existence and stability of periodic solutions for parabolic systems with time delays this paper is concerned with the existence and stability time periodic solutions for a class of coupled parabolic equations with time delay and time delays may appear in the nonlinear reaction functions the existence of time periodic solutions is for a class of locally lipschitz continuous reaction functions without any quasimonotone requirement our approach to the problem is by the method of upper and lower solution and using schauder fixed point theorem some methods for proving the stability of the periodic solution are also given the results for the general system can be applied to the standard parabolic equations without time delay and corresponding ordinary differential system finally a model arising from chemistry is used to illustrate the obtained results

centrosymmetric and centro skewsymmetric toeplitz plus hankel matrices and bezoutians abstract in the paper algebraic properties of centrosymmetric and centro skewsymmetric toeplitz plus hankel t h matrices are investigated it is shown that in the nonsingular case the inverse can be constructed with the help of the solutions of four related toeplitz systems with symmetric or skewsymmetric right hand sides in the singular case the related toeplitz matrices determine the kernel of the t h matrices finally centrosymmetric and centro skewsymmetric t h bezoutians are characterized

fast algorithms for close to toeplitz plus hankel systems and two sided linear prediction the low displacement rank definition of close to toeplitz ct matrices is extended to close to toeplitz plus hankel ctph matrices fast algorithms for solving ctph systems of equations are presented a matrix is defined as ctph if the sum of a ct matrix and a second ct matrix postmultiplied by an exchange matrix an equivalent definition in terms of uv rank is also given this definition is motivated by the application of the algorithms to two sided linear prediction tsp autocorrelation and covariance forms of tsp analogous to those for one sided linear prediction osp are defined the covariance form of tsp is solved using the ctph fast algorithms just as the covariance form of osp is solved using ct fast algorithms numerical examples show that tsp produces smaller residuals than osp and resolves sharp spectral peaks better than osp and that covariance tsp produces smaller residuals than autocorrelation tsp

matrix decompositions using displacement rank and classes of commutative matrix algebras using the notion of displacement rank we look for a unifying approach to representations of a matrix a as sums of products of matrices belonging to commutative matrix algebras these representations are then considered in case a is the inverse of a toeplitz or a toeplitz plus hankel matrix some well known decomposition formulas for a gohberg semencul or kailath et al gader bini pan and gohberg olshevsky turn out to be special cases of the above representations new formulas for a in terms of algebras of symmetric matrices are studied and their computational aspects are discussed

the split levinson algorithm the classical levinson algorithm for computing the predictor polynomial relative to a real positive definite toeplitz matrix is shown to be redundant in complexity it can be broken down into two simpler algorithms either of which needs only to be processed this result can be interpreted in the framework of the theory of orthogonal polynomials on the real line as follows the symmetric and antisymmetric parts of the predictors relative to the sequence of toeplitz matrices constitute two families of polynomials orthogonal on the interval 1 1 with respect to some even weight functions it turns out that the recurrence relations for these orthogonal polynomials can be used efficiently to compute the desired predictor the resulting split levinson algorithm requires roughly one half the number of multiplications and the same number of additions as the classical levinson algorithm a simple derivation of cybenko s method for computing the pisarenko frequencies is obtained from the recurrence relations underlying the split levinson algorithm

exterior problems for more general parabolic monge ampere equation in more general domain abstract in this paper we consider the exterior problem for more general parabolic monge ampere equation u t logdet d 2 u in more general domain r t 1 2 n 1 d r t 1 2 n 1 r n 1 t 1 t 2 d t 1 t 2 using the perron method we obtain the existence and uniqueness of viscosity solution with asymptotic behavior at infinity the results of exterior problems for the parabolic monge ampere equation u t logdet d 2 u 1 are extended

on a new class of reproducing kernel spaces and a new generalization of the iohvidov laws abstract a new class of finite dimensional reproducing kernel spaces of m 1 vector valued analytic functions on a fairly general domain is introduced the reproducing kernels of these spaces have a special form which is based on an m m matrix valued function which is j unitary on the boundary of every invertible hermitian matrix can be interpreted as the gram matrix of a suitably chosen basis in such a space where m is equal to the appropriately defined displacement rank of the given matrix orthogonal decompositions of the space are developed in terms of schur complements of the matrix and factorizations of the much as in the classical cases wherein is either the unit disc or the half plane finally a new generalization of the iohvidov laws which extends earlier generalizations by the same authors is deduced as an application of the theory

generalized inversion of block toeplitz matrices abstract an analog of a wiener hopf factorization method is proposed for finite block toeplitz matrices for an arbitrary rational matrix polynomial notions of essential indices and polynomials are introduced a connection between these notions and a wiener hopf factorization of some block triangular matrix functions is studied a formula for a generalized one sided two sided inversion of a block toeplitz matrix is found in terms of indices and essential polynomials of its symbol well known inversion formulas are obtained as special cases of this formula

hartley type algebras in displacement and optimization strategies abstract the hartley type ht algebras are used to face efficiently the solution of structured linear systems and to define low complexity methods for solving general nonstructured nonlinear problems displacement formulas for the inverse of a symmetric toeplitz matrix in terms of ht transforms are compared with the well known ammar gader formula the l qn unconstrained optimization methods which define hessian approximations by updating n n matrices from an algebra l can be implemented for l ht with an o n amount of memory allocations and o n log n arithmetic operations per step the l qn methods with the lowest experimental rate of convergence are shown to be linearly convergent

matrix representations of toeplitz plus hankel matrix inverses abstract inverses of toeplitz plus hankel matrices and more generally t h bezoutians are represented as sums of products of triangular toeplitz and hankel matrices the parameters occurring in these representations can be determined with the help of 1 solutions of fundamental equations 2 solutions of a certain homogeneous equation and 3 columns and rows of the inverse matrix

the exterior dirichlet problem for hessian quotient equations abstract in this paper using a reverse maclaurin inequality we established the existence theorem of the exterior dirichlet problem for hessian quotient equations with a stronger asymptotic behavior at infinity when k n 1 2 we improve the result in l m dai the dirichlet problem for hessian quotient equations in exterior domains j math anal appl 380 2011 87 93 for s k l with k l 3 to the result with any 0 l k

the dirichlet problem for hessian quotient equations in exterior domains in this paper we obtain the uniqueness and existence of viscosity solutions with prescribed asymptotic behavior at infinity to hessian quotient equations in exterior domains

bowtie factors of toeplitz matrices by means of split algorithms the bowtie factorization of a toeplitz correlation matrix r and its inverse is introduced the term bowtie describes the pattern of the factoring matrices several mathematical properties of bowtie matrices are described including their close relation with block triangular matrices the bowtie factors of the inverse r can be computed using a vector version of the split levinson algorithm restricted to orders with the same parity as n the dimension of r the bowtie factors of r can be computed using split vector versions of the schur algorithm the orthogonal factorization of the underlying data matrix y can also be computed by the split versions of the lattice algorithm producing a bowtie orthogonal factor of y

technical communique robust state estimation for uncertain discrete time stochastic systems with missing measurements in this paper results of robust estimation of zhou 2010a are extended to state estimation with missing measurements a new procedure is derived which inherits the main properties of that of zhou 2010a in this extension a covariance matrix used in the recursions is replaced by its estimate which makes its asymptotic property investigation mathematically difficult though introducing a monotonic function and using the so called squeeze rule this new robust estimator is proved to converge to a stable system numerical simulation results indicate that the proposed estimator may have an estimation accuracy better than the estimator of wang yang daniel and liu 2005

correspondence comments on robust state estimation for uncertain discrete time stochastic systems with missing measurements automatica 47 2011 1520 1524 we show that some results presented in the aforementioned article are rather limited one of the major restrictions on applicability of the obtained results is the ergodic requirement on the received plant output measurements which is generally not satisfied by a time varying system another major restriction is that the comparisons are not appropriate for its numerical simulations that is using the state estimation results of an estimator that utilizing current and past observations to compare with those of a one step state predictor is not appropriate to evaluate the filter performances objectively we do a numerical simulation with the robust state estimator and the well known kalman filter that does not take either parametric errors or random measurement loss into account and the simulation results show that there is no significant difference in filtering performance between these two estimators

on the exterior dirichlet problem for monge ampere equations abstract in this paper we use the perron method to prove the existence of viscosity solutions to a class of monge ampere equations on exterior domains in r n n 2 with prescribed asymptotic behavior at infinity this extends the caffarelli li s result on det d 2 u 1 we also obtain the existence of entire convex viscosity solutions to the monge ampere equations with prescribed asymptotic behavior at infinity

an error free levison algorithm to solve integer toeplitz system abstract this paper presents an error free form of levison algorithm to solve integer nonsymmetrical toeplitz system it removes roundoff errors their accumulation can when using classic algorithms deteriorate or destroy the solution it retains the speed of original levison algorithms its modular form can be easily implemented for parallel computer

fast inversion algorithms of toeplitz plus hankel matrices the paper deals with the problems of fast inversion of matricesa t h wheret is toeplitz andh is hankel several algorithms are presented and compared among them algorithms working for arbitrary strongly nonsingular matricesa t h

numerical solution of the eigenvalue problem for efficiently structured hermitian matrices abstract an iterative procedure is proposed for computing the eigenvalues and eigenvectors of a class of specially structured hermitian toeplitz matrices which includes hermitian toeplitz and toeplitz plus hankel matrices the computational cost per eigenvalue eigenvector for a matrix of order n is o n 2 in serial mode results of numerical experiments on kac murdock szego matrices and randomly generated real symmetric toeplitz matrices and toeplitz plus hankel matrices of orders as high as 2000 are included

the one dimensional inverse problem of reflection seismology an impulsive normal traction is applied uniformly over the surface of a perfectly stratified plane layered earth the ensuing particle velocity at the surface is assumed to be measured the problem of calculating the subsurface characteristic impedance from knowledge of the input pulse and the measured data is the one dimensional inverse problem of reflection seismology the problem is set up mathematically as an inverse problem for a first order 2 times 2 hyperbolic system the role of propagation of singularities is explained and the relation between the solution of the inverse problem and the cholesky factorization of certain matrices constructed from the data is elaborated for both the continuum problem and the related discrete problem it is found that recursive numerical techniques such as the levinson algorithm and certain other fast methods for cholesky factorization are intimately related to explicit finite difference schemes for solving hyperbolic systems the order of approximation between the

generalized lyapunov equations matrices with displacement structure and generalized bezoutians abstract matrix equations of the form i 0 r j 0 s f ij a i xb j c are consider ed it is shown that solving such equations can be reduced to finding the solutions of certain well chosen auxiliary equations the latter equations are essentially simpler than the original equation and their solutions are matrices with a displacement structure or connected with generalized bezoutians in the case r s 1 the approaches given here along with some results on structured matrices and generalized bezoutians lead to explicit formulas for the solution

an extension of a theorem by k jorgens and a maximum principle at infinity for parabolic affine spheres where is a planar domain and f is in the usual holder space c2 without loss of generality we shall consider only locally convex solutions of 1 this equation arises in the context of an affine differential geometric problem as the equation of a parabolic affine sphere in short pa sphere in the unimodular affine real 3 space see c1 c2 cy and lsz contrary to the case of smooth bounded convex domains little is known about solutions of 1 when the domain is unbounded here we recall a famous result by k jorgens which asserts that any solution of 1 on r2 is a quadratic polynomial see j and we also mention a previous paper see fmm where the authors study solutions of 1 on the exterior of a planar domain that are regular at infinity since the underlying almost complex structure of 1 is integrable one expects pa spheres with their canonical conformal structure to be conveniently described in terms of meromorphic functions the reader will find in sect 2 a complex representation of pa spheres and particularly a complex description for the solutions of 1

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

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

on the exterior dirichlet problem for the monge ampere equation in dimension two abstract in this paper we consider the dirichlet problem for the monge ampere equation on exterior domains in dimension two and prove a theorem on the existence of solutions with prescribed asymptotic behavior at infinity

kernel structure of toeplitz plus hankel matrices abstract the structure of the kernel of block toeplitz plus hankel matrices r aj k bj k where aj and bj are the given p q blocks with entries from a given field is investigated it is shown that r corresponds to two systems of at most p q vector polynomials from which a basis of the kernel of r and all other toeplitz plus hankel matrices with the same parameters aj and bj can be built the main result is an analogue of a known kernel structure theorem for block toeplitz and block hankel matrices

efficient solution of a toeplitz plus hankel coefficient matrix system of equations there are well known fast algorithms such as the levinson recursion for solving linear equations with a toeplitz or hankel coefficient matrix this paper extends the saving obtained by the levinson recursion for inversion of a toeplitz matrix to problems involving the inversion of matrices which are the sum of toeplitz and hankel matrices

schur algorithms for hermitian toeplitz and hankel matrices with singular leading principal submatrices it is shown how a simple matrix algebra procedure can be used to induce schur type algorithms for the solution of certain toeplitz and hankel linear systems of equations when given levinson durbin algorithms for such problems the algorithm of p delsarte et al 1985 for hermitian toeplitz matrices in the singular case is used to induce a schur algorithm for such matrices an algorithm due to g heinig and k rost 1984 for hankel matrices in the singular case is used to induce a schur algorithm for such matrices the berlekamp massey algorithm is viewed as a kind of levinson durbin algorithm and so is used to induce a schur algorithm for the minimal partial realization problem the schur algorithm for hermitian toeplitz matrices in the singular case is shown to be amenable to implementation on a linearly connected parallel processor array of the sort considered by kung and hu 1983 and in fact generalizes their result to the singular case

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

an extension to a theorem of jorgens calabi and pogorelov 1 1 det d2u 1 in rn must be a quadratic polynomial for n 2 a classical solution is either convex or concave the result holds without the convexity hypothesis a simpler and more analytical proof along the lines of affine geometry of the theorem was later given by cheng and yau 9 the first author extended the result for classical solutions to viscosity solutions 4 it was proven by trudinger and wang in 19 that the only open convex subset of rn which admits a convex c2 solution of det d2u 1 in with limx u x is rn in this paper we give the following extension to the theorem of jorgens calabi and pogorelov let u be a convex viscosity solution of det d2u 1 outside a bounded subset of

on the inverses of toeplitz plus hankel matrices abstract it is well known that the inverses of hankel and toeplitz matrices can be represented as bezoutians of polynomials in the present note a bezoutian type formula for the inverses of toeplitz plus hankel matrices and a complete characterization of toeplitz plus hankel matrix inverses are given

centro symmetric and centro skewsymmetric toeplitz matrices and bezoutians abstract it is shown how the property of a toeplitz matrix to be centro symmetric or centro skewsymmetric in particular symmetric or skewsymmetric is reflected in its fundamental system furthermore centro symmetric and centro skewsymmetric bezoutians are described finally the representation of inverses of symmetric and skewsymmetric toeplitz matrices by special fundamental equations is discussed

on generating polynomials which are orthogonal over several intervals we consider the problem of generating the recursion coefficients of orthogonal polynomials for a given weight function the weight function is assumed to be the weighted sum of weight functions each supported on its own interval some of these intervals may coincide overlap or are contiguous we discuss three algorithms two of them are based on modified moments whereas the other is based on an explicit expression for the desired coefficients several examples illustrating the numerical performance of the various methods are presented

exterior problems of parabolic monge ampere equations for n 2 abstract in this paper we study the parabolic monge ampere equations in exterior domains for n 2 using the perron method we establish the existence and uniqueness of viscosity solutions with asymptotic behavior we generalize the results of elliptic monge ampere equations in dimension two

a framework for state space estimation with uncertain models develops a framework for state space estimation when the parameters of the underlying linear model are subject to uncertainties compared with existing robust filters the proposed filters perform regularization rather than deregularization it is shown that under certain stabilizability and detectability conditions the steady state filters are stable and that for quadratically stable models the filters guarantee a bounded error variance moreover the resulting filter structures are similar to various time and measurement update prediction and information forms of the kalman filter albeit ones that operate on corrected parameters rather than on the given nominal parameters simulation results and comparisons with spl hscr sub spl infin guaranteed cost and set valued state estimation filters are provided

fast approximate joint diagonalization incorporating weight matrices we propose a new low complexity approximate joint diagonalization ajd algorithm which incorporates nontrivial block diagonal weight matrices into a weighted least squares wls ajd criterion often in blind source separation bss when the sources are nearly separated the optimal weight matrix for wls based ajd takes a nearly block diagonal form based on this observation we show how the new algorithm can be utilized in an iteratively reweighted separation scheme thereby giving rise to fast implementation of asymptotically optimal bss algorithms in various scenarios in particular we consider three specific yet common scenarios involving stationary or block stationary gaussian sources for which the optimal weight matrices can be readily estimated from the sample covariance matrices which are also the target matrices for the ajd comparative simulation results demonstrate the advantages in both speed and accuracy as well as compliance with the theoretically predicted asymptotic optimality of the resulting bss algorithms based on the weighted ajd both on large scale problems with matrices of the size 100times100

a schur algorithm and linearly connected processor array for toeplitz plus hankel matrices a levinson durbin type algorithm for solving toeplitz plus hankel t h linear systems of equations is used to induce a schur type algorithm for such systems a schur type algorithm is defined as one which efficiently computes the ldu decomposition of the matrix on the other hand levinson durbin type algorithms are defined as those algorithms which efficiently compute the udl decomposition of the inverse of a matrix it is shown that the schur algorithm so obtained is amenable to efficient implementation on a linearly connected array of processors in a manner which generalizes the results of s y kung and y h ku 1983 for symmetric toeplitz matrices specifically if t h is of order n then the schur algorithm runs on o n processors in o n time

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

the lanczos algorithm and hankel matrix factorization abstract we explore the connections between the lanczos algorithm for matrix tridiagonalization and two fast algorithms for hankel matrix factorization we show how the asymmetric lanczos process is related to the berlekamp massey algorithm and how the symmetrized lanczos process is related to the phillips algorithm we also discuss conditions under which the analysis of phillips applies

fast algorithms for toeplitz and hankel matrices abstract the paper gives a self contained survey of fast algorithms for solving linear systems of equations with toeplitz or hankel coefficient matrices it is written in the style of a textbook algorithms of levinson type and schur type are discussed their connections with triangular factorizations pade recursions and lanczos methods are demonstrated in the case in which the matrices possess additional symmetry properties split algorithms are designed and their relations to butterfly factorizations are developed

generalized companion matrices and matrix representations for generalized bezoutians abstract a generalization of companion matrices is presented as the starting point for developing a unified algebraic method to get matrix representations for generalized bezoutians special cases are discussed in particular well known formulas for hankel and toeplitz bezoutians are obtained also new matrix representations are given for example in a basis of orthogonal polynomials using corresponding christoffel darboux formulas

matrix representations of split bezoutians abstract inverses of symmetric or skewsymmetric toeplitz matrices as well as of centrosymmetric or centro skewsymmetric toeplitz plus hankel matrices can be represented as sums of two split bezoutians which are highly structured matrices since all of their rows and columns are symmetric or skewsymmetric vectors thus it is desirable to find matrix representations for split bezoutians b this is the main aim of the present paper recursion formulas for the entries of b are presented bases of very simple split bezoutians or of sparse matrices are constructed and b is represented as a corresponding linear combination moreover matrix representations of gohberg semencul type are established

the exterior dirichlet problem for special lagrangian equations in dimensions n 4 abstract in this paper we establish the existence theorem for the exterior dirichlet problem for special lagrangian equations with prescribed asymptotic behavior at infinity this extends the previous results on monge ampere equations and hessian equations to special lagrangian equations in dimensions n 4 which is from calibrated geometry more generally we prove that the result is also true for hessian quotient equations with 0 l k n in dimensions n 3

superfast tikhonov regularization of toeplitz systems toeplitz structured linear systems arise often in practical engineering problems correspondingly a number of algorithms have been developed that exploit toeplitz structure to gain computational efficiency when solving these systems the earliest fast algorithms for toeplitz systems required o n 2 operations while more recent superfast algorithms reduce the cost to o n log n 2 or below in this work we present a superfast algorithm for tikhonov regularization of toeplitz systems using an extension and transformation technique our algorithm translates a tikhonov regularized toeplitz system into a type of specialized polynomial problem known as tangential interpolation under this formulation we can compute the solution in only o n log n 2 operations we use numerical simulations to demonstrate our algorithm s complexity and verify that it returns stable solutions

kernel structure of block hankel and toeplitz matrices and partial realization abstract the kernel structure of block hankel and toeplitz matrices is studied this leads to the concept of a fundamental system which is basic in the theory and provides in particular a natural background for the partial realization problem

fast triangular factorization of the sum of quasi toeplitz and quasi hankel matrices abstract the literature contains several recent fast algorithms for the triangular factorization of strongly regular toeplitz plus hankel matrices in this paper we study the rather more general sum of quasi toeplitz and quasi hankel matrices both hermitian and non hermitian quasi toeplitz and quasi hankel matrices are those that are congruent to toeplitz and hankel matrices in a special sense the derivation is based on the concept of displacement structure and its intimate relation to the schur reduction procedure for triangular factorization various special cases covering displacement ranks from two to eight are considered several other problems e g factorization of the inverse matrix solution of exact or overdetermined linear systems can be reduced to the direct factorization problem

cepstral coefficients covariance lags and pole zero models for finite data strings one of the most widely used methods of spectral estimation in signal and speech processing is linear predictive coding lpc lpc has some attractive features which account for its popularity including the properties that the resulting modeling filter i matches a finite window of n 1 covariance lags ii is rational of degree at most n and iii has stable zeros and poles the only limiting factor of this methodology is that the modeling filter is all pole i e an autoregressive ar model in this paper we present a systematic description of all autoregressive moving average arma models of processes that have properties i iii in the context of cepstral analysis and homomorphic filtering we show that each such an arma model determines and is completely determined by its finite windows of cepstral coefficients and covariance lags we show that these nth order windows form local coordinates for all arma models of degree n and that the pole zero model can be determined from the windows as the unique minimum of a convex objective function we refine this optimization method by first noting that the maximum entropy design of an lpc filter is obtained by maximizing the zeroth cepstral coefficient subject to the constraint i more generally we modify this scheme to a more well posed optimization problem where the covariance data enter as a constraint and the linear weights of the cepstral coefficients are positive in a sense that a certain pseudo polynomial is positive rather succinctly generalizing the maximum entropy method this new problem is a homomorphic filter generalization of the maximum entropy method providing a procedure for the design of any stable minimum phase modeling filter of degree less or equal to n that interpolates the given covariance window we present an algorithm for realizing these filters in a lattice ladder form given the covariance window and the moving average part of the model while we also show how to determine the moving average part using cepstral smoothing one can make use of any good a priori estimate for the system zeros to initialize the algorithm we conclude the paper with an example of this method incorporating an example from the literature on arma modeling

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

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

multivariate frobenius pade approximants properties and algorithms the aim of this paper is to construct rational approximants for multivariate functions given by their expansion in an orthogonal polynomial system this will be done by generalizing the concept of multivariate pade approximation after defining the multivariate frobenius pade approximants we will be interested in the two following problems the first one is to develop recursive algorithms for the computation of the value of a sequence of approximants at a given point the second one is to compute the coefficients of the numerator and denominator of the approximants by solving a linear system for some particular cases we will obtain a displacement rank structure for the matrix of the system we have to solve the case of a tchebyshev expansion is considered in more detail

on the splitting of classical algorithms in linear prediction theory the levinson algorithm has been shown recently to be redundant in complexity it can be broken down into two simpler algorithms either of these split levinson algorithms needs only to be processed in the present paper it is shown that the schur algorithm the lattice algorithm and the normalized lattice algorithm for computing reflection coefficients can be broken down in a similar manner the number of multiplications in each of the resulting split algorithms is reduced with respect to the classical version by a factor lying between 1 5 and 2

not every matrix is similar to a toeplitz matrix abstract a negative answer to the question is every matrix similar to a toeplitz matrix raised by d s mackey n mackey and s petrovic linear algebra appl 297 1999 87 is given the tool of investigation is a result concerning the structure of the intersection of kernel and range of a toeplitz matrix which is of independent interest