a fast and accurate algorithm for solving bernstein vandermonde linear systems abstract a fast and accurate algorithm for solving a bernstein vandermonde linear system is presented the algorithm is derived by using results related to the bidiagonal decomposition of the inverse of a totally positive matrix by means of neville elimination the use of explicit expressions for the determinants involved in the process serves to make the algorithm both fast and accurate

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

error analysis of the bjo rck pereyra algorithms for solving vandermonde systems a forward error analysis is presented for the bjorck pereyra algorithms used for solving vandermonde systems of equations this analysis applies to the case where the points defining the vandermonde matrix are nonnegative and are arranged in increasing order it is shown that for a particular class of vandermonde problems the error bound obtained depends on the dimensionn and on the machine precision only being independent of the condition number of the coefficient matrix by comparing appropriate condition numbers for the vandermonde problem with the forward error bounds it is shown that the bjorck pereyra algorithms introduce no more uncertainty into the numerical solution than is caused simply by storing the right hand side vector on the computer a technique for computing running a posteriori error bounds is derived several numerical experiments are presented and it is observed that the ordering of the points can greatly affect the solution accuracy

spectral methods for nonstationary spatial processes summary we propose a nonstationary periodogram and various parametric approaches for estimating the spectral density of a nonstationary spatial process we also study the asymptotic properties of the proposed estimators via shrinking asymptotics assuming the distance between neighbouring observations tends to zero as the size of the observation region grows without bound with this type of asymptotic model we can uniquely determine the spectral density avoiding the aliasing problem we also present a new class of nonstationary processes based on a convolution of local stationary processes this model has the advantage that the model is simultaneously defined everywhere unlike moving window approaches but it retains the attractive property that locally in small regions it behaves like a stationary spatial process applications include the spatial analysis and modelling of air pollution data provided by the us environmental protection agency

accurate bidiagonal decomposition of totally positive cauchy vandermonde matrices and applications abstract cauchy vandermonde matrices play a fundamental role in rational interpolation theory and in other fields when all their corresponding nodes are different and positive and all poles are different and negative and follow adequate orderings these matrices are totally positive in this paper we provide fast algorithms for computing bidiagonal factorizations of these matrices and their inverses with high relative accuracy these algorithms can be used to solve with high relative accuracy other algebraic problems such as the computation of all singular values all eigenvalues or the solution of certain linear systems the error analysis of the algorithm for computing the bidiagonal factorization and the corresponding perturbation theory are also performed

a chebychev vandermonde solver abstract let t j n j 0 be a family of chebyshev polynomials for a finite interval a b let x k n k 0 be a set of distinct nodes on a b and define the transposed chebyshev vandermonde matrix w w kj n k j 0 w kj t j x k we describe a new fast algorithm for the solution of linear systems of equations of the form w a f and compare it with a fast solution scheme recently proposed by higham the latter scheme is a modification of the bjorck pereyra algorithm and the computed solution depends on the ordering of the nodes x k our algorithm is designed so that the computed solution is essentially independent of the ordering of the x k extensive computational experience indicates that our algorithm generally yields higher accuracy in the computed solution than higham s

a fast parallel bjorck pereyra type algorithm for solving cauchy linear equations abstract we propose a new fast o n 2 parallel algorithm for solving cauchy systems of linear equations we perform an a priori rounding error analysis and obtain componentwise bounds for the forward backward and residual errors these bounds indicate that for the class of totally positive cauchy matrices the new algorithm is forward and backward stable producing a remarkably high relative accuracy in particular hilbert linear systems often considered to be too ill conditioned to be attacked can be rapidly solved with high precision the results indicate a close resemblance between the numerical properties of cauchy matrices and the much studied vandermonde matrices in fact our proposed cauchy solver is an analog of the well known bjorck pereyra algorithm for vandermonde systems as a by product we obtain favorably backward error bounds for the original bjorck pereyra algorithm several computed examples illustrate a connection of high relative accuracy to the concepts of effective well conditioning and total positivity

accurate bidiagonal decomposition of totally positive h bernstein vandermonde matrices and applications abstract an algorithm for the computation of the bidiagonal decomposition of strictly totally positive h bernstein vandermonde matrices is presented these matrices are collocation matrices of h bernstein bases a generalization of the bernstein basis of the space of polynomials of degree less than or equal to n the algorithm is fast its computational complexity is o n 2 and has high relative accuracy the computation of this bidiagonal decomposition is used as the initial stage for the accurate and efficient solution of several linear algebra problems with these matrices linear system solving eigenvalue computation and inverse computation numerical experiments are also included

factorizations of cauchy vandermonde matrices abstract this paper analyzes the factorization of the inverse of a cauchy vandermonde matrix as a product of bidiagonal matrices factorizations of cauchy vandermonde matrices are also considered the results are applied to derive a fast algorithm for solving linear systems whose coefficient matrices are cauchy vandermonde matrices

an efficient implementation of a conformal mapping method based on the szego kernel an implementation based on iterative techniques of a method to compute the riemann mapping function is presented the method has been recently introduced by n kerzman and the author it expresses the szego kernel as the solution of an integral equation of the second kind it is shown how to treat symmetric regions the algorithm is tested on five examples the numerical results show that the method is competitive with respect to accuracy stability and efficiency

backup 2 center on interval graphs a graph g is called an interval graph if there exists a set of intervals corresponding to the vertex set of g and two vertices are adjacent to each other if and only if the two corresponding intervals are intersecting with each other in this paper we apply the reliability based backup 2 center modal proposed by wang wu and chao in which each server may fail with a given probability to interval graphs and present an o n time algorithm

solving np hard problems on graphs that are almost trees and an application to facility location problems description d une technique generale pour resoudre certains problemes de graphes np durs en un temps qui est exponentiel dans un parametre k defini comme le maximum sur tous les composants c non separables du graphe d un nombre de bords qui peut etre ajoute a l arbre pour produire c on donne un algorithme de temps de complexite o e 6r rh 27

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

diagonal pivoting for partially reconstructible cauchy like matrices with applications to toeplitz like linear equations and to boundary rational matrix interpolation problems abstract in an earlier paper we exploited the displacement structure of cauchy like matrices to derive for them a fast o n 2 implementation of gaussian elimination with partial pivoting one application is to the rapid and numerically accurate solution of linear systems with toeplitz like coefficient matrices based on the fact that the latter can be transformed into cauchy like matrices by using the fast fourier sine or cosine transform however symmetry is lost in the process and the algorithm given is not optimal for hermitian coefficient matrices in this paper we present a new fast o n 2 implementation of symmetric gaussian elimination with partial diagonal pivoting for hermitian cauchy like matrices and show how to transform hermitian toeplitz like matrices to hermitian cauchy like matrices obtaining algorithms that are twice as fast as those in the earlier work numerical experiments indicate that in order to obtain not only fast but also numerically accurate methods it is advantageous to explore the important case in which the corresponding displacement operators have nontrivial kernels this situation gives rise to what we call partially reconstructible matrices which are introduced and studied in the present paper we extend the transformation technique and the generalized schur algorithms i e fast displacement based implementations of gaussian elimination to partially reconstructible matrices we show by a variety of computed examples that the incorporation of diagonal pivoting methods leads to high accuracy we focused in this paper on the design of new numerically reliable algorithms for hermitian toeplitz like matrices however the proposed algorithms have other important applications in particular we briefly describe how they recursively solve a boundary interpolation problem for j unitary rational matrix functions

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

fast algorithms for multiple evaluations of the riemann zeta function the best previously know algorithm for evaluating the riemann zeta function it with bounded and t large to moderate accuracy was based on the rienmann siegel formula and required on the order of t 1 2 operations for each value that was computed new algorithms are presented in this paper which enable one to compute any single value of it with fixed and t t t t 1 2 in o te operations on numbers of o log t bits for any e 0 for example these algorithms lead to methods for numerically verifying the riemann hypothesis for the first n zeros in what is expected to be o n 1 e operations

fast gaussian elimination with partial pivoting for matrices with displacement structure fast o n 2 implementation of gaussian elimination with partial pivoting is designed for matrices possessing cauchy like displacement structure we show how toeplitz like toeplitz plus hankel like and vandermonde like matrices can be transformed into cauchy like matrices by using discrete fourier cosine or sine transform matrices in particular this allows us to propose a new fast o n 2 toeplitz solver gko which incorporates partial pivoting a large set of numerical examples showed that gko demonstrated stable numerical behavior and can be recommended for solving linear systems especially with nonsymmetric indefinite and ill conditioned positive definite toeplitz matrices it is also useful for block toeplitz and mosaic toeplitz toeplitz block matrices the algorithms proposed in this paper suggest an attractive alternative to look ahead approaches where one has to jump over ill conditioned leading submatrices which in the worst case requires o n 3 operations

a lagrange matrices approach to confluent cauchy matrices we present a new approach to study inversion and factorization properties of confluent cauchy matrices we consider a class of generalized vandermonde matrices called lagrange matrices that are connected in a simple way with the cauchy matrices we apply the methods introduced in adv appl math 10 1989 348 to obtain inversion and factorization theorems for lagrange matrices and their confluent forms and then derive corresponding results for cauchy matrices a lagrange matrix v is associated with a pair of vectors x0 x1 xn and y0 y1 yn v is the matrix representation of the substitution operator that sends the vector p x0 p x1 p xn t to the vector p y0 p y1 p yn t for any polynomial p of degree at most n

accurate computations with said ball vandermonde matrices abstract a generalization of the vandermonde matrices which arise when the power basis is replaced by the said ball basis is considered when the nodes are inside the interval 0 1 then those matrices are strictly totally positive an algorithm for computing the bidiagonal decomposition of those said ball vandermonde matrices is presented which allows us to use known algorithms for totally positive matrices represented by their bidiagonal decomposition the algorithm is shown to be fast and to guarantee high relative accuracy some numerical experiments which illustrate the good behaviour of the algorithm are included

an overview of representative problems in location research we present a survey of over 50 representative problems in location research our goal is not to review all variants of different location models or to describe solution results but rather to provide a broad overview of major location problems that have been studied indicating briefly how they are formulated and how they relate to one another we review standard problems such as median center and warehouse location problems as well as less traditional location problems which have emerged in recent years our primary focus is on problems for which operations research type models have been developed most of the problems we review have been formulated as optimization problems

factorizations of cauchy matrices the development of fast algorithms for the solution of linear systems of equations with a cauchy matrix has recently received considerable attention several of these algorithms factor a cauchy matrix or its inverse into triangular and possibly diagonal matrices the numerical properties of the factorization methods depend on the selection of pivots this note presents elementary derivations of some factorization methods and describes a new strategy for searching both rows and columns for suitable pivots

computing the singular value decomposition with high relative accuracy we analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy this means that each computed singular value is guaranteed to have some correct digits even if the singular values have widely varying magnitudes this is in contrast to the absolute accuracy provided by conventional backward stable algorithms which in general only guarantee correct digits in the singular values with large enough magniturds it is of interest to compute the tiniest singular values with several correct digits because in some cases such as finite element problems and quantum mechanics it is the smallest singular values that have physical meaning and should be determined accurately by the data many recent papers have identified special classes of matrices where high relative accuracy is possible since it is not possible in general the perturbation theory and algorithms for these matrix classes have been quite different motivating us to seek a common perturbation theory and common algorithm we provide these in this paper and show that high relative accuracy is possible in many new cases as well the briefest way to describe our results is that we can compute the svd to high relative accuracy provided we can compute a high accuracy pivoted ldu decomposition we provide many examples of matrix classes permitting such an ldu decomposition

a bjorck pereyra type algorithm for szego vandermonde matrices based on properties of unitary hessenberg matrices abstract in this paper we carry over the bjorck pereyra algorithm for solving vandermonde linear systems to what we suggest to call szego vandermonde systems v x i e polynomial vandermonde systems where the corresponding polynomial system is the szego polynomials the properties of the corresponding unitary hessenberg matrix allow us to derive a fast o n 2 computational procedure we present numerical experiments that indicate that for ill conditioned matrices the new algorithm yields better forward accuracy than gaussian elimination

matrices with higher order displacement structure abstract a matrix a is said to possess a displacement structure if the rank of a au va is smalled compared with the order of a in the note the problem is raised to which extend well known theory and algorithms for matrices with a displacement structure can be generalized to matrices a for which the rank of m a is small for m 1

accurate solutions of product linear systems associated with rank structured matrices abstract in this paper we consider how to accurately solve linear systems associated with a wide class of rank structured matrices containing the well known vandermonde and cauchy matrices i e consecutive rank descending crd matrices we provide a mechanism to guarantee that the inverse of any product of crd matrices is generated in a subtraction free manner with the mechanism the solutions of linear systems associated with such products are accurately determined by the parameters of crd factors and we then accurately compute the solutions as warranted by these parameters in particular linear systems associated with products of vandermonde and cauchy matrices whose nodes satisfy certain positive or negative properties are solved to high relative accuracy error analysis and numerical experiments are provided to confirm the high accuracy

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

an o n log 2 n algorithm for the kth longest path in a tree with applications to location problems many known algorithms are based on selection in a set whose cardinality is superlinear in terms of the input length it is desirable in these cases to have selection algorithms that run in sublinear time in terms of the cardinality of the set this paper presents a successful development in this direction the methods developed here are applied to improve the previously known upper bounds for the time complexity of various location problems

new results on the complexity of p center problems an o n log 3 n algorithm for the continuous p center problem on a tree is presented following a sequence of previous algorithms ours is the first one whose time bound in uniform in p and less than quadratic in n we also present an o n log 2 n log log n algorithm for a weighted discrete p center problem