moments of spectral functions monte carlo evaluation and verification the subject of the present study is the monte carlo path integral evaluation of the moments of spectral functions such moments can be computed by formal differentiation of certain estimating functionals that are infinitely differentiable against time whenever the potential function is arbitrarily smooth here i demonstrate that the numerical differentiation of the estimating functionals can be more successfully implemented by means of pseudospectral methods e g exact differentiation of a chebyshev polynomial interpolant which utilize information from the entire interval ensuremath ensuremath beta ensuremath hbar 2 ensuremath beta ensuremath hbar 2 the algorithmic detail that leads to robust numerical approximations is the fact that the path integral action and not the actual estimating functional are interpolated although the resulting approximation to the estimating functional is nonlinear the derivatives can be computed from it in a fast and stable way by contour integration in the complex plane with the help of the cauchy integral formula e g by lyness method an interesting aspect of the present development is that hamburger s conditions for a finite sequence of numbers to be a moment sequence provide the necessary and sufficient criteria for the computed data to be compatible with the existence of an inversion algorithm finally the issue of appearance of the sign problem in the computation of moments albeit in a milder form than for other quantities is addressed

an improved algorithm for the multidimensional moment constrained maximum entropy problem the maximum entropy principle is a versatile tool for evaluating smooth approximations of probability density functions with the least bias beyond specified constraints in the recent paper we introduced new computational framework for the moment constrained maximum entropy problem in a multidimensional domain and developed a simple numerical algorithm capable of computing maximum entropy problem in a two dimensional domain with moment constraints of order up to 4 here we design an improved numerical algorithm for computing the maximum entropy problem in a two and higher dimensional domain with higher order moment constraints the algorithm features multidimensional orthogonal polynomial basis in the dual space of lagrange multipliers to achieve numerical stability and rapid convergence of newton iterations the new algorithm is found to be capable of solving the maximum entropy problem in the two dimensional domain with moment constraints of order up to 8 in the three dimensional domain with moment constraints of order up to 6 and in the four dimensional domain with moment constraints of order up to 4 corresponding to the total number of moment constraints of 44 83 and 69 respectively the two and higher dimensional maximum entropy test problems in the current work are based upon long term statistics of numerical simulation of the real world geophysical model for wind stress driven oceanic currents such as the gulf stream and the kuroshio

lower bounds for the condition number of vandermonde matrices we derive lower bounds for the condition number of then n vandermonde matrixvn x in the cases where the node vectorxt x1 x2 xn has positive elements or real elements located symmetrically with respect to the origin the bounds obtained grow exponentially inn witho 2n ando 2n 2 respectively we also compute the optimal spectral condition numbers ofvn x for the two node configurations including the optimal nodes and compare them with the bounds obtained

hausdorff moment problem and fractional moments a simplified procedure abstract the purpose of this paper is the recovering of a probability density function with support 0 1 from the knowledge of its sequence of moments i e the classical hausdorff moment problem to avoid the well known ill conditioning firstly the moment curve is calculated from the assigned sequence of moments next the unknown density is approximated by maximum entropy maxent technique selecting some proper points on the moment curve exploiting convergence in entropy a simplified quick procedure is suggested to recover the approximate density an application to laplace transform inversion is illustrated

hausdorff moment problem and maximum entropy a unified approach hausdorff moment problem in the approach of maximum entropy is reconsidered some aspects as existence condition entropy convergence entropy decreasing stability are studied through a unified approach results known in literature and some new ones are illustrated

accurate solution of dense linear systems part i algorithms in rounding to nearest we investigate how extra precise accumulation of dot products can be used to solve ill conditioned linear systems accurately for a given p bit working precision extra precise evaluation of a dot product means that the products and summation are executed in 2p bit precision and that the final result is rounded into the p bit working precision denote by u 2 p the relative rounding error unit in a given working precision we treat two types of matrices first up to condition number u 1 and second up to condition number u 2 for both types of matrices we present two types of methods first for calculating an approximate solution and second for calculating rigorous error bounds for the solution together with the proof of non singularity of the matrix of the linear system in the first part of this paper we present algorithms using only rounding to nearest in part ii we use directed rounding to obtain better results all algorithms are given in executable matlab code and are available from my homepage

periodic van der pol equation with state dependent impulses abstract conditions are given for the existence of a solution of given period to the impulsively driven van der pol equation results are obtained for impulses of varying degrees of state dependence

a piecewise quadratic maximum entropy method for the statistical study of chaos abstract let the frobenius perron operator p s l 1 0 1 l 1 0 1 related to a nonsingular transformation s 0 1 0 1 have an invariant density f we propose a piecewise quadratic maximum entropy method for the numerical approximation of f the role of the partition of unity property of the quadratic functions for the numerical recovery of f has been depicted the proposed algorithm overcomes the ill conditioning shortage of the traditional maximum entropy method which only employs polynomials so that any number of moments can be used to increase the accuracy of the computed invariant density the convergence rate of the method is shown to be of order 3 numerical results are presented to justify the theoretical analysis of the method

lyapunov exponents and the natural invariant density determination of chaotic maps an iterative maximum entropy ansatz we apply the maximum entropy principle to construct the natural invariant density and the lyapunov exponent of one dimensional chaotic maps using a novel function reconstruction technique that is based on the solution of the hausdorff moment problem via maximizing shannon entropy we estimate the invariant density and the lyapunov exponent of nonlinear maps in one dimension from a knowledge of finite number of moments the accuracy and the stability of the algorithm are illustrated by comparing our results to a number of nonlinear maps for which the exact analytical results are available furthermore we also consider a very complex example for which no exact analytical result for the invariant density is available a comparison of our results to those available in the literature is also discussed

periodic duffing delay equations with state dependent impulses conditions are obtained for the duffing equation with delay and state dependent impulses to admit a periodic solution

vandermonde systems on gauss lobatto chebyshev nodes abstract this paper deals with vandermonde matrices v n whose nodes are the gauss lobatto chebyshev nodes also called extrema chebyshev nodes we give an analytic factorization and explicit formula for the entries of their inverse and explore its computational issues we also give asymptotic estimates of the frobenius norm of both v n and its inverse and present an explicit formula for the determinant of v n

lower bounds for the condition number of a real confluent vandermonde matrix lower bounds on the condition numberp vc of a real con uent vander monde matrix vc are established in terms of the dimension n or n and the largest absolute value among all nodes that deflne the con uent vander monde matrix or n and the interval that contains the nodes in particular it is proved that for any modest kmax the largest number of equal nodes p vc behaves no smaller than on 1 p 2 n and than on 1 p 2 2n if all nodes are nonnegative it is not clear whether those bounds are asymp totically sharp for modest kmax

condition numbers of hankel matrices for exponential weights abstract we obtain the rate of growth of the largest eigenvalues and euclidean condition numbers of the hankel matrices i t j k w 2 t d t j k 0 n for a general class of even exponential weights w 2 exp 2 q on an interval i as particular examples we discuss q x x on i r and q x d 2 x 2 on i d d

approximate solution of fredholm integral equations by the maximum entropy method an approximate means of solving fredholm integral equations by the maximum entropy method is developed the fredholm integral equation is converted to a generalized moment problem whose approximate solution by maximum entropy methods has been successfully implemented in a previous paper by mead and papanicolaou l r mead and n papanicolaou j math phys 25 2404 1984 several explicit examples are given of approximate maximum entropy solutions of fredholm integral equations of the first and second kinds and of the wiener hopf type both the weaknesses and strengths of the method are discussed

moment information for probability distributions without solving the moment problem ii main mass tails and shape approximation how much information does a small number of moments carry about the unknown distribution function is it possible to explicitly obtain from these moments some useful information e g about the support the modality the general shape or the tails of a distribution without going into a detailed numerical solution of the moment problem in this previous and subsequent papers clear and easy to implement answers will be given to some questions of this type first the question of how to distinguish between the main mass interval and the tail regions in the case we know only a number of moments of the target distribution function will be addressed the answer to this question is based on a version of the chebyshev stieltjes markov inequality which provides us with upper and lower moment based bounds for the target distribution then exploiting existing asymptotic results in the main mass region an explicit moment based approximation of the target probability density function is provided although the latter cannot be considered in general as a satisfactory solution it can always serve as an initial approximation in any iterative scheme for the numerical solution of the moment problem numerical results illustrating all the theoretical statements are also presented

the numerical rank of krylov matrices abstract in this paper we expose interesting features of krylov matrices and vandermonde matrices let s be a large symmetric matrix of order n let x be a real n vector and let k l denote the related n l krylov matrix the question considered in this paper is how the numerical rank of k l grows as l increases the key for answering this question lies in the close link between k l and the n l vandermonde matrix which is generated by the eigenvalues of s analysis of large vandermonde matrices shows that the numerical rank is expected to remain much smaller than l the proof is based on partition theorems and clustering theorems the basic tool is a new matrix equality the vandermonde pascal toeplitz equality the actual numerical rank of a vandermonde or krylov matrix depends on the distribution of the eigenvalues but often the rank is remarkable small numerical experiments illustrate these points the observation that the numerical rank of krylov matrices stays small has important practical consequences

the condition number of real vandermonde krylov and positive definite hankel matrices we show that the euclidean condition number of any positive definite hankel matrix of order n geq 3 may be bounded from below by gamma n 1 16n with gamma exp 4 cdot it catalan pi approx 3 210 and that this bound may be improved at most by a factor 8 gamma n similar estimates are given for the class of real vandermonde matrices the class of row scaled real vandermonde matrices and the class of krylov matrices with hermitian argument improved bounds are derived for the case where the abscissae or eigenvalues are included in a given real interval our findings confirm that all such matrices including for instance the famous hilbert matrix are ill conditioned already for moderate order as application we describe implications of our results for the numerical condition of various tasks in numerical analysis such as polynomial and rational i nterpolation at real nodes determination of real roots of polynomials computation of coefficients of orthogonal polynomials or the iterative solution of linear systems of equations

vandermonde matrices on integer nodes the rectangular case we consider a rectangular vandermonde matrix v on integer nodes using combinatorial identities we give an explicit cholesky factorization of the matrix b v rm t v a factorization of the pseudo inverse of v and asymptotic estimates of the extremal eigenvalues of b we also discuss the numerical properties of the proposed formulas

on the class of chebyshev approximation problems which arise in connection with a conjugate gradient type method we investigate approximation problems of the form mathop min limits p p 0 1 mathop max p lambda limits lambda in s p withp ranging over all polynomials of degree k ands any line segment of the complex plane problems of this type appear in connection with error bounds for the minimum residual algorithm for solving linear systems with coefficient matrixa ei di n wheren nh andd besides a few general results on p we present the explicit solution of p for segmentss which are parallel to the imaginary and symmetric to the real axis this result leads to an optimal semi iterative method for the class of matricesa i n wheren nt

determination of the distribution of total loss from the fractional moments of its exponential in the insurance and banking industries as in many branches of engineering in many instances a model can be proposed to describe either the accumulated losses or the accumulated damage or stress such models are of the compound type namely random sums of independent positive random variables and a reliable statistical model is available for both but computing the distribution of such random sums is a hard problem that has received much attention here we propose a technique based on the maximum entropy method which makes use of the fractional moments of the exponential of the accumulated loss which is available since for many models the laplace transform can be exactly determined here we provide an answer to the question of how to determine distribution of a total severity when the only information that we have consists of the laplace transform of the compound distribution our technique yields fairly good reconstructions from a low number of fractional moments

spectral properties of hankel matrices and numerical solutions of finite moment problems abstract after proving that any hankel matrix generated by moments of positive functions is conditioned essentially the same as the hilbert matrix of the same size we show a preconditioning technique i e a congruence transform of the original hankel matrix that drastically reduces its ill conditioning applications of this result to classical orthogonal polynomial sequences and to modified moment problems are given also we outline an efficient algorithm for the computation of the function f x w x exp p x where w x is positive and p x is a polynomial of degree n 1 from the knowledge of its first n moments

minimizing the condition number of a gram matrix the condition number of a gram matrix defined by a polynomial basis and a set of points is often used to measure the sensitivity of the least squares polynomial approximation given a polynomial basis we consider the problem of finding a set of points and or weights which minimizes the condition number of the gram matrix the objective function f in the minimization problem is nonconvex and nonsmooth we present an expression of the clarke generalized gradient of f and show that f is clarke regular and strongly semismooth moreover we develop a globally convergent smoothing method to solve the minimization problem by using the exponential smoothing function to illustrate applications of minimizing the condition number we report numerical results for the gram matrix defined by the weighted vandermonde like matrix for least squares approximation on an interval and for the gram matrix defined by an orthonormal set of real spherical harmonics for least squares approximation on the sphere

matrix methods for pade approximation numerical calculation of poles zeros and residues abstract a representation of the pade approximation of the z transform of a signal as a resolvent of a tridiagonal matrix j n is given several formulas for the poles zeros and residues of the pade approximation in terms of the matrix j n are proposed their numerical stability is tested and compared methods for computing forward and backward errors are presented

on the numerical condition of a generalized hankel eigenvalue problem the generalized eigenvalue problem widetilde h y lambda h y with h a hankel matrix and widetilde h the corresponding shifted hankel matrix occurs in number of applications such as the reconstruction of the shape of a polygon from its moments the determination of abscissa of quadrature formulas of poles of pade approximants or of the unknown powers of a sparse black box polynomial in computer algebra in many of these applications the entries of the hankel matrix are only known up to a certain precision we study the sensitivity of the nonlinear application mapping the vector of hankel entries to its generalized eigenvalues a basic tool in this study is a result on the condition number of vandermonde matrices with not necessarily real abscissas which are possibly row scaled

on the application of maximum entropy to the moments problem the maximum entropy approach to the solution of classical inverse problem of moments in which one seeks to reconstruct a function p x where x 0 from the values of a finite set n 1 of its moments is studied it is shown that for n 4 such a function always exists while for n 2 and n 3 the acceptable values of the moments are singled out analytically the paper extends to the general case where the results were previously bounded to the case n 2

periodic solution of differential equation with ϕ laplacian and state dependent impulses abstract sufficient conditions for the existence of at least one periodic solution of differential equation with laplacian z t f t z t z t for a e t 0 t r and state dependent impulses z t m i t z t t i t z t i 1 p are given here t 0 r r is an increasing homeomorphism r r 0 0 f 0 t r 2 r satisfies caratheodory conditions m i 0 t r r are continuous and i 0 t r 0 t are continuous for i 1 p p n z t lim s t z s lim s t z s the results are obtained via lower and upper functions method and schauder fixed point theorem

minimal condition number for positive definite hankel matrices using semidefinite programming abstract we present a semidefinite programming approach for computing optimally conditioned positive definite hankel matrices of order n unlike previous approaches our method is guaranteed to find an optimally conditioned positive definite hankel matrix within any desired tolerance since the condition number of such matrices grows exponentially with n this is a very good test problem for checking the numerical accuracy of semidefinite programming solvers our tests show that semidefinite programming solvers using fixed double precision arithmetic are not able to solve problems with n 30 moreover the accuracy of the results for 24 n 30 is questionable in order to accurately compute minimal condition number positive definite hankel matrices of higher order we use a mathematica 6 0 implementation of the sdpha solver that performs the numerical calculations in arbitrary precision arithmetic by using this code we have validated the results obtained by standard codes for n 24 and we have found optimally conditioned positive definite hankel matrices up to n 100

smallest eigenvalue of large hankel matrices at critical point comparing conjecture with parallelised computation abstract we propose a novel parallel numerical algorithm for calculating the smallest eigenvalues of highly ill conditioned hankel matrices it is based on the ldlt decomposition and involves finding a k k sub matrix of the inverse of the original n n hankel matrix h n 1 the computation involves extremely high precision arithmetic message passing interface and shared memory parallelisation we demonstrate that this approach achieves good scalability on a high performance computing cluster hpcc which constitutes a major improvement of the earlier approaches we use this method to study a family of hankel matrices generated by the weight w x e x supported on 0 and 0 such weight generates a hankel determinant a fundamental object in random matrix theory in the situation where 1 2 the smallest eigenvalue tends to 0 exponentially fast if 1 2 it is conjectured that the smallest eigenvalue tends to 0 algebraically with a precise exponent the algorithm run on the hpcc producing a fantastic match between the theoretical value of 2 and the numerical result

stieltjes moment problem via fractional moments stieltjes moment problem is considered to recover a probability density function from the knowledge of its infinite sequence of ordinary moments the approximate density is obtained through maximum entropy technique under the constraint of few fractional moments the latter are numerically obtained from the infinite sequence of ordinary moments and are chosen in such a way as to convey the maximum information content carried by the ordinary moments as a consequence a model with few parameters is obtained and intrinsic numerical instability is avoided it is proved that the approximate density is useful for calculating expected values and some other interesting probabilistic quantities

discrete probability distributions and moment problem numerical aspects discrete probability distributions with finite range generated by maximum entropy technique and constrained by first moments have been considered numerical aspects as stability of parameter s computation entropy variation spectral properties of involved hankel s matrices have been studied the ill conditioning of hankel s matrices suggests minimization methods like newton s should be unreliable as a high number of moments are kept while methods relying upon relative error estimate between assigned and calculated moments should be possible candidates

how bad are hankel matrices considered are hankel vandermonde and krylov basis matrices it is proved that for any real positive definite hankel matrix of order n its spectral condition number is bounded from below by 3 cdot 2 n 6 also proved is that the spectral condition number of a krylov basis matrix is bounded from below by 3 frac 1 2 cdot 2 frac n 2 3 for v v x 1 ldots x n a vandermonde matrix with arbitrary but pairwise distinct nodes x 1 ldots x n we show that mbox rm cond 2 v geq 2 n 2 n frac 1 2 if either x j leq 1 or x j geq 1 for all j then mbox rm cond 2 v geq 2 n 2

algebraic description of the finite stieltjes moment problem abstract the stieltjes problem of moments seeks for a nondecreasing positive distribution function on the semi axis 0 so that its moments match a given infinite sequence of positive real numbers m 0 m 1 in his seminal paper investigations on continued fractions published in 1894 stieltjes gave a complete solution including the conditions for the existence and uniqueness in relation to his main goal the convergence theory of continued fractions one can also reformulate the stieltjes problem of moments as looking for a sequence of positive distribution functions 1 2 where the nth distribution function has n points of increase and m 0 m 1 m 2 n 1 represent its first 2n moments i e as the sequence of the finite stieltjes moment problems this view can be linked to iterative solution of large linear algebraic systems providing that m 0 m 1 are moments of some linear self adjoint and coercive operator a on a hilbert space with respect to a given vector f the finite stieltjes moment problems determine the iterations of the conjugate gradient method applied for solving a u f and vice versa here the existence and uniqueness is guaranteed by the properties of the operator a reformulation for finite sequences matrices and finite vectors is obvious this fundamental link raises a question on how the solution of the finite stieltjes moment problem can be described purely algebraically this has motivated the presented exposition built upon ideas published previously by several authors since the description uses matrices of moments it is not intended for numerical computations

hamburger moment problem and maximum entropy on the existence conditions the existence of maximum entropy solution for the reduced hamburger moment problem is reconsidered existence conditions previously appeared in literature are revisited allowing an easy way to identifying the existence of maximum entropy solution the obtained results suggest that except for special sequences of moments unknown a priori the maximum entropy solution for the non symmetric reduced hamburger moment problem exists for practical purposes the replacing of the support r with a large enough finite interval finds a theoretical warranty the symmetric case may be formulated as follows once assigned the first 2m moments if maxent density does not exist conclusion drawn uniquely from numerical evidence maxent density with the first 2m 2 moments exists in such a case even if the first 2m moments are known we have to settle for a density which carries less information theoretical results are illustrated through some numerical examples

vandermonde systems on equidistant nodes in 0 1 accurate computation this paper deals with vandermonde matrices v whose nodes are the equidistant points in 0 1 we give an analytic factorization and explicit formula for the entries of their inverse and explore its computational issues we also give asymptotic estimates of the frobenius norm of both v and its inverse and show that a new representation of the floating point number system allows one to build an accurate algorithm for the interpolation problem on equidistant nodes in 0 1

semi infinite programming using high degree polynomial interpolants and semidefinite programming in a common formulation of semi infinite programs the infinite constraint set is a requirement that a function parametrized by the decision variables is nonnegative over an interval if this function is sufficiently closely approximable by a polynomial or a rational function then the semi infinite program can be reformulated as an equivalent semidefinite program which in turn can be solved with interior point methods very efficiently to high accuracy on the other hand solving this semidefinite program is challenging if the polynomials involved are of high degree due to numerical difficulties and bad scaling arising both from the polynomial approximations and from the fact that the semidefinite programming constraints coming from the sum of squares representation of nonnegative polynomials are badly scaled we combine polynomial function approximation techniques and polynomial programming to overcome these numerical difficulties using sum of squares interpolants specifically it is shown that the c

chaotic systems and maximum entropy formalism we show how the maximum entropy formalism can be applied in nonlinear chaotic systems in particular we show how one can use a few moments of the time evolution of a dynamical variable to infer the probability density and hence the lyapunov exponent

perturbation bounds for polynomials using two different elementary approaches we derive a global and a local perturbation theorem on polynomial zeros that significantly improve the results of ostrowski acta math 72 99 257 1940 elsner et al linear algebra appl 142 195 209 1990 a comparison of different perturbation bounds shows that our results are better in many cases than the similar local result of beauzamy can math bull 42 1 3 12 1999 using the matrix theoretical approach we also improve the backward stability result of edelman and murakami proceedings of the fifth siam conference on applied linear algebra siam philapdelphia 1994 math comput 64 210 763 1995

displacement structure approach to polynomial vandermonde and related matrices abstract we introduce a new class of what we call polynomial vandermonde like matrices this class generalizes the polynomial vandermonde matrices studied earlier by various authors who derived explicit inversion formulas and fast algorithms for inversion and for solving the associated linear systems a displacement structure approach allows us to carry over all these results to the wider class of polynomial vandermonde like matrices

additive preconditioning and aggregation in matrix computations we combine our novel svd free additive preconditioning with aggregation and other relevant techniques to facilitate the solution of a linear system of equations and other fundamental matrix computations our analysis and experiments show the power of our algorithms guide us in selecting most effective policies of preconditioning and aggregation and provide some new insights into these and related subjects compared to the popular svd based multiplicative preconditioners our additive preconditioners are generated more readily and for a much larger class of matrices furthermore they better preserve matrix structure and sparseness and have a wider range of applications e g they facilitate the solution of a consistent singular linear system of equations and of the eigenproblem

moment information and entropy evaluation for probability densities abstract how much information does the sequence of integer moments carry about the corresponding unknown absolutely continuous distribution we prove that a reliable evaluation of the corresponding shannon entropy can be done by exploiting some known theoretical results on the entropy convergence uniquely involving exact moments without solving the underlying moment problem all the procedure essentially rests on the solution of linear systems with nearly singular matrices and hence it requires both calculations in high precision and a pre conditioning technique numerical examples are provided to support the theoretical results

quadrature based vector fitting for discretized mathcal h _2 approximation vector fitting is a popular method of constructing rational approximants designed to fit given frequency response measurements the original method which we refer to as vf is based on a least squares fit to the measurements by a rational function using an iterative reallocation of the poles of the approximant we show that one can improve the performance of vf significantly by using a particular choice of frequency sampling points and properly weighting their contribution based on quadrature rules that connect the least squares objective with an mathcal h 2 error measure our modified approach designated here as quadvf helps recover the original transfer function with better global fidelity as measured with respect to the mathcal h 2 norm than the localized least squares approximation implicit in vf we extend the new framework also to incorporate derivative information leading to rational approximants that minimize system error with respect to a discrete sobolev norm we consider the

on reflexive modules and abelian groups the class of a reflexive modules is investigated where a is a bimodule over rings s and r conditions are studied under which this class is closed under taking direct products or direct sums respectively furthermore it is shown that if a is free over a suitable ring r and s endr a then for every right r module m the natural map m m m is the completion of m in a certain functorial topology applications are given in the case that a is a free abelian group

periodic lienard type delay equations with state dependent impulses abstract conditions are obtained for lienard type equations with delay and state dependent impulses to admit an absolutely continuous periodic solution with first derivative of bounded variation and consequently with lebesgue integrable second derivative the results are applied to josephson s equation and the nonconservative forced pendulum equation

regularized nonlinear acceleration we describe a convergence acceleration technique for generic optimization problems our scheme computes estimates of the optimum from a nonlinear average of the iterates produced by any optimization method the weights in this average are computed via a simple linear system whose solution can be updated online this acceleration scheme runs in parallel to the base algorithm providing improved estimates of the solution on the fly while the original optimization method is running numerical experiments are detailed on classical classification problems

a fast algorithm for generalized hankel matrices arising in finite moment problems abstract consider an n n lower triangular matrix l whose i 1 st row is defined by the coefficients of the real polynomial p i x of degree i such that p i x is a set of orthogonal polynomials satisfying a standard three term recurrence relation if h is an n n real hankel matrix with nonsingular leading principal submatrices then h lhl t will be referred to as a strongly nonsingular hankel matrix with respect to the orthogonal polynomial basis p i x in this paper we develop an efficient o n 2 algorithm for the solution of a system of linear equations with a real symmetric coefficient matrix h which is a hankel matrix with respect to a suitable orthogonal polynomial basis this leads to an efficient method for computing polynomial approximations of an unknown function given its modified moments

structured low rank decomposition of multivariate hankel matrices abstract we present a new algorithm for the decomposition of a multivariate hankel matrix h as a sum of hankel matrices of small rank this decomposition corresponds to the decomposition of its symbol as a sum of polynomial exponential series by exploiting the properties of the associated artinian gorenstein quotient algebra a we obtain new ways to compute the frequencies and the weights of the decomposition from generalized eigenvectors of sub matrices of h the new method is a multivariate generalization of the so called pencil method for solving prony type problems we analyse its numerical behaviour in the presence of noisy input moments we describe rescaling techniques and newton iterations which improve the numerical quality of the reconstruction and show their impact for correcting errors on input moments

on the accuracy of solving confluent prony systems in this paper we consider several nonlinear systems of algebraic equations which can be called prony type these systems arise in various reconstruction problems in several branches of theoretical and applied mathematics such as frequency estimation and nonlinear fourier inversion consequently the question of stability of solution with respect to errors in the right hand side becomes critical for the success of any particular application we investigate the question of maximal possible accuracy of solving prony type systems putting stress on the local behavior which approximates situations with low absolute measurement error the accuracy estimates are formulated in very simple geometric terms shedding some light on the structure of the problem numerical tests suggest that global solution techniques such as prony s algorithm and the esprit method are suboptimal when compared to this theoretical best local behavior

a fast iterative method for determining the stability of a polynomial we present an iterative numerical method for solving two classical stability problems for a polynomial p x of degree n the routh hurwitz and the schur cohn problems this new method relies on the construction of a polynomial sequence p k x k in bf n p 0 x p x such that p k x quadratically converges to x 1 p x 1 n p whenever the starting polynomial p x has p zeros with positive real parts and n p zeros with negative real parts by combining some new results on structured matrices with the fast polynomial arithmetic we prove that the coefficients of p k x can be computed starting from the coefficients of p k 1 x at the computational cost of o n log 2 n arithmetical operations moreover by means of numerical experiments we show that the o n log n bit precision of computations suffices to support the stated computational properties in this way apart from a logarithmic factor we arrive at the current best upper bound of o n 3 log 4 n for the bit complexity of the mentioned stability problems

periodic boundary value problem for first order differential equations with impulses at variable times in this paper we show the validity of the method of upper and lower solutions to obtain an existence result for a periodic boundary value problem of first order impulsive differential equations at variable times

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

the hausdorff entropic moment problem our aim in this paper is twofold first to find the necessary and sufficient conditions to be satisfied by a given sequence of real numbers n n 0 to represent the entropic moments 0 a x ndx of an unknown non negative decreasing and differentiable a e density function x with a finite interval support these moments are called entropic moments because they are closely connected with various information entropies renyi tsallis second we outline an efficient method for the reconstruction of the density function from the knowledge of its first n entropic moments

maximum entropy and the problem of moments a stable algorithm we present a technique for entropy optimization to calculate a distribution from its moments the technique is based upon maximizing a discretized form of the shannon entropy functional by mapping the problem onto a dual space where an optimal solution can be constructed iteratively we demonstrate the performance and stability of our algorithm with several tests on numerically difficult functions we then consider an electronic structure application the electronic density of states of amorphous silica and study the convergence of the fermi level with increasing number of moments

maxentropic solution of fractional moment problems the problem of reconstructing a function from the knowledge of a few of its moments is a special case of the problem inverting an integral transform when there are finitely many data points the classical method of maximum entropy and the method of maximum entropy in the mean are well suited to attack these problems what is interesting is that when we are free to choose the data points the choice can be made to improve the quality of the reconstruction when the method of maximum entropy is used too in this note we review how to apply these methods and compare them in a couple of examples

detecting rigid convexity of bivariate polynomials abstract given a polynomial x r n p x in n 2 variables a symbolic numerical algorithm is first described for detecting whether the connected component of the plane sublevel set p x p x 0 containing the origin is rigidly convex or equivalently whether it has a linear matrix inequality lmi representation or equivalently if polynomial p x is hyperbolic with respect to the origin the problem boils down to checking whether a univariate polynomial matrix is positive semidefinite an optimization problem that can be solved with eigenvalue decomposition when the variety c x p x 0 is an algebraic curve of genus zero a second algorithm based on bezoutians is proposed to detect whether p has an lmi representation and to build such a representation from a rational parametrization of c finally some extensions to positive genus curves and to the case n 2 are mentioned

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

maximum entropy in the problem of moments the maximum entropy approach to the solution of underdetermined inverse problems is studied in detail in the context of the classical moment problem in important special cases such as the hausdorff moment problem we establish necessary and sufficient conditions for the existence of a maximum entropy solution and examine the convergence of the resulting sequence of approximations a number of explicit illustrations are presented in addition to some elementary examples we analyze the maximum entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems we also briefly indicate possible applications to the lee yang theory of ising models to the summation of divergent series and so on the general conclusion is that maximum entropy provides a valuable approximation scheme a serious competitor of traditional pade like procedures

maximum entropy approximation for lyapunov exponents of chaotic maps let s 0 1 0 1 be a nonsingular chaotic map that preserves an integrable density f that describes the statistics of the orbits in this article we use the maximum entropy approach to approximate the density f and the corresponding lyapunov exponent

on the inversion of the vandermonde matrix abstract the inversion of the vandermonde matrix has received much attention for its role in the solution of some problems of numerical analysis and control theory this work deals with the problem of getting an explicit formula for the generic element of the inverse we derive two algorithms in o n2 and o n3 and compare them with the parker traub and the bjorck pereyra algorithms

entropy and density approximation from laplace transforms how much information does the laplace transforms on the real line carry about an unknown absolutely continuous distribution if we measure that information by the boltzmann gibbs shannon entropy the original question becomes how to determine the information in a probability density from the given values of its laplace transform we prove that a reliable evaluation both of the entropy and density can be done by exploiting some theoretical results about entropy convergence that involve only finitely many real values of the laplace transform without having to invert the laplace transform we provide a bound for the approximation error of in terms of the kullback leibler distance and a method for calculating the density to arbitrary accuracy

on the decay rate of hankel singular values and related issues this paper investigates the decay rate of the hankel singular values of linear dynamical systems this issue is of considerable interest in model reduction by means of balanced truncation for instance since the sum of the neglected singular values provides an upper bound for an appropriate norm of the approximation error the decay rate involves a new set of invariants associated with a linear system which are obtained by evaluating a modified transfer function at the poles of the system these considerations are equivalent to studying the decay rate of the eigenvalues of the product of the solutions of two lyapunov equations the related problem of determining the decay rate of the eigenvalues of the solution to one lyapunov equation will also be addressed very often these eigenvalues like the hankel singular values are rapidly decaying this fact has motivated the development of several algorithms for computing low rank approximate solutions to lyapunov equations however until now conditions assuring rapid decay have not been well understood such conditions are derived here by relating the solution to a numerically low rank cauchy matrix determined by the poles of the system bounds explaining rapid decay rates are obtained under some mild conditions

rectangular vandermonde matrices on chebyshev nodes abstract we consider a rectangular vandermonde matrix v on chebyshev nodes using combinatorial identities we give an explicit factorization of the matrix v a factorization of thepseudo inverse of v and prove that the condition number 2 v does not depend onthe dimension of nodes set we also discuss the numerical properties of the proposed formulas

bounds on elementary symmetric functions abstract tight bounds on an elementary symmetric function are established under the assumption that the values of the elementary symmetric functions of lower orders are given the explicit form of the inverse hankel moment matrix leads to inequalities for moments and for elementary symmetric polynomials

convergence of best entropy estimates given a finite number of moments of an unknown density on a finite measure space the best entropy estimate that nonnegative density x with the given moments which minimizes the boltzmann shannon entropy i x x log x is considered a direct proof is given that i has the kadec property in l if yn converges weakly to 37 and i yn converges to i 37 then yn converges to 37 in norm as a corollary it is obtained that as the number of given moments increases the best entropy estimates converge in li norm to the best entropy estimate of the limiting problem which is simply in the determined case furthermore for classical moment problems on intervals with strictly positive and sufficiently smooth error bounds and uniform convergence are actually obtained

a marginal fractional moments based strategy for points selection in seismic response analysis of nonlinear structures with uncertain parameters abstract the present paper proposes a new strategy for selecting representative points in the probability density evolution method pdem to conduct stochastic seismic response analysis of nonlinear structures with uncertain parameters in pdem the strategy for selecting representative points in random variate space is of critical importance to the efficiency and accuracy the proposed strategy is established based on the marginal fractional moments of input random variables which can be evaluated both analytically and numerically without difficulty before performing stochastic analysis in this strategy an optimization problem is actually involved first the initial points are generated by a low discrepancy sequence and the corresponding assigned probabilities can be computed accordingly then the initial points are rearranged to minimize the index which is adopted as the maximum relative error between the estimated marginal moments and the exact ones the rearranged points are accepted as the representative points in pdem when the index reaches the prescribed tolerance numerical example is investigated showing that the proposed strategy can achieve the good tradeoff of efficiency and accuracy in pdem for seismic response analysis of structures with uncertain parameters

fractional calculus approach to the statistical characterization of random variables and vectors fractional moments have been investigated by many authors to represent the density of univariate and bivariate random variables in different contexts fractional moments are indeed important when the density of the random variable has inverse power law tails and consequently it lacks integer order moments in this paper starting from the mellin transform of the characteristic function and by fractional calculus method we present a new perspective on the statistics of random variables introducing the class of complex moments that include both integer and fractional moments we show that every random variable can be represented within this approach even if its integer moments diverge applications to the statistical characterization of raw data and in the representation of both random variables and vectors are provided showing that the good numerical convergence makes the proposed approach a good and reliable tool also for practical data analysis

convergence analysis for lasserre s measure based hierarchy of upper bounds for polynomial optimization we consider the problem of minimizing a continuous function f over a compact set mathbf k k we analyze a hierarchy of upper bounds proposed by lasserre siam j optim 21 3 864 885 2011 obtained by searching for an optimal probability density function h on mathbf k k which is a sum of squares of polynomials so that the expectation int mathbf k f x h x dx kf x h x dx is minimized we show that the rate of convergence is no worse than o 1 sqrt r o 1 r where 2r is the degree bound on the density function this analysis applies to the case when f is lipschitz continuous and mathbf k k is a full dimensional compact set satisfying some boundary condition which is satisfied e g for convex bodies the rth upper bound in the hierarchy may be computed using semidefinite programming if f is a polynomial of degree d and if all moments of order up to 2r d 2r d of the lebesgue measure on mathbf k k are known which holds for example if mathbf k k is a simplex hypercube or a euclidean ball

eigenvalue clustering control energy and logarithmic capacity abstract we prove two bounds showing that if the eigenvalues of a matrix are clustered in a region of the complex plane then the corresponding discrete time linear system requires significant energy to control a curious feature of one of our bounds is that the dependence on the region is via its logarithmic capacity which is a measure of how well a unit of mass may be spread out over the region to minimize a logarithmic potential

a maximum entropy method based on piecewise linear functions for the recovery of a stationary density of interval mappings let s 0 1 0 1 be a nonsingular transformation such that the corresponding frobenius perron operator p s l 1 0 1 l 1 0 1 has a stationary density f we propose a maximum entropy method based on piecewise linear functions for the numerical recovery of f an advantage of this new approximation approach over the maximum entropy method based on polynomial basis functions is that the system of nonlinear equations can be solved efficiently because when we apply newton s method the jacobian matrices are positive definite and tri diagonal the numerical experiments show that the new maximum entropy method is more accurate than the markov finite approximation method which also uses piecewise linear functions provided that the involved moments are known this is supported by the convergence rate analysis of the method

the maximum entropy method applied to stationary density computation abstract the maximum entropy method maxent is widely used in the context of the moment problem which appears naturally in many branches of physics and engineering it is used to numerically recover the density with least bias from finitely many known moments we introduce the basic idea behind this method and apply this method to approximating fixed densities of markov operators in stochastic analysis and frobenius perron operators in ergodic theory of chaotic dynamics

abstract impulsive differential equations without predefined time impulses abstract we introduce and study a general class of abstract impulsive differential equations with non instantaneous impulses at non predefined times we study the existence uniqueness and qualitative properties of different types of solutions in the last section some examples are presented

hausdorff moment problem via fractional moments we outline an efficient method for the reconstruction of a probability density function from the knowledge of its infinite sequence of ordinary moments the approximate density is obtained resorting to maximum entropy technique under the constraint of some fractional moments the latter ones are obtained explicitly in terms of the infinite sequence of given ordinary moments it is proved that the approximate density converges in entropy to underlying density so that it turns out to be useful for calculating expected values

reconstruction of conditional expectations from product moments with applications in this paper it is shown under conditions associated with the moment problem that a sequence of product moments uniquely determines a conditional expectation of two random variables then a numerical procedure is derived to reconstruct a conditional expectation in terms of a sequence of its product moments

positive definite hankel matrices of minimal condition abstract real positive definite hankel matrices h n have spectral condition numbers which are exponentially increasing with n this paper attempts to characterize those matrices h n in this class which are minimally conditioned and to describe some of their properties we also compute h n explicitly for n 16

entropy convergence in stieltjes and hamburger moment problem in the classical stieltjes and hamburger moment problem the sequence of maximum entropy approximants whose first m moments are equal to given ones is considered it is proved that whenever an infinite moment problem is determined then maximum entropy approximants converge in entropy to the function characterized by given moments entropy convergence is proved by using exclusively existence and uniqueness conditions

smallest eigenvalues of hankel matrices for exponential weights abstract we obtain the rate of decay of the smallest eigenvalue of the hankel matrices i t j k w 2 t dt j k 0 n for a general class of even exponential weights w2 exp 2q on an interval i more precise asymptotics for more special weights have been obtained by many authors