the eigenvalues of second order spectral differentiation matrices the eigenvalues of the pseudospectral second derivative matrix with homogeneous dirichlet boundary conditions are important in many applications of spectral methods this paper investigates some of their properties numerical results show that a certain fraction of the eigenvalues approximate the eigenvalues of the continuous operator very accurately but the errors in the remaining ones are large it is demonstrated that the inaccurate eigenvalues correspond to those eigenfunctions of the continuous operator that cannot be resolved by polynomial interpolation in the spectral grid in particular it is proved that 7r points on average per wavelength are sufficient for successful interpolation of the eigenfunctions of the continuous operator in a chebyshev distribution of nodes and six points per wavelength for a uniform distribution these results are in agreement with the observed fractions of accurate eigenvalues by using the characteristic polynomial a bound on the spectral radius of the differentiation matrix is derived that is accurate to 2 or better the effect of filtering on the eigenvalues is studied numerically

computation of rational interpolants with prescribed poles a constructive proof for existence and unicity of the rational r belonging to gm v m 0 n 0 having prescribed n poles and interpolating m 1 hermite data is given it is based upon explicit computation of the confluent cauchy vandermonde determinant in terms of the nodes and the poles an algorithm to compute r z is presented it calculates this value from a triangular field of values of rational interpolants of lowest possible orders the algorithm is based upon a neville aitken interpolation formula and has arithmetical complexity 0 l l max m 1 n

some schur complement identities and applications to matrix extrapolation methods the schur complement has many applications in linear algebra and in numerical analysis schur complement identities have been used for implementing some scalar and vector extrapolation methods in this paper we define some matrix sequence transformations and give two new schur complement identities that will be used to derive algorithms for implementing these transformations

a block pseudospectral method for maxwell s equations a block pseudospectral bps method is proposed as a new way to couple pseudospectral discretizations across interfaces in computations for a linear hyperbolic system the coupling is achieved via discretized derivative matching conditions obtained from the system compared to the standard technique of imposing compatibility conditions based on characteristics of the system the bps method offers better stability and accuracy especially in the case where equation coefficients are discontinuous computational examples for maxwell s equations in nonhomogeneous media demonstrate that bps retains high accuracy over times that are orders of magnitude larger than those for not only low order methods such as yee s but also high order methods such as characteristic based spectral elements

the principal rank characteristic sequence of a real symmetric matrix abstract given a vector u r 2 n the principal minor assignment problem asks when is there an n n matrix having its 2 n principal minors given by u this paper explores the following related problem given a sequence r 0 r 1 r n of 0s and 1s does there exist an n n real symmetric matrix that has a principal submatrix of rank k if and only if r k 1 for all 0 k n certain conditions are shown to be necessary in order for this question to have an affirmative answer several families of matrices are constructed to attain certain classes of sequences the problem is solved completely for n 6 and for 7 n 10 in the case of sequences beginning with 010

high order fdtd methods via derivative matching for maxwell s equations with material interfaces this paper introduces a series of novel hierarchical implicit derivative matching methods to restore the accuracy of high order finite difference time domain fdtd schemes of computational electromagnetics cem with material interfaces in one 1d and two spatial dimensions 2d by making use of fictitious points systematic approaches are proposed to locally enforce the physical jump conditions at material interfaces in a preprocessing stage to arbitrarily high orders of accuracy in principle while often limited by numerical instability orders up to 16 and 12 are achieved respectively in 1d and 2d detailed stability analyses are presented for the present approach to examine the upper limit in constructing embedded fdtd methods as natural generalizations of the high order fdtd schemes the proposed derivative matching methods automatically reduce to the standard fdtd schemes when the material interfaces are absent an interesting feature of the present approach is that it encompasses a variety of schemes of different orders in a single code another feature of the present approach is that it can be robustly implemented with other high accuracy time domain approaches such as the multiresolution time domain method and the local spectral time domain method to cope with material interfaces numerical experiments on both 1d and 2d problems are carried out to test the convergence examine the stability access the efficiency and explore the limitation of the proposed methods it is found that operating at their best capacity the proposed high order schemes could be over 2000 times more efficient than their fourth order versions in 2d in conclusion the present work indicates that the proposed hierarchical derivative matching methods might lead to practical high order schemes for numerical solution of time domain maxwell s equations with material interfaces

ect b splines defined by generalized divided differences ect spline curves are generated from different local ect systems via connection matrices if they are nonsingular lower triangular and totally positive there is a basis of the space of ect splines consisting of functions having minimal compact supports normalized either to form a nonnegative partition of unity or to have integral one in this paper such ect b splines are defined by generalized divided differences this definition reduces to the classical one in case of a schoenberg space under suitable assumptions it leads to a recursive method for computing the ect b splines that reduces to the de boor mansion cox recursion in case of ordinary polynomial splines and to lyche s recursion in case of tchebycheff splines muhlbach and tang calculation of ect b splines and of ect spline curves recursively in preparation there is an ect spline space naturally adjoint to every ect spline space we also construct b splines via generalized divided differences for this space and study relations between the two adjoint spaces

linear extrapolation by rational functions exponentials and logarithmic functions abstract in this paper linear extrapolation by rational functions with given poles is considered from an arithmetical point of view it is shown that the classical interpolation algorithms of lagrange neville aitken and newton which are well known for polynomial interpolation can be extended in a natural way to this problem yielding recursive methods of nearly the same complexity the proofs are based upon explicit representations of generalized vandermonde determinants which are calculated by the elimination method combined with analytical considerations as an application a regularity criterion for certain linear sequence transformations is given also by the same method simplified recurrence relations for linear extrapolation by exponentials and logarithmic functions at special knots are derived

interpolation by cauchy vandermonde systems and applications abstract cauchy vandermonde systems consist of rational functions with prescribed poles they are complex ect systems allowing hermite interpolation for any dimension of the basic space a survey of interpolation procedures using cv systems is given some equipped with short new proofs which generalize the well known formulas of lagrange neville aitken and newton for interpolation by algebraic polynomials the arithmetical complexitiy is o n 2 for n hermite data also inversion formulas for the cauchy vandermonde matrix are surveyed moreover a new algorithm solving the system of n linear cauchy vandermonde equations for multiple nodes and multiple poles recursively is given which does not require additional partial fraction decompositions as an application construction of rational b splines with prescribed poles is discussed

second order spectral differentiation matrices the second order spectral differentiation matrix is required when spectral methods are used to solve second order differential equations for methods based on polynomials the precise form of this matrix will depend on the method used in each case however the eigenvalue with largest magnitude grows as o n 4 which has implications for the stability when explicit methods are used to solve time dependent problems this paper investigates the relationship between the different matrices and shows that they are related by a rank two update furthermore it is shown that the matrix can be chosen so that the magnitude of the largest eigenvalue grows as o n 2 rather than o n 4 and the properties of the underlying polynomials in this case are discussed

the pseudospectral method comparisons with finite differences for the elastic wave equation the pseudospectral or fourier method has been used recently by several investigators for forward seismic modeling the method is introduced here in two different ways as a limit of finite differences of increasing orders and by trigonometric interpolation an argument based on spectral analysis of a model equation shows that the pseudospectral method for the accuracies and integration times typical of forward elastic seismic modeling may require in each space dimension as little as a quarter the number of grid points compared to a fourth order finite difference scheme and one sixteenth the number of points as a second order finite difference scheme for the total number of points in two dimensions these factors become 1 16 and 1 256 respectively in three dimensions they become 1 64 and 1 4 096 respectively in a series of test calculations on the two dimensional elastic wave equation only minor degradations are found in cases with variable coefficients and discontinuous interfaces

le probleme de lehmer en dimension superieure we study a higher dimensional lehmer problem or alternatively the lehmer problem for a power of the multiplicative group more precisely if 1 n are multiplicatively independent algebraic numbers we provide a lower bound for the product of the heights of the i s in terms of the degree d of the number field generated by the i s this enables us to study the successive minima for the height function in a given number field our bound is a generalisation of an earlier result of dobrowolski and is best possible up to a power of log d this in particular shows that the lehmer problem is true for number fields having a small galois group

intersection de sous groupes et de sous varietes i we study the intersection of a subvariety x of an abelian variety a over open image in new window with the union of all the algebraic subgroups of a of given dimension d our main result states that if we remove a suitable exceptional subset from x and if d is small enough then the intersection enjoys a northcott like property the points of bounded height on it form a finite set the condition on d involves only the dimension of x and the structure of a up to isogenies we show how it can be weakened if we assume certain conjectures in the direction of an abelian version of lehmer s problem the theorem is especially meaningful when x is a curve since it is then possible to bound the height and hence to prove finiteness of the set under consideration this generalises the result of e viada on powers of elliptic curves and is analogous to work of e bombieri d masser and u zannier on tori whose general strategy we follow

can sylvester s determinantal identity equivalently muir s law of extensible minors be generalized abstract we show that sylvester s classical determinantal identity is equivalent to its generalizations given by the muhlbach gasca lopez carmona ramirez identity and the beckermann muhlbach identity we also show that the generating extension principles associated with each of these three identities muir s law of extensible minors muhlbach s extension principle and the extension principle given in camargo 2013 3 are equivalent and we use this fact to conclude that all these six statements and also jacobi s identity on the minors of the adjugate and cayley s law of complementaries are actually equivalent these findings lead naturally to enquire whether there would exist a generalization of sylvester s identity or an extension principle which could not be derived from sylvester s identity itself

elimination technique from extrapolation to totally positive matrices and cagd in this survey we will show some connections between several mathematical problems such as extrapolation linear systems totally positive matrices and computer aided geometric design with elimination techniques as the common tool to deal with all of them

an instability phenomenon in spectral methods the eigenvalues of chebyshev and legendre spectral differentiation matrices which determine the allowable time step in an explicit time integration are extraordinarily sensitive to rounding errors and other perturbations on a grid of n points per space dimension machine rounding leads to errors in the eigenvalues of size o n 2 this phenomenon may lead to inconsistency between predicted and observed time step restrictions one consequence of it is that spectral differentiation by interpolation in legendre points which has a favorable o n 1 time step restriction for the model problem u t u x in theory is subject to an o n 2 restriction in practice the same effect occurs with chebyshev points for the model problem u t xu x another consequence is that a spectral calculation with a fixed time step may be stable in double precision but unstable in single precision we know of no other examples in numerical computation of this kind of precision dependent stability

intersecting subvarieties of abelian varieties with algebraic subgroups of complementary dimension in fact both authors stated stronger conjectures zilber conjectured a finiteness statement for subvarieties that are contained in an improper intersection of x with an algebraic subgroup of s pink was motivated by a more general conjecture in the context of mixed shimura varieties he also showed that conjecture 1 implies the celebrated mordell lang conjecture an important precursor was set in 1999 by bombieri masser and zannier 3 who proved the following result for an algebraic curve c embedded in the algebraic torus gm not contained in the translate of a proper algebraic subgroup and defined over q an algebraic closure of q the set of algebraic points of c which are contained in the union of all algebraic subgroups of codimension at least 1 has bounded absolute weil height they then used their result to show that there are only finitely many points on c contained in the union of all algebraic subgroups of codimension at least 2

high order finite differences and the pseudospectral method on staggered grids finite difference approximations for the first derivative valid halfway between equidistant gridpoints are in general much more accurate than the corresponding approximations which are valid at gridpoints the pseudospectral fourier method can be viewed as the limit of finite difference approximations when the order of accuracy tends to infinity a fundamentally different and more accurate pseudospectral method is obtained from the halfway approximations this study derives these and related methods and discusses their accuracies

schur complements and statistics abstract in this paper we discuss various properties of matrices of the type s h ge 1 f which we call the schur complement of e in a e f g h the matrix e is assumed to be nonsingular when e is singular or rectangular we consider the generalized schur complements s h ge f where e is a generalized inverse of e a comprehensive account of results pertaining to the determinant the rank the inverse and generalized inverses of partitioned matrices and the inertia of a matrix is given both for schur complements and for generalized schur complements we survey the known results in a historical perspective and obtain several extensions numerous applications in numerical analysis and statistics are included the paper ends with an exhaustive bibliography of books and articles related to schur complements

a generalization of sylvester s identity on determinants and some applications abstract a generalization of sylvester s identity on determinants is proved by elimination techniques as an application some inequalities for determinants of totally positive or positive definite matrices are derived

fast accurate and small scale direct trajectory optimization using a gegenbauer transcription method this paper reports a novel direct gegenbauer ultraspherical transcription method gtm for solving continuous time optimal control oc problems ctocps with linear nonlinear dynamics and path constraints in elgindy et al 2012 1 we presented a gtm for solving nonlinear ctocps directly for the state and the control variables and the method was tailored to find the best path for an unmanned aerial vehicle mobilizing in a stationary risk environment this article extends the gtm to deal further with problems including higher order time derivatives of the states by solving the ctocp directly for the control u t and the highest order time derivative x n t n z the state vector and its derivatives up to the n 1 th order derivative can then be stably recovered by successive integration moreover we present our solution method for solving linear quadratic regulator lqr problems as we aim to cover a wider collection of ctocps with the concrete aim of comparing the efficiency of the current work with other classical discretization methods in the literature the proposed numerical scheme fully parameterizes the state and the control variables using gegenbauer expansion series for problems with various order time derivatives of the state variables arising in the cost function dynamical system or path terminal constraints the gtm seeks to fully parameterize the control variables and the highest order time derivatives of the state variables the time horizon is mapped onto the closed interval 0 1 the dynamical system characterized by differential equations is transformed into its integral formulation through direct integration the resulting problem on the finite interval is then transcribed into a nonlinear programming nlp problem through collocation at the gegenbauer gauss gg points the integral operations are approximated by optimal gegenbauer quadratures in a certain optimality sense the reduced nlp problem is solved in the gegenbauer spectral space and the state and the control variables are approximated on the entire finite horizon the proposed method achieves discrete solutions exhibiting exponential convergence using relatively small scale number of collocation points the advantages of the proposed direct gtm over other traditional discretization methods are shown through four well studied oc test examples the present work is a major breakthrough in the area of computational oc theory as it delivers significantly more accurate solutions using considerably smaller numbers of collocation points states and controls expansion terms moreover the gtm produces very small scale nlp problems which can be solved very quickly using the modern nlp software

an improved pseudospectral method for fluid dynamics boundary value problems abstract pseudospectral methods are commonly used to obtain accurate solutions to initial boundary value problems especially with second or higher order derivatives in space the corresponding differentiation matrices tend to have large spurious eigenvalues leading for example to severe time step restrictions in case of explicit time stepping methods we introduce here a new procedure for incorporating boundary conditions which reduces spurious eigenvalues by an order of magnitude or more also allowing more freedoms in the choice of grid point locations

global properties of pseudospectral methods abstract we apply polynomial interpolation methods both to the approximation of functions and to the numerical solutions of hyperbolic and elliptic partial differential equations the derivative matrix for a general sequence of collocation points is explicitly constructed we explore the effect of several factors on the performance of these methods we show that global methods cannot be interpreted in terms of local methods for example the accuracy of the approximation differs when large gradients of the function occur near the center of the region or in the vicinity of the boundary this difference does not depend on the density of the collocation points near the boundaries hence intuition based on finite difference methods can lead to false results

on deriving the inverse of a sum of matrices available expressions are reviewed and new ones derived for the inverse of the sum of two matrices one of them being nonsingular particular attention is given to text bf a text bf ubv 1 where textbf a is nonsingular and textbf u b and textbf v may be rectangular generalized inverses of textbf a textbf ubv are also considered several statistical applications are discussed

a generalized nonisospectral camassa holm equation and its multipeakon solutions abstract motivated by the paper of beals sattinger and szmigielski 2000 3 we propose an extension of the camassa holm equation which also admits the multipeakon solutions the novel aspect is that our approach is mainly based on classic determinant technique furthermore the proposed equation is shown to possess a nonisospectral lax pair

convergence properties of some block krylov subspace methods for multiple linear systems in the present paper we give some convergence results of the global minimal residual methods and the global orthogonal residual methods for multiple linear systems using the schur complement formulae and a new matrix product we give expressions of the approximate solutions and the corresponding residuals we also derive some useful relations between the norm of the residuals

other manifestations of the schur complement abstract many series and sequence transformations used in numerical analysis are defined as ratios of determinants they are implemented by recursive algorithms based on determinantal identities the aim of this paper is to study the connections of these transformations and algorithms with the schur complement of a matrix the schur s formula is extended to the vector case thus providing the same treatment for vector sequence transformations and the corresponding recursive algorithms thanks to these connections particular rules for avoiding division by zero or numerical instability are obtained for these algorithms some fixed point methods and continued fractions also fit in this framework

the approximation of a totally positive band matrix by a strictly banded totally positive one every nonsingular totally positive m banded matrix is shown to be the product of m totally positive one banded matrices and therefore the limit of strictly m banded totally positive matrices this result is then extended to bi infinite m banded totally positive matrices with linearly independent rows and columns in the process such matrices are shown to possess at least one diagonal whose principal sections are all nonzero as a consequence such matrices are seen to be approximable by strictly m banded totally positive ones

a general determinantal identity of sylvester type and some applications abstract a general determinantal identity of sylvester type over arbitrary commutative fields is derived while its proof is rather short and conceptually simpler than earlier attempts the result contains sylvester s classical determinantal identity and more recent extensions as well as some old and some new determinantal identities an application to the e algorithm is added

matrix extrapolation algorithms abstract extrapolation algorithms are used for accelerating scalar or vector sequences they are also used for solving systems of linear and nonlinear equations these algorithms are expressed in terms of a ratio of two determinants as the e algorithm and the general recursive projection algorithm grpa in this paper we define the matrix extrapolation problem and we use the schur complement and the sylvester identity for solving this problem we give two transformations equivalent to the e algorithm and the grpa for the matrix case we give also a characterization of their kernels

finiteness of hilbert functions and bounds for castelnuovo mumford regularity of initial ideals bounds for the castelnuovo mumford regularity and hilbert coefficients are given in terms of the arithmetic degree if the ring is reduced or in terms of the defining degrees from this it follows that there exists only a finite number of hilbert functions associated with reduced algebras over an algebraically closed field with a given arithmetic degree and dimension a good bound is also given for the castelnuovo mumford regularity of initial ideals which depends neither on term orders nor on the coordinates and holds for any field

on extending determinantal identities abstract a new principle for extending determinantal identities is established which generalizes muir s classical law of extensible minors the proof makes use of general elimination strategies and of generalized schur complements this principle allows either the list of columns or the list of rows extending the corresponding list in the given identity to depend on the latter as applications of this technique karlin s and a generalization of sylvester s identity are derived

two composition methods for solving certain systems of linear equations this note is concerned with the following problem given a systema x b of linear equations and knowing that certains of its subsystemsa 1 x 1 b 1 a m x m b m can be solved uniquely what can be said about the regularity ofa and how to find the solutionx fromx 1 x m this question is of particular interest for establishing methods computing certain linear or quasilinear sequence transformations recursively 7 13 15 in dieser arbeit wird die folgende fragestellung behandelt wenn man wei da gewisse teilsystemea 1 x 1 b 1 a m x m b m eines linearen gleichungssystemsa x b eindeutig losbar sind was kann man uber die losbarkeit des systemsa x b sagen und wie kann man gegebenenfallsx ausx 1 x m berechnen die losung dieses problems erlaubt die rekursive berechnung gewisser linearer oder quasilinearer folgentransformationen 7 13 15

stability of pseudospectral and finite difference methods for variable coefficient problems it is shown that pseudospectral approximation to a special class of variable coefficient one dimensional wave equations is stable and convergent even though the wave speed changes sign within the domain computer experiments indicate similar results are valid for more general problems similarly computer results indicate that the leapfrog finite difference scheme is stable even though the wave speed changes sign within the domain however both schemes can be asymptotically unstable in time when a fixed spatial mesh is used

determinantal identities gauss schur cauchy sylvester kronecker jacobi binet laplace muir and cayley abstract we give a common concise derivation of some important determinantal identities attributed to the mathematicians in the title we also give a formal treatment of determinantal identities of the minors of a matrix

generation of finite difference formulas on arbitrarily spaced grids on etablit des recurrences simples pour le calcul des poids dans les formules aux differences finies compactes pour les derivees de tous ordres avec une precision d ordre arbitraire sur des grilles a une dimension d espacement arbitraire

on the computation of determinants arising in some bivariate rational interpolation problems abstract bivariate cauchy vandermonde determinants arise in some bivariate rational interpolation problems these determinants are easily computed by means of the technique of interpolation systems numer math 39 1 14 1982 when it is applied to prove the unisolvence of those interpolation problems the solution of the problem is given in a newton form

optimal gegenbauer quadrature over arbitrary integration nodes this paper treats definite integrations numerically using gegenbauer quadratures the novel numerical scheme introduces the idea of exploiting the strengths of the chebyshev legendre and gegenbauer polynomials through a unified approach and using a unique numerical quadrature in particular the developed numerical scheme employs the gegenbauer polynomials to achieve rapid rates of convergence of the quadrature for the small range of the spectral expansion terms for a large scale number of expansion terms the numerical quadrature has the advantage of converging to the optimal chebyshev and legendre quadratures in the l norm and l 2 norm respectively the key idea is to construct the gegenbauer quadrature through discretizations at some optimal sets of points of the gegenbauer gauss gg type in a certain optimality sense we show that the gegenbauer polynomial expansions can produce higher order approximations to the definite integrals 1 x if x dx of a smooth function f x c 1 1 for the small range by minimizing the quadrature error at each integration point x i through a pointwise approach the developed gegenbauer quadrature can be applied for approximating integrals with any arbitrary sets of integration nodes exact integrations are obtained for polynomials of any arbitrary degree n if the number of columns in the developed gegenbauer integration matrix gim is greater than or equal to n the error formula for the gegenbauer quadrature is derived moreover a study on the error bounds and the convergence rate shows that the optimal gegenbauer quadrature exhibits very rapid convergence rates faster than any finite power of the number of gegenbauer expansion terms two efficient computational algorithms are presented for optimally constructing the gegenbauer quadrature we illustrate the high order approximations of the optimal gegenbauer quadrature through extensive numerical experiments including comparisons with conventional chebyshev legendre and gegenbauer polynomial expansion methods the present method is broadly applicable and represents a strong addition to the arsenal of numerical quadrature methods

on interpolation by rational functions with prescribed poles with applications to multivariate interpolation abstract this paper is concerned with interpolation in the sense of hermite by certain rational functions of one or several complex variables in the univariate setting the interpolants are generalized polynomials of a cauchy vandermonde space whereas in the multivariate setting the interpolants are elements of suitable subspaces of tensor products of cauchy vandermonde spaces a newton type algorithm is given computing an interpolating univariate rational function with prescribed poles with no more than o m 2 arithmetical operations where m is the number of nodes it is proved that the generalized divided differences are analytic functions of the nodes if the function to be interpolated is analytic the algorithm will be extended to the multivariate setting for subsets of grids possessing the rectangular property and for certain subspaces of a tensor product of cauchy vandermonde spaces an algorithm computing an interpolating rational function of two variables is given whose complexity is o m 2 n n 2 m where m and n are the numbers of interpolation points in the x and y direction respectively