generalized confluent cauchy vandermonde matrices displacement structures inversion formulas and tangential interpolations generalized confluent cauchy and cauchy vandermonde matrices are introduced these two kinds of matrices generalize the ordinary cauchy and cauchy vandermonde matrices with multiple nodes studied earlier by various authors by using displacement structure theory fast inversion formulas for these matrices are derived the tangential interpolation interpretations for associated linear systems with such matrices are given the fast algorithm for solving this kind of linear systems are also considered

weak maximum principles and geometric estimates for spacelike hypersurfaces in generalized robertson walker spacetimes abstract in this paper we deal with complete spacelike hypersurfaces in a generalized robertson walker spacetime using as main analytical tool a new local form of the weak maximum principle for a class of operators including the lorentzian mean curvature operator we obtain some mean curvature estimates and height estimates for spacelike graphs with nice bernstein type consequences we also give height estimates for spacelike hypersurfaces with constant higher order mean curvature which are obtained via the local form of the weak maximum principle recently given in alias et al in press for a class of operators including those constructed from the newton tensors of a hypersurface

a fast hankel solver based on an inversion formula for loewner matrices abstract we propose a new o n2 algorithm for solving complex n n linear systems that have hankel structure via ffts the hankel system is transformed into a loewner system an inversion formula enables us to calculate the inverse of the loewner matrix explicitely the parameters that occur in this inversion formula are calculated by solving two rational interpolation problems on the unit circle we present an o n2 algorithm to solve these interpolation problems one of the advantages of this algorithm is that it incorporates pivoting we have implemented our hankel solver in fortran 90 numerical examples are included they show the effectiveness of our pivoting strategy

linear complexity parallel algorithms for linear systems of equations with recursive structure abstract a method of derivation of parallel algorithms for n 1 n 1 matrices with recursive structure is presented and applied to toeplitz hankel and other toeplitz like matrices the derived algorithms executed on o n parallel processors require o n arithmetic operations per processor

hankel and loewner matrices abstract mutual relations between the hankel toeplitz bezout and loewner matrices as well as further connections to rational interpolation and projective geometry are investigated

some aspects of circulant preconditioners this paper studies circulant approximations of hermitian toeplitz matrices that are solutions of certain minimization problems relative to the frobenius norm the lub 1 or the lub infty norms the examinations supplement the family of the so called optimal and superoptimal preconditioners that have been proposed for the preconditioned conjugate gradient method for solving toeplitz systems t n x b for the new frobenius norm approximations it is shown that they can be computed in o n log n arithmetic operations and that the eigenvalues of the preconditioned linear equations are asymptotically clustered around 1 if t n is a leading principal submatrix of an infinite toeplitz matrix connected with a positive function of the wiener class

a stabilized superfast solver for indefinite hankel systems abstract we present a stabilized superfast solver for indefinite hankel systems whose size is a power of 2 the hankel system is transformed into a loewner system which is solved by using an inversion formula for loewner matrices this explicit formula for the inverse of a loewner matrix contains certain parameters that are computed by solving two linearized rational interpolation problems on the unit circle the heart of our hankel solver is a superfast algorithm to solve these interpolation problems this algorithm is stabilized via pivoting iterative improvement and by giving the so called difficult interpolation points an adequate treatment we have implemented our algorithm in fortran 90 numerical examples illustrate the effectiveness of our approach

representations of toeplitz plus hankel matrices using trigonometric transformations with application to fast matrix vector multiplication abstract representations of real toeplitz and toeplitz plus hankel matrices are presented that involve real trigonometric transformations dct dst dht and diagonal matrices these representations can be used for fast matrix vector multiplication in particular it is shown that the multiplication of an n n toeplitz plus hankel matrix by a vector requires only 4 transformations of length n plus o n 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

efficient solution of linear systems of equations with recursive structure abstract inversion algorithms of order n2 for n n matrices with recursive structure are derived in a unified way several known algorithms as well as some new ones for toeplitz type hilbert type and vandermonde type matrices are given

fast computation of determinants of bezout matrices and application to curve implicitization when using bivariate polynomial interpolation for computing the implicit equation of a rational plane algebraic curve given by its parametric equations the generation of the interpolation data is the most costly of the two stages of the process in this work a new way of generating those interpolation data with less computational cost is presented the method is based on an efficient computation of the determinants of certain constant bezout matrices

on bezoutian reduction with the vandermonde matrix abstract the vandermonde matrix reduces by congruence the bezoutian matrix when the zeros of the highest degree associated polynomial are simple in this paper a generalization of this result to the case of multiple zeros is presented

dft representations of toeplitz plus hankel bezoutians with application to fast matrixvector multiplication abstract representations for inverses of toeplitz plus hankel matrices and more general t h bezoutians with discrete fourier transformations are presented which can be used for fast matrix vector multiplication with the help of some of the formulas multiplication by a toeplitz plus hankel matrix inverse can be carried out with six dft s plus eight dft s for preprocessing which improves the so far best known result of seven dft s plus 10 dft s for preprocessing in e bozzo linear algebra appl 230 1995 127 150

transformation techniques for toeplitz and toeplitz plus hankel matrices ii algorithms abstract this paper is a continuation of g heinig a bojanczyk linear algebra appl 254 1997 193 226 where transformations mapping toeplitz and toeplitz plus hankel matrices into generalized cauchy matrices were studied in the present paper fast algorithms for lu factorization and inversion of generalized cauchy matrices are discussed it is shown that the combination of transformation pivoting techniques leads to algorithms for indefinite toeplitz and toeplitz plus hankel matrices that are more stable than the classical ones special attention is paid to the symmetric and hermitian cases

a superfast method for solving toeplitz linear least squares problems in this paper we develop a superfast o m n log 2 m n complexity algorithm to solve a linear least squares problem with an m n toeplitz coefficient matrix the algorithm is based on the augmented matrix approach the augmented matrix is further extended to a block circulant matrix and dft is applied this leads to an equivalent tangential interpolation problem where the nodes are roots of unity this interpolation problem can be solved by a divide and conquer strategy in a superfast way to avoid breakdowns and to stabilize the algorithm pivoting is used and a technique is applied that selects difficult points and treats them separately the effectiveness of the approach is demonstrated by several numerical examples 2003 elsevier science inc all rights reserved

generalized confluent cauchy and cauchy vandermonde matrices abstract the so called generalized confluent cauchy vandermonde gccv matrices of the form c v consisting of a generalized confluent cauchy part c and a generalized confluent vandermonde part v are considered a simple relationship between gccv and classical confluent cauchy vandermonde ccv matrices is given this leads to the reduction of the displacement structure inversion formulas and factorizations of gccv matrices and the interpolation interpretations of linear systems with such gccv coefficient matrices as tangential interpolation problems to the corresponding results of ccv matrices the criteria of invertibility and left right inversion formulas for such matrices are given all results are stated for the general case of multiple interpolation nodes extending the work of heinig and vavrin among others

stable row recurrences for the pade table and generically superfast lookahead solvers for non hermitian toeplitz systems abstract we present general recurrences for the pade table that allow us to skip ill conditioned pade approximants while we proceed along a row of the table in conjunction with a certain inversion formula for toeplitz matrices these recurrences form the basis for fast algorithms for solving non hermitian toeplitz systems under the assumption that the lookahead step size i e the number of successive skipped approximants remains bounded we give both o n 2 and o n log 2 n algorithms which are presumably weakly stable with little additional work still in o n 2 operations we can also obtain a decomposition of the toeplitz matrix t according to tr ld where r is upper triangular l is unit lower triangular and d is block diagonal the relation to continued fractions is also discussed

generalized cauchy vandermonde matrices abstract matrices of the form c v consisting of a generalized cauchy matrix and a generalized vandermonde matrix are considered using the displacement structure of these matrices inversion formulas and criteria are presented the interpretation of linear systems with such a coefficient matrix as tangential interpolation problems leads to the concept of fundamental matrix which is basic in this approach for fundamental matrices recursion formulas are established from them fast inversion algorithms emerge that work for arbitrary nonsingular matrices of this kind

wiener hopf and spectral factorization of real polynomials by newton s method abstract we present a new method for factoring a real polynomial into the product of two polynomials which have their zeros inside and outside the unit circle respectively the approach is based on solving a nonlinear system by newton s method the jacobi matrix is a polynomial of a companion matrix and thus a bezoutian times the inverse of a triangular toeplitz matrix after getting deeper understanding of the displacement structure of the bezoutian each newton step can be reduced to a few applications of discrete fourier or cosine transforms and the lu decomposition of a conceived linear system with a cauchy like matrix these structural features are employed to design a fast algorithm which shows excellent numerical behavior for example in the case of the extremely ill conditioned test polynomial 2 z n k 0 n 1 z k 2 k 1 n z k of the degree 2 n 20000 we obtain the factorization after 22 newton steps with an error of 3 4 10 8 in the euclidean norm the execution time on a laptop being a couple of minutes the local quadratic convergence of our newton method is proved and explicit convergence balls are given on the basis of estimates for the lipschitz constant which occurs in kantorovich s theorem we also design and test two superfast versions of our method which for polynomials of degree 20000 typically need about half a minute to produce an initial coefficient vector and then perform the newton iteration within 1 s

transformation techniques for toeplitz and toeplitz plus hankel matrices part i transformations transformations of the form a c1ac2 are investigated that transform toeplitz and toeplitz plus hankel matrices into generalized cauchy matrices c1 and c2 are matrices related to the discrete fourier transformation or to various real trigonometric transformations combining these results with pivoting techniques in part ii algorithms for toeplitz and toeplitz plus hankel systems will be presented that are more stable than classical algorithms

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

look ahead methods for block hankel systems in this paper the updating formulas used by three look ahead methods for solving hankel systems are generalized to the square block case each of the original methods was described in the literature using the terminology of different but strongly related fields formal orthogonal polynomials pade approximants structured matrices this paper gives several of these connections generalized to the block case and shows that each viewpoint has its own merits

fast direct solution methods for symmetric banded toeplitz systems based on the sine transform we present new fast direct methods for solving a large symmetric banded toeplitz system of order n with bandwidth p we make use of structured matrices which can be diagonalized by the discrete sine transform matrix sometimes called matrices a first method writes the toeplitz matrix as the sum of a matrix and a low rank matrix a second method embeds the toeplitz matrix in a larger matrix of order m the methods are similar to jain ieee trans acoust speech signal process 26 1978 121 and linzer linear algebra appl 170 1992 1 who worked with circulant matrices both algorithms consist in solving two systems and two smaller systems a system of order n can be solved in o n log n by using a discrete sine transform if n 1 has small prime factors therefore the second algorithm is preferable since we can choose m such that m 1 has small prime factors on the other hand in the second method the smaller systems can become large when m differs too much from n w hile in the first method the order is always p 1 in both methods the small systems have low displacement rank so we can use fast methods to solve them 2002 elsevier science inc all rights reserved

pivoted cauchy like preconditioners for regularized solution of ill posed problems many ill posed problems are solved using a discretization that results in a least squares problem or a linear system involving a toeplitz matrix the exact solution to such problems is often hopelessly contaminated by noise since the discretized problem is quite ill conditioned and noise components in the approximate null space dominate the solution vector therefore we seek an approximate solution that does not have large components in these directions we use a preconditioned conjugate gradient algorithm to compute such a regularized solution a unitary change of coordinates transforms the toeplitz matrix to a cauchy like matrix and we choose our preconditioner to be a low rank cauchy like matrix determined in the course of gu s fast modified complete pivoting algorithm we show that if the kernel of the ill posed problem is smooth then this preconditioner has desirable properties the largest singular values of the preconditioned matrix are clustered around one the smallest singular values corresponding to the lower subspace remain small and the upper and lower spaces are relatively unmixed the preconditioned algorithm costs only o n lg n operations per iteration for a problem with n variables the effectiveness of the preconditioner for filtering noise is demonstrated on three examples

a lookahead algorithm for the solution of block toeplitz systems abstract we give a weakly stable algorithm to solve a block toeplitz system of linear equations if the lookahead steps taken to compute the parameters of the inversion formula for the block toeplitz matrix are small compared to the order n of the matrix the algorithm requires o n2 floating point operations the parameters of the inversion formula are interpreted and computed in a recursive way as solutions of certain interpolation problems given the formal laurent series based on the data of the block toeplitz matrix

the pearson random walk with unequal step sizes there have been many analyses of pearson random walks with equal step sizes several applications suggest the importance of these walks with unequal step sizes numerical comparisons are made of approximations obtained using the central limit theorem and the steepest descents method these suggest that the latter generally but not always leads to more accurate results the authors also derive an approximation that is extremely accurate at near maximal extensions

optimal trigonometric preconditioners for nonsymmetric toeplitz systems abstract this paper is concerned with the solution of systems of linear equations tnxn bn where tn n n denotes a sequence of nonsingular nonsymmetric toeplitz matrices arising from a generating function of the wiener class we present a technique for the fast construction of optimal trigonometric preconditioners mn mn t ntn of the corresponding normal equation which can be extended to toeplitz least squares problems in a straightforward way moreover we prove that the spectrum of the preconditioned matrix mn1t ntn is clustered at 1 such that the pcg method applied to the normal equation converges superlinearly numerical tests confirm the theoretical expectations

a decomposition of toeplitz matrices and optimal circulant preconditioning abstract a factorization of the given toeplitz matrix t obtained in this work is used to derive an optimal circulant preconditioner c in the sense of minimizing the frobenius norm of i ct by use of the fast fourier transform the computation of the new preconditioner can be performed in o n log n operations