohibited interval in which p x 0 our simulations indicate that the bak sneppen universality class is robust with regard to changes in the updating rule we find the same value for the exponent for all variants mean field theory which furnishes an exact description for the model on a complete graph reproduces the character of the probability distribution found in simulations for the modified processes mean field theory takes the form of a functional equation for p x

on the thresholds probability densities and critical exponents of bak sneppen like models abstract we report a simple method to accurately determine the threshold and the exponent of the bak sneppen bs model and also investigate the bs universality class for the random neighbor version of the bs model we find the threshold x 0 33332 3 in agreement with the exact result x 1 3 given by mean field theory for the one dimensional original model we find x 0 6672 2 in good agreement with the results reported in the literature for the anisotropic bs model we obtain x 0 7240 1 we study the finite size effect x l x l l observed in a system with l sites and find 1 00 1 for the random neighbor version 1 40 1 for the original model and 1 58 1 for the anisotropic case finally we discuss the effect of defining the extremal site as the one which minimizes a general function f x instead of simply f x x as in the original updating rule we emphasize that models with extremal dynamics have singular stationary probability distributions p x our simulations indicate the existence of two symmetry based universality classes

on the near interaction elements in integral equation solvers for electromagnetic scattering by three dimensional thin objects the accuracy of near interaction ni evaluation is investigated for hypersingular and strongly singular kernels in solving electromagnetic em integral equations we first define the ni between a field point and a source triangular patch using a nearness factor nf and show the accuracy of numerical integrations for the ni integrals with hypersingular or strongly singular kernels it is found that there exists a transition area or turning point before and after which the convergence behaviors of those integrations are greatly different we define the nf value at the 1 relative error as the turning point nf 0 and define nf les nf 0 as the ni range we find that nf 0 1 4 could be a bound of ni range for the triangles with a quality factor q 0 35 and analytical formulas should be used in evaluating ni elements within the bound numerical examples for em scattering by three dimensional 3d thin objects are used to demonstrate the significance of ni treatments for the accurate solutions

branching process in a stochastic extremal model we considered a stochastic version of the bak sneppen model sbsm of ecological evolution where the number m of sites mutated in a mutation event is restricted to only two here the mutation zone consists of only one site and this site is randomly selected from the neighboring sites at every mutation event in an annealed fashion the critical behavior of the sbsm is found to be the same as the bs model in dimensions d 1 and 2 however on the scale free graphs the critical fitness value is nonzero even in the thermodynamic limit but the critical behavior is mean field like finally m has been made even smaller than two by probabilistically updating the mutation zone which also shows the original bs model behavior we conjecture that a sbsm on any arbitrary graph with any small branching factor greater than unity will lead to a self organized critical state

on the numerical evaluation of the singular integrals of scattering theory in a previous work the authors introduced a scheme for the numerical evaluation of the singular integrals which arise in the discretization of certain weakly singular integral operators of acoustic and electromagnetic scattering that scheme is designed to achieve high order algebraic convergence and high accuracy when applied to operators given on smoothly parameterized surfaces this paper generalizes the approach to a wider class of integral operators including many defined via the cauchy principal value operators of this type frequently occur in the course of solving scattering problems involving boundary conditions on tangential derivatives the resulting scheme achieves high order algebraic convergence and approximately 12 digits of accuracy

a nonlinear optimization procedure for generalized gaussian quadratures we present a new nonlinear optimization procedure for the computation of generalized gaussian quadratures for a broad class of square integrable functions on intervals while some of the components of this algorithm have been previously published we present a simple and robust scheme for the determination of a sparse solution to an underdetermined nonlinear optimization problem which replaces the continuation scheme of the previously published works the new algorithm successfully computes generalized gaussian quadratures in a number of instances in which the previous algorithms fail four applications of our scheme to computational physics are presented the construction of discrete plane wave expansions for the helmholtz green s function the design of linear array antennae the computation of a quadrature for the discretization of laplace boundary integral equations on certain domains with corners and the construction of quadratures for the discretization of laplace and helmholtz boundary integral equations on smooth surfaces

a numerical method for computing the hamiltonian schur form we derive a new numerical method for computing the hamiltonian schur form of a hamiltonian matrix mathcal m in mathbb r 2n times 2n that has no purely imaginary eigenvalues we demonstrate the properties of the new method by showing its performance for the benchmark collection of continuous time algebraic riccati equations despite the fact that no complete error analysis for the method is yet available the numerical results indicate that if no eigenvalues of mathcal m are close to the imaginary axis then the method computes the exact hamiltonian schur form of a nearby hamiltonian matrix and thus is numerically strongly backward stable the new method is of complexity mathbf o n 3 and hence it solves a long standing open problem in numerical analysis

on the numerical integration of the linear shape functions times the 3 d green s function or its gradient on a plane triangle formulas for the static and dynamic potentials due to linearly varying source distributions defined on a planar triangle are presented the general linear variation of the source is represented in terms of three linear basis functions each one associated with a different vertex of the triangular domain the singular kernels considered are given by the 3 d green s function and its gradient both for the static and dynamic case in the static case the evaluation of potentials is performed analytically and compact form results are given in the dynamic case the results are given in a form suited for numerical integration

bak sneppen model near zero dimension we consider the bak sneppen model near zero dimension where the avalanche exponent tau is close to 1 and the exponents m and sigma are close to 0 we demonstrate that tau 1 m sigma exp m 1 gamma small middle dot small middle dot small middle dot in this limit where gamma is euler s constant the avalanche hierarchy equation is rewritten in a form that makes it possible to find the relation between the critical exponents sigma and m with high accuracy we obtain precise values of the critical exponents for the one and two dimensional bak sneppen model and for the one dimensional anisotropic bak sneppen model

computing the generalized singular values vectors of large sparse or structured matrix pairs we present a numerical algorithm for computing a few extreme generalized singular values and corresponding vectors of a sparse or structured matrix pair a b the algorithm is based on the cs decomposition and the lanczos bidiagonalization process at each iteration step of the lanczos process the solution to a linear least squares problem with a rm t b rm t rm t as the coefficient matrix is approximately computed and this consists the only interface of the algorithm with the matrix pair a b numerical results are also given to demonstrate the feasibility and efficiency of the algorithm

machine precision evaluation of singular and nearly singular potential integrals by use of gauss quadrature formulas for rational functions a new technique for machine precision evaluation of singular and nearly singular potential integrals with 1 r singularities is presented the numerical quadrature scheme is based on a new rational expression for the integrands obtained by a cancellation procedure in particular by using library routines for gauss quadrature of rational functions readily available in the literature this new expression permits the exact numerical integration of singular static potentials associated with polynomial source distributions the rules to achieve the desired numerical accuracy for singular and nearly singular static and dynamic potential integrals are presented and discussed and several numerical examples are provided