miniversal deformations of pairs of skew symmetric matrices under congruence for each pair of complex symmetric matrices a b we provide a normal form with a minimal number of independent parameters to which all pairs of complex symmetric matrices a over tilde b ove

integral and integrable algorithms for a nonlinear shallow water wave equation an asymptotic higher order model of wave dynamics in shallow water is examined in a combined analytical and numerical study with the aim of establishing robust and efficient numerical solution methods based on the hamiltonian structure of the nonlinear equation an algorithm corresponding to a completely integrable particle lattice is implemented first each particle in the particle method travels along a characteristic curve the resulting system of nonlinear ordinary differential equations can have solutions that blow up in finite time we isolate the conditions for global existence and prove l1 norm convergence of the method in the limit of zero spatial step size and infinite particles the numerical results show that this method captures the essence of the solution without using an overly large number of particles a fast summation algorithm is introduced to evaluate the integrals of the particle method so that the computational cost is reduced from o n2 to o n where n is the number of particles the method possesses some analogies with point vortex methods for 2d euler equations in particular near singular solutions exist and singularities are prevented from occurring in finite time by mechanisms akin to those in the evolution of vortex patches the second method is based on integro differential formulations of the equation two different algorithms are proposed based on different ways of extracting the time derivative of the dependent variable by an appropriately defined inverse operator the integro differential formulations reduce the order of spatial derivatives thereby relaxing the stability constraint and allowing large time steps in an explicit numerical scheme in addition to the cauchy problem on the infinite line we include results on the study of the nonlinear equation posed in the quarter space time plane we discuss the minimum number of boundary conditions required for solution uniqueness and illustrate this with numerical examples

tikhonov type regularization method for the cauchy problem of the modified helmholtz equation abstract in this paper we propose a numerical method for solving the cauchy problem of the modified helmholtz equation by using green s formula the cauchy problem is transformed to a moment problem then we propose a tikhonov type regularization algorithm for obtaining an approximate solution to the neumann date on the unspecified boundary convergence analysis and error estimation are also discussed for numerical verification several numerical examples in the two dimensional case are presented

on a three dimensional cauchy problem for inhomogeneous helmholtz equation associated with perturbed wave number abstract in recent years the cauchy problem for inhomogeneous helmholtz equation cphe is often considered as the wave number k is a constant number at present we investigate a three dimensional cphe with inhomogeneous cauchy conditions given at z 0 while wave number k is perturbed the problem is well known to be ill posed in the sense of hadamard therefore we regularize the problem by applying the truncation method and possess error estimate between the exact solution and the regularized solution a numerical experiment is given for the purpose of illustrating our method

generic skew symmetric matrix polynomials with fixed rank and fixed odd grade we show that the set of m m complex skew symmetric matrix polynomials of odd grade d i e of degree at most d and normal rank at most 2r is the closure of the single set of matrix polynomials

an improved fast multipole algorithm for potential fields a new version of the fast multipole method fmm for potential fields is presented we introduce a new representation of potentials in which most translation operators are diagonal as a result for double precision calculations in two dimensions we obtain an improvement of a factor of two to four in speed compared to previously published algorithms the improvement is expected to be much greater in three dimensions the performance of the method is illustrated with several numerical examples

on the inverse eigenvalue problem for t alternating and t palindromic matrix polynomials abstract the inverse eigenvalue problem for t alternating matrix polynomials over arbitrary algebraically closed fields of characteristic different from two is considered the main result shows that the necessary conditions obtained in 9 for a matrix polynomial to be the smith form of a t alternating matrix polynomial are under mild conditions also sufficient to be the smith form of a t alternating matrix polynomial with invertible leading coefficient which is additionally in anti triangular form in particular this result implies that any t alternating matrix polynomial with invertible leading coefficient is equivalent to a t alternating matrix polynomial in anti triangular form that has the same finite and infinite elementary divisors as the original matrix polynomial finally the inverse eigenvalue problem for t palindromic matrix polynomials is considered excluding the case that both 1 and 1 are eigenvalues

regular article a fast adaptive multipole algorithm in three dimensions we present an adaptive fast multipole method for the laplace equation in three dimensions it uses both new compression techniques and diagonal forms for translation operators to achieve high accuracy at a reasonable cost

coulomb interactions on planar structures inverting the square root of the laplacian we present an adaptive fast multipole method for inverting the square root of the laplacian in two dimensions solving this problem is the dominant computational cost in many applications arising in electrical engineering geophysical fluid dynamics and the study of thin films it corresponds to the evaluation of the field induced by a planar distribution of charge or vorticity our algorithm is direct and assumes only that the source distribution is discretized using an adaptive quad tree the amount of work grows linearly with the number of mesh points

fast integral equation methods for rothe s method applied to the isotropic heat equation we present an efficient integral equation approach to solve the forced heat equation u t x email protected x f x u t in a two dimensional multiply connected domain with dirichlet boundary conditions instead of using an integral equation formulation based on the heat kernel we discretize in time first this approach known as rothe s method leads to a non homogeneous modified helmholtz equation that is solved at each time step we formulate the solution to this equation as a volume potential plus a double layer potential and both of these potentials are calculated with available tools accelerated by the fast multipole method for a total of n points in the discretization of the boundary and the domain the total computational cost per time step is o n we demonstrate our approach on the heat equation and the allen cahn equation

congruences of a square matrix and its transpose abstract it is known that any square matrix a over any field is congruent to its transpose at stas for some nonsingular s moreover s can be chosen such that s2 i that is s can be chosen to be involutory we show that a and at are congruent over any field f of characteristic not two with involution a a the involution can be the identity a t s t as for some nonsingular s moreover s can be chosen such that s s i that is s can be chosen to be coninvolutory the short and simple proof is based on sergeichuk s canonical form for congruence math ussr izvestiya 31 3 1988 481 it follows that any matrix a over f can be represented as a eb in which e is coninvolutory and b is symmetric

jordan structures of alternating matrix polynomials alternating matrix polynomials that is polynomials whose coefficients alternate between symmetric and skew symmetric matrices generalize the notions of even and odd scalar polynomials we investigate the smith forms of alternating matrix polynomials showing that each invariant factor is an even or odd scalar polynomial necessary and sufficient conditions are derived for a given smith form to be that of an alternating matrix polynomial these conditions allow a characterization of the possible jordan structures of alternating matrix polynomials and also lead to necessary and sufficient conditions for the existence of structure preserving strong linearizations most of the results are applicable to singular as well as regular matrix polynomials

direct complements of invariant lagrangian subspaces and minimal factorizations of skew symmetric rational matrix functions abstract symmetric factorizations of skew symmetric real rational matrix functions are studied in particular it is proved that generically a proper skew symmetric real rational matrix mxm matrix function w w t with w invertible can be factorized minimally as w l tdl for a real rational mxm matrix function l applications are given concerning symmetric solutions of matrix quadratic equations

implementing the fast multipole method in three dimensions the rokhlin greengard fast multipole algorithm for evaluating coulomb and multipole potentials has been implemented and analyzed in three dimensions the implementation is presented for bounded charged systems and systems with periodic boundary conditions the results include timings and error characterizations

emergence of quasiunits in the one dimensional zhang model we study the zhang model of sandpile on a one dimensional chain of length l where a random amount of energy is added at a randomly chosen site at each time step we show that in spite of this randomness in the input energy the probability distribution function of energy at a site in the steady state is sharply peaked and the width of the peak decreases as l 1 2 for large l we discuss how the energy added at one time is distributed among different sites by topplings with time we relate this distribution to the time dependent probability distribution of the position of a marked grain in the one dimensional abelian model with discrete heights we argue that in the large l limit the variance of energy at site x has a scaling form l 1 g x l where g xi varies as ln 1 xi for small xi which agrees very well with the results from numerical simulations

quadratic realizability of palindromic matrix polynomials let cl cl 1 cl 2 be a list consisting of a sublist cl 1 of powers of irreducible monic scalar polynomials over an algebraically closed field ff and a sublist cl 2 of nonnegative integers for an arbitrary such list cl we give easily verifiable necessary and sufficient conditions for cl to be the list of elementary divisors and minimal indices of some t palindromic quadratic matrix polynomial with entries in the field ff for cl satisfying these conditions we show how to explicitly construct a t palindromic quadratic matrix polynomial having cl as its structural data that is we provide a t palindromic quadratic realization of cl our construction of t palindromic realizations is accomplished by taking a direct sum of low bandwidth t palindromic blocks closely resembling the kronecker canonical form of matrix pencils an immediate consequence of our in depth study of the structure of t palindromic quadratic polynomials is that all even grade t palindromic matrix polynomials have a t palindromic strong quadratification finally using a particular m o bius transformation we show how all of our results can be easily extended to quadratic matrix polynomials with t even structure

on hamiltonian and symplectic lanczos processes abstract large sparse hamiltonian eigenvalue problems arise in a variety of contexts these problems can be attacked directly or they can first be transformed to problems having some related structure such as symplectic or skew hamiltonian in the interest of efficiency stability and accuracy such problems should be solved by methods that preserve the structure whether it be hamiltonian skew hamiltonian or symplectic the present work outlines krylov subspace methods for computing partial eigensystems of skew hamiltonian hamiltonian and symplectic matrices and records some of the relationships between them the ordinary unsymmetric lanczos process is a structure preserving method for skew hamiltonian matrices the hamiltonian and symplectic lanczos processes developed here are condensed versions of the processes that have been published previously the condensed hamiltonian lanczos process applied to h is equivalent to the unsymmetric lanczos process applied to the skew hamiltonian h 2 but costs half as much to execute the condensed symplectic lanczos process applied to s is equivalent to the unsymmetric lanczos process applied to the skew hamiltonian matrix s s 1 but also costs half as much to execute implicit restarts of the hamiltonian and symplectic lanczos processes can be effected by the sr algorithm because of the known relationship between the sr and hr algorithms the much simpler hr algorithm can be used to restart the condensed symplectic and hamiltonian lanczos processes each of these restart procedures is equivalent to restarting the unsymmetric lanczos process using the hr algorithm the hr algorithm is most effectively implemented as an implicit bulge chasing hz algorithm on a symmetric tridiagonal diagonal pencil t d

hamiltonian square roots of skew hamiltonian matrices abstract we present a constructive existence proof that every real skew hamiltonian matrix w has a real hamiltonian square root the key step in this construction shows how one may bring any such w into a real quasi jordan canonical form via symplectic similarity we show further that every w has infinitely many real hamiltonian square roots and give a lower bound on the dimension of the set of all such square roots some extensions to complex matrices are also presented

structured strong linearizations from fiedler pencils with repetition ii abstract in this paper we give strong linearizations of a matrix polynomial p preserving the skew symmetry or t alternating structure of p the linearizations obtained are of the form s l where l is a block symmetric fiedler pencil with repetition and s is a direct sum of blocks of the form i or i with i the identity matrix this paper is a continuation of 2 where the corresponding problem for p with a symmetric structure was studied and as a consequence the block symmetric fiedler pencils with repetition were characterized

stratification of full rank polynomial matrices we show that perturbations of polynomial matrices of full normal rank can be analyzed viathe study of perturbations of companion form linearizations of such polynomial matrices it is proved that a

a fourier series based kernel independent fast multipole method we present in this paper a new kernel independent fast multipole method fmm named as fki fmm for pairwise particle interactions with translation invariant kernel functions fki fmm creates using numerical techniques sufficiently accurate and compressive representations of a given kernel function over multi scale interaction regions in the form of a truncated fourier series it provides also economic operators for the multipole to multipole multipole to local and local to local translations that are typical and essential in the fmm algorithms the multipole to local translation operator in particular is readily diagonal and does not dominate in arithmetic operations fki fmm provides an alternative and competitive option among other kernel independent fmm algorithms for an efficient application of the fmm especially for applications where the kernel function consists of multi physics and multi scale components as those arising in recent studies of biological systems we present the complexity analysis and demonstrate with experimental results the fki fmm performance in accuracy and efficiency

a boundary integral method for computing the dynamics of an epitaxial island in this paper we present a boundary integral method for computing the quasi steady evolution of an epitaxial island the problem consists of an adatom diffusion equation with desorption on terrace and a kinetic boundary condition at the step island boundary the normal velocity for step motion is determined by a two sided flux the integral formulation of the problem involves both double and single layer potentials due to the kinetic boundary condition numerical tests on a growing shrinking circular or a slightly perturbed circular island are in excellent agreement with the linear analysis demonstrating that the method is stable efficient and spectrally accurate in space nonlinear simulations for the growth of perturbed circular islands show that sharp tips and flat edges will form during growth instead of the usual dense branching morphology seen throughout physical and biological systems driven out of equilibrium in particular bales zangwill instability is manifested in the form of wave like fronts meandering instability around the tip regions the numerical techniques presented here can be applied generally to a class of free moving boundary problems in physical and biological science

a renormalization method for the evaluation of lattice sums a number of problems in solid state physics and materials science can be resolved by the evaluation of certain lattice sums sums over all sites of an infinite perfect lattice of some potential energy function one classical example the calculation of lattice sums of circular and spherical harmonics dates back to the last century to lord rayleigh s work on computing the effective conductivity of a simple composite while lord rayleigh presented an efficient asymptotic method for two dimensional problems he resorted to direct evaluation of the lattice sums in the three dimensional case more recent methods based on ewald summation have been developed by nijboer and de wette schmidt and lee and others in this article a fast method for evaluating lattice sums which is based on a new renormalization identity is described

an implicitly restarted krylov subspace method for real symmetric skew symmetric eigenproblems abstract a new implicitly restarted krylov subspace method for real symmetric skew symmetric generalized eigenvalue problems is presented the new method improves and generalizes the shira method of mehrmann and watkins 2001 37 to the case where the skew symmetric matrix is singular it computes a few eigenvalues and eigenvectors of the matrix pencil close to a given target point several applications from control theory are presented and the properties of the new method are illustrated by benchmark examples

large scale simulations of the zhang sandpile model we consider the non abelian sandpile model introduced by y c zhang on a two dimensional square lattice the static and dynamical properties of the model are investigated and compared to the abelian sandpile model of bak tang and wiesenfeld a detailed analysis which takes the finite size effects into account yields that the exponents of the avalanche probability distribution are the same as in the abelian model

normal forms of elements of classical real and complex lie and jordan algebras elements of the classical complex and real lie and jordan algebras with involutions are classified into conjugacy classes under the action of the corresponding classical lie group normal forms of representatives of each conjugacy class are chosen so as to resemble the jordan normal forms of n n complex matrices for completeness similar results are given for gl n c gl n r and gl n h

fast integral equation methods for the modified helmholtz equation we present integral equation methods for the solution to the two dimensional modified helmholtz equation u x email protected email protected x 0 in bounded or unbounded multiply connected domains we consider both dirichlet and neumann problems we derive well conditioned fredholm integral equations of the second kind which are discretized using high order hybrid gauss trapezoid rules our fast multipole based iterative solution procedure requires only o n operations where n is the number of nodes in the discretization of the boundary we demonstrate the performance of our methods on several numerical examples and we show that they have both the ability to handle highly complex geometry and the potential to solve large scale problems

structure preserving stratification of skew symmetric matrix polynomials we study how elementary divisors and minimal indices of a skew symmetric matrix polynomial of odd degree may change under small perturbations of the matrix coefficients we investigate these change

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

second kind integral equation formulation for the modified biharmonic equation and its applications a system of fredholm second kind integral equations skies is constructed for the modified biharmonic equation in two dimensions with gradient boundary conditions such boundary value problem arises naturally when the incompressible navier stokes equations are solved via a semi implicit discretization scheme and the resulting boundary value problem at each time step is then solved using the pure stream function formulation the advantages of such an approach greengard and kropinski 1998 14 are two fold first the velocity is automatically divergence free and second complicated nonlocal boundary conditions for the vorticity are avoided our construction though similar to that of farkas 1989 10 for the biharmonic equation is modified such that the skie formulation has low condition numbers for large values of the parameter we illustrate the performance of our numerical scheme with several numerical examples finally the scheme can be easily coupled with standard fast algorithms such as fft fast multipole methods greengard and rokhlin 1987 15 or fast direct solvers ho and greengard 2012 martinsson and rokhlin 2005 17 25 to achieve optimal complexity bringing accurate large scale long time fluid simulations within practical reach

the fast multipole method i error analysis and asymptotic complexity this paper is concerned with the application of the fast multipole method fmm to the maxwell equations this application differs in many aspects from other applications such as the n body problem laplace equation and quantum chemistry etc the fmm leads to a significant speed up in cpu time with a major reduction in the amount of computer memory needed when performing matrix vector products this leads to faster resolution of scattering of harmonic plane waves from perfectly conducting obstacles emphasis here is on a rigorous mathematical approach to the problem we focus on the estimation of the error introduced by the fmm and a rigorous analysis of the complexity o n log n of the algorithm we show that error estimates reported previously are not entirely satisfactory and provide sharper and more precise estimates

a cartesian grid embedded boundary method for poisson s equation on irregular domains we present a numerical method for solving poisson s equation with variable coefficients and dirichlet boundary conditions on two dimensional regions the approach uses a finite volume discretization which embeds the domain in a regular cartesian grid we treat the solution as a cell centered quantity even when those centers are outside the domain cells that contain a portion of the domain boundary use conservative differencing of second order accurate fluxes on each cell volume the calculation of the boundary flux ensures that the conditioning of the matrix is relatively unaffected by small cell volumes this allows us to use multigrid iterations with a simple point relaxation strategy we have combined this with an adaptive mesh refinement amr procedure we provide evidence that the algorithm is second order accurate on various exact solutions and compare the adaptive and nonadaptive calculations

skew symmetric matrix polynomials and their smith forms we characterize the smith form of skew symmetric matrix polynomials over an arbitrary field f showing that all elementary divisors occur with even multiplicity restricting the class of equivalence transformations to unimodular congruences a smith like skew symmetric canonical form for skew symmetric matrix polynomials is also obtained these results are used to analyze the eigenvalue and elementary divisor structure of matrices expressible as products of two skew symmetric matrices as well as the existence of structured linearizations for skew symmetric matrix polynomials by contrast with other classes of structured matrix polynomials e g alternating or palindromic polynomials every regular skew symmetric matrix polynomial is shown to have a structured strong linearization while there are singular skew symmetric polynomials of even degree for which a structured linearization is impossible for each odd degree we develop a skew symmetric companion form that uniformly provides a structured linearization for every regular and singular skew symmetric polynomial of that degree finally the results are applied to the construction of minimal symmetric factorizations of skew symmetric rational matrices

nonlinear optimization quadrature and interpolation we present a nonlinear optimization procedure for the design of generalized gaussian quadratures for a fairly broad class of functions while some of the components of the algorithm have been published previously we introduce an improved procedure for the determination of an acceptable initial point for the continuation scheme that stabilizes the newton type process used to find the quadratures the resulting procedure never failed when applied to chebyshev systems for which the existence and uniqueness of generalized gaussian quadratures are well known it also worked for many non chebyshev systems for which the generalized gaussian quadratures are not guaranteed to exist the performance of the algorithm is illustrated with several numerical examples some of the presented quadratures integrate efficiently large classes of singular functions

toppling distributions in one dimensional abelian sandpiles we consider toppling distributions for the abelian sandpile model in one dimension we study the avalanche mass and duration distributions in the thermodynamic limit we also investigate their dependence on the seeding distribution which describes how sand is dropped none of the models shows criticality

hermitian matrix polynomials with real eigenvalues of definite type part i classification the spectral properties of hermitian matrix polynomials with real eigenvalues have been extensively studied through classes such as the definite or definitizable pencils definite hyperbolic or quasihyperbolic matrix polynomials and overdamped or gyroscopically stabilized quadratics we give a unified treatment of these and related classes that uses the eigenvalue type or sign characteristic as a common thread equivalent conditions are given for each class in a consistent format we show that these classes form a hierarchy all of which are contained in the new class of quasidefinite matrix polynomials as well as collecting and unifying existing results we make several new contributions we propose a new characterization of hyperbolicity in terms of the distribution of the eigenvalue types on the real line by analyzing their effect on eigenvalue type we show that homogeneous rotations allow results for matrix polynomials with nonsingular or definite leading coefficient to be translated into results with no such requirement on the leading coefficient which is important for treating definite and quasidefinite polynomials we also give a sufficient condition for a quasihyperbolic matrix polynomial to be diagonalizable by structure preserving congruence and show that this condition is always satisfied in the quadratic case and for any hyperbolic matrix polynomial thereby identifying an important new class of diagonalizable matrix polynomials

eulerian walkers as a model of self organized criticality we propose a new model of self organized criticality a particle is dropped at random on a lattice and moves along directions specified by arrows at each site as it moves it changes the direction of the arrows according to fixed rules on closed graphs these walks generate euler circuits on open graphs the particle eventually leaves the system and a new particle is then added the operators corresponding to particle addition generate an abelian group same as the group for the abelian sandpile model on the graph we determine the critical steady state and some critical exponents exactly using this equivalence

palindromic companion forms for matrix polynomials of odd degree the standard way to solve polynomial eigenvalue problems p x 0 is to convert the matrix polynomial p into a matrix pencil that preserves its spectral information a process known as linearization when p is palindromic the eigenvalues elementary divisors and minimal indices of p have certain symmetries that can be lost when using the classical first and second frobenius companion linearizations for numerical computations since these linearizations do not preserve the palindromic structure recently new families of pencils have been introduced with the goal of finding linearizations that retain whatever structure the original p might possess with particular attention to the preservation of palindromic structure however no general construction of palindromic linearizations valid for all palindromic polynomials has as yet been achieved in this paper we present a family of linearizations for odd degree polynomials p which are palindromic whenever p is and which are valid for all palindromic polynomials of odd degree we illustrate our construction with several examples in addition we establish a simple way to recover the minimal indices of the polynomial from those of the linearizations in the new family

hybrid asymptotic numerical approach for estimating first passage time densities of the two dimensional narrow capture problem a hybrid asymptotic numerical method is presented for obtaining an asymptotic estimate for the full probability distribution of capture times of a random walker by multiple small traps located inside a bounded two dimensional domain with a reflecting boundary as motivation for this study we calculate the variance in the capture time of a random walker by a single interior trap and determine this quantity to be comparable in magnitude to the mean this implies that the mean is not necessarily reflective of typical capture times and that the full density must be determined to solve the underlying diffusion equation the method of laplace transforms is used to obtain an elliptic problem of modified helmholtz type in the limit of vanishing trap sizes each trap is represented as a dirac point source that permits the solution of the transform equation to be represented as a superposition of helmholtz green s functions using this solution we construct asymptotic short time solutions of the first passage time density which captures peaks associated with rapid capture by the absorbing traps when numerical evaluation of the helmholtz green s function is employed followed by numerical inversion of the laplace transform the method reproduces the density for larger times we demonstrate the accuracy of our solution technique with a comparison to statistics obtained from a time dependent solution of the diffusion equation and discrete particle simulations in particular we demonstrate that the method is capable of capturing the multimodal behavior in the capture time density that arises when the traps are strategically arranged the hybrid method presented can be applied to scenarios involving both arbitrary domains and trap shapes

on computing the eigenvalues of a symplectic pencil this paper presents an algorithm for computing the eigenvalues of a symplectic pencil that arises in one of the commonly used approaches for solving the discrete time algebraic riccati equation the algorithm is numerically efficient and reliable in that it employs only orthogonal transformations and makes use of the structure of the symplectic pencil it requires about one fourth the number of floating point operations that the qz algorithm uses to compute the eigenvalues of the pencil directly the proposed method can be regarded as being analogous for the case of symplectic pencils to the method developed by van loan for computing the eigenvalues of hamiltonian matrices

hamiltonian square roots of skew hamiltonian matrices revisited abstract recently h fassbender et al linear algebra appl 287 1999 125 proved the following theorem every real skew hamiltonian matrix w has a real hamiltonian square root h i e h 2 w we prove an analog of this theorem for complex matrices our approach may be of independent interest namely we use the polar decomposition of a nonsingular operator acting in a space with the symplectic inner product

an integral equation method for the cahn hilliard equation in the wetting problem abstract we present an integral equation approach to solving the cahn hilliard equation equipped with boundary conditions that model solid surfaces with prescribed young s angles the discretization of the system in time using convex splitting leads to a modified biharmonic equation at each time step to solve it we split the solution into a volume potential computed with free space kernels plus the solution to a second kind integral equation skie the volume potential is evaluated with the help of a box based volume fmm method for non box domains the source density is extended by solving a biharmonic dirichlet problem the near singular boundary integrals are computed using quadrature by expansion qbx with fmm acceleration our method has linear complexity in the number of surface volume degrees of freedom and can achieve high order convergence in space with adaptive refinement to manage error from function extension

large vector spaces of block symmetric strong linearizations of matrix polynomials abstract given a matrix polynomial p i 0 k i a i of degree k where a i are n n matrices with entries in a field f the development of linearizations of p that preserve whatever structure p might posses has been a very active area of research in the last decade most of the structure preserving linearizations of p discovered so far are based on certain modifications of block symmetric linearizations the block symmetric linearizations of p available in the literature fall essentially into two classes linearizations based on the so called fiedler pencils with repetition which form a finite family and a vector space of dimension k of block symmetric pencils called dl p such that most of its pencils are linearizations one drawback of the pencils in dl p is that none of them is a linearization when p is singular in this paper we introduce new vector spaces of block symmetric pencils most of which are strong linearizations of p the dimensions of these spaces are o n 2 which for n k are much larger than the dimension of dl p when k is odd many of these vector spaces contain linearizations also when p is singular the coefficients of the block symmetric pencils in these new spaces can be easily constructed as k k block matrices whose n n blocks are of the form 0 i n a i or arbitrary n n matrices where is an arbitrary nonzero scalar

block kronecker linearizations of matrix polynomials and their backward errors we introduce a new family of strong linearizations of matrix polynomials which we call block kronecker pencils and perform a backward stability analysis of complete polynomial eigenproblems these problems are solved by applying any backward stable algorithm to a block kronecker pencil such as the staircase algorithm for singular pencils or the qz algorithm for regular pencils this analysis allows us to identify those block kronecker pencils that yield a computed complete eigenstructure which is exactly that of a slightly perturbed matrix polynomial these favorable pencils include the famous fiedler linearizations which are just a very particular case of block kronecker pencils thus our analysis offers the first proof available in the literature of global backward stability for fiedler pencils in addition the theory developed for block kronecker pencils is much simpler than the theory available for fiedler pencils especially in the case of rectangular matrix polynomials the global backward error analysis in this work presents for the first time the following key properties it is a rigorous analysis valid for finite perturbations i e it is not a first order analysis it provides precise bounds it is valid simultaneously for a large class of linearizations and it establishes a framework that may be generalized to other classes of linearizations these features are related to the fact that block kronecker pencils are a particular case of the new family of strong block minimal bases pencils which include certain perturbations of block kronecker pencils this allows us to extend the results in this paper to other contexts

composite overlapping meshes for the solution of partial differential equations abstract we discuss the generation of curvilinear composite overlapping grids and the numerical solution of partial differential equations on them a composite overlapping grid consists of a set of curvilinear component grids that cover a region and overlap where they meet continuity conditions interpolation are imposed at the overlapping boundaries the principal advantage of composite grids is in the generation of grids for regions with complicated geometries the grid construction program cmpgrd is used to create composite grids with any number of component grids we describe some techniques for the solution of elliptic and time dependent pdes on composite meshes applications to the solution of the compressible navier stokes equations are presented

mobius transformations of matrix polynomials we discuss mobius transformations for general matrix polynomials over arbitrary fields analyzing their influence on regularity rank determinant constructs such as compound matrices and on structural features including sparsity and symmetry results on the preservation of spectral information contained in elementary divisors partial multiplicity sequences invariant pairs and minimal indices are presented the effect on canonical forms such as smith forms and local smith forms on relationships of strict equivalence and spectral equivalence and on the property of being a linearization or quadratification are investigated we show that many important transformations are special instances of mobius transformations and analyze a mobius connection between alternating and palindromic matrix polynomials finally the use of mobius transformations in solving polynomial inverse eigenproblems is illustrated

a direct adaptive poisson solver of arbitrary order accuracy we present a direct adaptive solver for the poisson equation which can achieve any prescribed order of accuracy it is based on a domain decomposition approach using local spectral approximation as well as potential theory and the fast multipole method in two space dimensions the algorithm requireso nk work wherenis the number of discretization points andkis the desired order of accuracy

lattice sums and the two dimensional periodic green s function for the helmholtz equation many algorithms that are currently used for the solution of the helmholtz equation in periodic domains require the evaluation of the green s function g x x0 the fact that the natural representat

canonical forms for hamiltonian and symplectic matrices and pencils abstract we study canonical forms for hamiltonian and symplectic matrices or pencils under equivalence transformations which keep the class invariant in contrast to other canonical forms our forms are as close as possible to a triangular structure in the same class we give necessary and sufficient conditions for the existence of hamiltonian and symplectic triangular jordan kronecker and schur forms the presented results generalize results of lin and ho w w lin t c ho on schur type decompositions for hamiltonian and symplectic pencils technical report institute of applied mathematics national tsing hua university taiwan 1990 and simplify the proofs presented there

krylov deferred correction accelerated method of lines transpose for parabolic problems in this paper a new class of numerical methods for the accurate and efficient solutions of parabolic partial differential equations is presented unlike traditional method of lines mol the new krylov deferred correction kdc accelerated method of lines transpose mol t first discretizes the temporal direction using gaussian type nodes and spectral integration and symbolically applies low order time marching schemes to form a preconditioned elliptic system which is then solved iteratively using newton krylov techniques such as newton gmres or newton bicgstab method each function evaluation in the newton krylov method is simply one low order time stepping approximation of the error by solving a decoupled system using available fast elliptic equation solvers preliminary numerical experiments show that the kdc accelerated mol t technique is unconditionally stable can be spectrally accurate in both temporal and spatial directions and allows optimal time stepsizes in long time simulations

integral representations of harmonic lattice sums we present new integral formulas for coulombic lattice sums that arise in solid state physics material science and complex analysis unlike ewald summation which can yield approximate integral representations the formulas described here are exact simple quadrature rules with modest numbers of nodes yield highly accurate results

palindromic quadratization and structure preserving algorithm for palindromic matrix polynomials of even degree in this paper we propose a palindromic quadratization approach transforming a palindromic matrix polynomial of even degree to a palindromic quadratic pencil based on the mathcal s mathcal s 1 transform and patel s algorithm the structure preserving algorithm can then be applied to solve the corresponding palindromic quadratic eigenvalue problem numerical experiments show that the relative residuals for eigenpairs of palindromic polynomial eigenvalue problems computed by palindromic quadratized eigenvalue problems are better than those via palindromic linearized eigenvalue problems or texttt polyeig in matlab

a wavelet collocation method for boundary integral equations of the modified helmholtz equation abstract a wavelet collocation method is to proposed for solving the linear boundary integral equations reformulated from the modified helmholtz equation with robin boundary conditions to deal with the difficulties caused by robin boundary conditions we provide an improved version of wavelet collocation method by employing a matrix compression strategy and augmentation method we obtain fully discrete system and solve efficiently the resulting systems at last we point out that the proposed method employs an optimal convergence order and a nearly linear computational complexity numerical experiments are presented to demonstrate its approximation accuracy and computational efficiency

general matrix pencil techniques for the solution of algebraic riccati equations a unified approach we present a unified theory of matrix pencil techniques for solving both continuous and discrete time algebraic riccati equations ares under fairly general conditions on the coefficient matrices the theory applies to a large class of ares and riccati like equations arising from the singular h sup spl infin and h sup 2 control problems singular linear quadratic control the 4 block nehari problem or from singular j spectral factorizations the underlying concept is the so called proper deflating subspace of a possibly singular matrix pencil in terms of which necessary and sufficient conditions for the solvability of riccati equations are given it is shown that these conditions can be checked and the solutions computed by a numerically sound algorithm

a new fast multipole accelerated poisson solver in two dimensions we present an adaptive fast multipole method for solving the poisson equation in two dimensions the algorithm is direct assumes that the source distribution is discretized using an adaptive quad tree and allows for dirichlet neumann periodic and free space conditions to be imposed on the boundary of a square the amount of work per grid point is comparable to that of classical fast solvers even for highly nonuniform grids

an adaptive fast solver for the modified helmholtz equation in two dimensions in this paper we present a fast multipole accelerated integral equation method for solving the modified helmholtz equation du x b 2u x f x in two dimensions the method is direct and unlike classical fft based fast solvers it allows for adaptive mesh refinement but with comparable amount of work per grid point when the computational domain is rectangular dirichlet neumann periodic and free space boundary conditions can be imposed analytically without the need to solve a system of linear equations several important features of the algorithm are discussed including the use of precomputed tables diagonal translation operators and lattice sums to impose periodic boundary conditions numerical experiments show that for a wide range of the parameter b the algorithm is stable and high order accurate

polynomial zigzag matrices dual minimal bases and the realization of completely singular polynomials minimal bases of rational vector spaces are a well known and important tool in systems theory if minimal bases for two subspaces of rational n space are displayed as the rows of polynomial matrices z 1 lambda k times n and z 2 lambda m times n respectively then z 1 and z 2 are said to be dual minimal bases if the subspaces have complementary dimension i e k m n and z 1 lambda z 2 t lambda 0 in other words each z j lambda provides a minimal basis for the nullspace of the other it has long been known that for any dual minimal bases z 1 lambda and z 2 lambda the row degree sums of z 1 and z 2 are the same in this paper we show that this is the only constraint on the row degrees thus characterizing the possible row degrees of dual minimal bases the proof is constructive making extensive use of a new class of sparse structured polynomial matrices that we have baptized zigzag matrices another application of these polynomial zigzag matrices is the constructive solution of the following inverse problem for minimal indices given a list of left and right minimal indices and a desired degree d does there exist a completely singular matrix polynomial i e a matrix polynomial with no elementary divisors whatsoever of degree d having exactly the prescribed minimal indices we show that such a matrix polynomial exists if and only if d divides the sum of the minimal indices the constructed realization is simple and explicitly displays the desired minimal indices in a fashion analogous to the classical kronecker canonical form of singular pencils

the abelian sandpile and related models abstract the abelian sandpile model is the simplest analytically tractable model of self organized criticality this paper presents a brief review of known results about the model the abelian group structure of the algebra of operators allows an exact calculation of many of its properties in particular when there is a preferred direction one can calculate all the critical exponents characterizing the distribution of avalanche sizes in all dimensions for the undirected case the model is related to q 0 state potts model this enables exact calculation of some exponents in two dimensions and there are some conjectures about others we also discuss a generalization of the model to a network of communicating reactive processors this includes sandpile models with stochastic toppling rules as a special case we also consider a nondashabelian stochastic variant which lies in a different universality class related to directed percolation

the continuing influence of fiedler s work on companion matrices this is a reconstruction in article like form of a talk given at the minisymposium in honor of miroslav fiedler at the 17th ilas conference held at tu braunschweig germany on thurs 25 aug 2011

a symplectic method for approximating all the eigenvalues of a hamiltonian matrix a fast method for computing all the eigenvalues of a hamiltonian matrix m is given the method relies on orthogonal symplectic similarity transformations which preserve structure and have desirable numerical properties the algorithm is about four times faster than the standard q r algorithm the computed eigenvalues are shown to be the exact eigenvalues of a matrix m e where vert e vert depends on the square root of the machine precision the accuracy of a computed eigenvalue depends on its condition and its magnitude larger eigenvalues typically being more accurate

fast adaptive high order accurate discretization of the lippmann schwinger equation in two dimensions we present a fast direct solver for two dimensional scattering problems where an incident wave impinges on a penetrable medium with compact support we represent the scattered field using a volume potential whose kernel is the outgoing green s function for the exterior domain inserting this representation into the governing partial differential equation we obtain an integral equation of lippmann schwinger type the principal contribution here is the development of an automatically adaptive high order accurate discretization based on a quad tree data structure which provides rapid access to arbitrary elements of the discretized system matrix this permits the straightforward application of state of the art algorithms for constructing compressed versions of the solution operator these solvers typically require o n 3 2 work where n denotes the number of degrees of freedom we demonstrate the performance of the method for a variety of problems in both low and high frequency regimes

spectral equivalence of matrix polynomials and the index sum theorem abstract the concept of linearization is fundamental for theory applications and spectral computations related to matrix polynomials however recent research on several important classes of structured matrix polynomials arising in applications has revealed that the strategy of using linearizations to develop structure preserving numerical algorithms that compute the eigenvalues of structured matrix polynomials can be too restrictive because some structured polynomials do not have any linearization with the same structure this phenomenon strongly suggests that linearizations should sometimes be replaced by other low degree matrix polynomials in applied numerical computations motivated by this fact we introduce equivalence relations that allow the possibility of matrix polynomials with coefficients in an arbitrary field to be equivalent with the same spectral structure but have different sizes and degrees these equivalence relations are directly modeled on the notion of linearization and consequently inherit the simplicity applicability and most relevant properties of linearizations simultaneously though they are much more flexible in the possible degrees of equivalent polynomials this flexibility allows us to define in a unified way the notions of quadratification and l ification to introduce the concept of companion form of arbitrary degree and to provide concrete and simple examples of these notions that generalize in a natural and smooth way the classical first and second frobenius companion forms the properties of l ifications are studied in depth in this process a fundamental result on matrix polynomials the index sum theorem is recovered and extended to arbitrary fields although this result is known in the systems theory literature for real matrix polynomials it has remained unnoticed by many researchers it establishes that the sum of the finite and infinite partial multiplicities together with the left and right minimal indices of any matrix polynomial is equal to the rank times the degree of the polynomial the index sum theorem turns out to be a key tool for obtaining a number of significant results on the possible sizes and degrees of l ifications and companion forms on the minimal index preservation properties of companion forms of arbitrary degree as well as on obstructions to the existence of structured companion forms for structured matrix polynomials of even degree this paper presents many new results blended together with results already known in the literature but extended here to the most general setting of matrix polynomials of arbitrary sizes and degrees over arbitrary fields therefore we have written the paper in an expository and self contained style that makes it accessible to a wide variety of readers

a dynamical system approach to soc models of zhang s type we discuss zhang s model of soc in the framework of hyperbolic dynamical systems with singularities the fractal structure of the invariant energy distribution correlation decay like phenomena and symbolic coding are discussed

a numerically stable structure preserving method for computing the eigenvalues of real hamiltonian or symplectic pencils a new method is presented for the numerical computation of the generalized eigenvalues of real hamiltonian or symplectic pencils and matrices the method is numerically backward stable and preserves the structure i e hamiltonian or symplectic in the case of a hamiltonian matrix the method is closely related to the square reduced method of van loan but in contrast to that method which may suffer from a loss of accuracy of order sqrt varepsilon where varepsilon is the machine precision the new method computes the eigenvalues to full possible accuracy

introduction to the sandpile model abstract this article is based on a talk given by one of us evi at the conference statphys taipei 1997 it overviews the exact results in the theory of the sandpile model and discusses briefly the yet unsolved problem of calculation of avalanche distribution exponents the key ingredients include the analogy with the critical reaction diffusion system the spanning tree representation of height configurations and the decomposition of the avalanche process into waves of topplings

theoretical studies of self organized criticality these notes are intended to provide a pedagogical introduction to the abelian sandpile model of self organized criticality and its related models the abelian group the algebra of particle addition operators the burning test for recurrent states equivalence to the spanning trees problem are described the exact solution of the directed version of the model in any dimension is explained the model s equivalence to scheidegger s model of river basins takayasu s aggregation model and the voter model is discussed for the undirected case the solution for one dimensional lattices and the bethe lattice is briefly described known results about the two dimensional case are summarized generalization to the abelian distributed processors model is discussed time dependent properties and the universality of critical behavior in sandpiles are briefly discussed i conclude by listing some still unsolved problems

structured strong linearizations from fiedler pencils with repetition i abstract in many applications the polynomial eigenvalue problem p x 0 arises with p being a structured matrix polynomial for example palindromic symmetric skew symmetric in order to solve a structured polynomial eigenvalue problem it is convenient to use strong linearizations with the same structure as p to ensure that the symmetries in the eigenvalues due to that structure are preserved in numerical computations in this paper we characterize all the pencils in the family of the fiedler pencils with repetition introduced by vologiannidis and antoniou 25 associated with a square matrix polynomial p that are block symmetric for every matrix polynomial p we show that this family of pencils is precisely the set of all fiedler pencils with repetition that are symmetric when p is when some generic nonsingularity conditions are satisfied these pencils are strong linearizations of p in particular our family strictly contains the standard basis for dl p a k dimensional vector space of symmetric pencils introduced by mackey mackey mehl and mehrmann 20

universality classes in isotropic abelian and non abelian sandpile models universality in isotropic abelian and non abelian sandpile models is examined using extensive numerical simulations to characterize the critical behavior we employ an extended set of critical exponents geometric features of the avalanches as well as scaling functions describing the time evolution of average quantities such as the area and size during the avalanche comparing between the abelian bak tang wiesenfeld model p bak c tang and k wiensenfeld phys rev lett 59 381 1987 and the non abelian models introduced by manna s s manna j phys a 24 l363 1991 and zhang y c zhang phys rev lett 63 470 1989 we find strong indications that each one of these models belongs to a distinct universality class

dynamical properties of the zhang model of self organized criticality critical exponents of the infinitely slowly driven zhang model of self organized criticality are computed for d52 and 3 with particular emphasis devoted to the various roughening exponents besides confirming recent estimates of some exponents new quantities are monitored and their critical exponents computed among other results it is shown that the three dimensional exponents do not coincide with the bak tang wiesenfeld phys rev lett 59 381 1987 phys rev a 38 364 1988 abelian model and that the dynamical exponent as computed from the correlation length and from the roughness of the energy profile do not necessarily coincide as is usually implicitly assumed an explanation for this is provided the possibility of comparing these results with those obtained from renormalization group arguments is also briefly addressed s1063 651x 98 06107 8 pacs number s 64 60 lx