on the partial stochastic realization problem we describe a complete parameterization of the solutions to the partial stochastic realization problem in terms of a nonstandard matrix riccati equation our analysis of this covariance extension equation cee is based on a complete parameterization of all strictly positive real solutions to the rational covariance extension problem answering a conjecture due to georgiou 1987 in the affirmative we also compute the dimension of partial stochastic realizations in terms of the rank of the unique positive semidefinite solution to the cee yielding some insights into the structure of solutions to the minimal partial stochastic realization problem by combining this parameterization with some of the classical approaches in partial realization theory we are able to derive new existence and robustness results concerning the degrees of minimal stochastic partial realizations as a corollary to these results we note that in sharp contrast with the deterministic case there is no generic value of the degree of a minimal stochastic realization of partial covariance sequences of fixed length

equivalence between stability of partial realizations and feedback stabilization applications to reduced order stabilization abstract given a transfer function b z a z a classical feedback problem is to find a dynamic feedback d z c z such that the closed loop characteristic polynomial ca db z becomes stable antoulas has proved that this is equivalent to finding a stable partial realization of a certain sequence of real numbers the converse is also true every partial realization problem can be transformed into a feedback problem this paper gives a new formulation of the equivalence between feedback and realization of scalar systems as an application algebraic conditions for existence of reduced order stabilizing feedback are derived

on the partial realization problem abstract in this paper we take a unified approach to the partial realization problem in which we seek to incorporate ideas from numerical linear algebra most of which were originally developed in other contexts we approach the partial realization problem from several different angles and explore the connections to such topics as factoriza tion of hankel matrices block tridiagonalization generalizations of the lanczos process for biorthogonalization the euclidean algorithm and the principal part con tinued fractions of arne magnus the pade table and the berlekamp masseyalgo rithm in this way we are able to clarify some previous results by rissanen kalman and others and place them in a broader context this leads to several results and concepts which we think are new our analysis is restricted to the scalar case but some definitions and formulations have been rigged to facilitate an extension to the matrix case

a cell decomposition of the space of real hankel matrices of rank n and some applications abstract we construct a cellular decomposition of the space of all infinite real hankel matrices of rank n it induces cell decompositions on each manifold h m of hankel matrices of fixed rank m n the relationship between the cellular decomposition of h n and the decomposition of rat n into continued fraction cells is determined finally the cell decomposition is applied to derive some topological properties of the manifolds h n and rat n

parametrising the solutions of the generalised partial realisation problem through an atlas of coordinates given a finite nice sequence of matrices and a positive integer d the set of generalised partial realisations of of order d is a stratified smooth manifold an atlas of coordinate charts for each stratum is provided that locally parametrises it in particular the set of all generalised partial realisations of order d of any given nice sequence will be described

computationally efficient applications of the euclidean algorithm to zero location abstract we characterize the inertia of the n n hankel matrix associated with a pair of real polynomials in terms of the quotients generated by the euclidean algorithm applied to these polynomials in this way we solve certain classical zero location problems at the computational cost of o n 2 arithmetic operations which can be reduced to o n log 2 n if fast polynomial arithmetic is used

the stability and instability of partial realizations a finite or infinite sequence of real numbers is said to be stable if it admits minimal realization by a stable linear system it is shown that the preservation of stability as such a sequence is truncated or extended is not a generic property even among stable sequences this is of interest for identification of linear systems from partial data and for partial realization of random systems in which cases it constitutes a negative result certain finite sequences have infinitely many minimal realizations each having different stability properties in this case a graphical criterion in the spirit of the nyquist criterion is derived to exploit this lack of uniqueness in order to determine whether one can achieve a stable pole placement by a judicious choice of partial realization

from finite covariance windows to modeling filters a convex optimization approach the trigonometric moment problem is a classical moment problem with numerous applications in mathematics physics and engineering the rational covariance extension problem is a constrained version of this problem with the constraints arising from the physical realizability of the corresponding solutions although the maximum entropy method gives one well known solution in several applications a wider class of solutions is desired in a seminal paper georgiou derived an existence result for a broad class of models in this paper we review the history of this problem going back to carath eodory as well as applications to stochastic systems and signal processing in particular we present a convex optimization problem for solving the rational covariance extension problem with degree constraint given a partial covariance sequence and the desired zeros of the shaping filter the poles are uniquely determined from the unique minimum of the corresponding optimization problem in this way we obtain an algorithm for solving the covariance extension problem as well as a constructive proof of georgiou s existence result and his conjecture a generalized version of which we have recently resolved using geometric methods we also survey recent related results on constrained nevanlinna pick interpolation in the context of a variational formulation of the general moment problem

multiple solutions for boundary value problems of second order difference equations with resonance in this paper we study the existence of multiple solutions for boundary value problems of second order difference equations with resonance at both infinity and zero by using morse theory critical point theory minimax methods and bifurcation theory

an implementation of the look ahead lanczos algorithm for non hermitian matrices the nonsymmetric lanczos method can be used to compute eigenvalues of large sparse non hermitian matrices or to solve large sparse non hermitian linear systems however the original lanczos algorithm is susceptible to possible breakdowns and potential instabilities an implementation of a look ahead version of the lanczos algorithm is presented that except for the very special situation of an incurable breakdown overcomes these problems by skipping over those steps in which a breakdown or near breakdown would occur in the standard process the proposed algorithm can handle look ahead steps of any length and requires the same number of matrix vector products and inner products as the standard lanczos process without look ahead

matrix continued fractions are directly related to the maximal a b invariant subspace in ker c the maximal a b invariant subspace in ker c is characterized in terms of polynomial models by its properties pertaining to the hankel map this characterization is directly related to the first stage in a continued fraction representation of the transfer function the atoms of the continued fraction are given explicitly in terms of tocplitz operators we also obtain a complete characterization and parametrization of eligible first atoms based on weiner hopf indexes we show how to construct a first atom which is column proper and determines a feedback equivalent feedback irreducible transfer function

solving nonlinear integral equations of fredholm type with high order iterative methods the application of high order iterative methods for solving nonlinear integral equations is not usual in mathematics but in this paper we show that high order iterative methods can be used to solve a special case of nonlinear integral equations of fredholm type and second kind in particular those that have the property of the second derivative of the corresponding operator have associated with them a vector of diagonal matrices once a process of discretization has been done

computation of matrix valued formally orthogonal polynomials and applications we present a computational procedure for generating formally orthogonal polynomials associated with a given bilinear hankel form with rectangular matrix valued moments our approach covers the most general case of moments of any size and is not restricted to square moments moreover our algorithm has a built in deflation procedure to handle linearly dependent or almost linearly dependent columns and rows of the block hankel matrix associated with the bilinear form possible singular or close to singular leading principal submatrices of the deflated block hankel matrix are avoided by means of look ahead techniques applications of the computational procedure to eigenvalue computations reduced order modeling the solution of multiple linear systems and the fast solution of block hankel systems are also briefly described

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

a directional partial realization problem abstract this paper introduces and solves a new version of the minimal partial realization problem concerning a finite sequence of markow parameters m1 mr for the case when each mj is only specified on a subspace l j such that l 1 l r the results include a state space criterion for minimality a procedure for constructing a minimal realization and a formula for the minimal state space dimension in terms of the original data

a review of the parameter estimation problem of fitting positive exponential sums to empirical data exponential sum models are used frequently in heat diffusion diffusion of chemical compounds time series in medicine economics physical sciences and technology thus it is important to find good methods for the estimation of parameters in exponential sums in this paper we review and discuss results from the last forty years of research there are many different ways of estimating parameters in exponential sums and model a fit criterion which gives a valid result from the fit we find that a good choice is a weighted two norm objective function with weights based on the maximum likelihood ml criterion if the number of exponential terms is unknown statistical methods based on an information criterion or cross validation can be used to determine the optimal number it is suitable to use hybrid gauss newton gn and quasi newton algorithms to find the unknown parameters in the constrained weighted nonlinear least squares nlls problem formulated using an maximal likelihood ml objective function the problem is highly ill conditioned and it is crucial to find good starting values for the parameters to find the initial parameter values a modified prony method or a method based upon rewriting partial sums as geometrical sums is proposed

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

application of pade via lanczos approximations for efficient multifrequency solution of helmholtz problems this paper addresses the efficient solution of acoustic problems in which the primary interest is obtaining the solution only on restricted portions of the domain but over a wide range of frequencies the exterior acoustics boundary value problem is approximated using the finite element method in combination with the dirichlet to neumann dtn map the restriction domain problem is formally posed in transfer function form based on the finite element solution in order to obtain the solution over a range of frequencies a matrix valued pade approximation of the transfer function is employed using a two sided block lanczos algorithm this approach provides a stable and efficient representation of the pade approximation in order to apply the algorithm it is necessary to reformulate the transfer function due to the frequency dependency in the nonreflecting boundary condition this is illustrated for the case of the dtn boundary condition but there is no restriction on the approach which can also be applied to other radiation boundary conditions numerical tests confirm that the approach offers significant computational speed up

on moments based pade approximations of ruin probabilities in this paper we investigate the quality of the moments based pade approximation of ultimate ruin probabilities by exponential mixtures we present several numerical examples illustrating the quick convergence of the method in the case of gamma processes while this is not surprising in the completely monotone case which holds when the shape parameter is less than 1 it is more so in the opposite case for which we improve even further the performance by a fix up which may be of special importance due to its potential use in the four moments gamma approximation we also review the connection of the exponential mixtures approximation to pade approximation orthogonal polynomials and gaussian quadrature these connections may turn out useful for providing rates of convergence

modified moments for indefinite weight functions the problem of generating the recurrence coefficients of orthogonal polynomials from the moments or from modified moments of the weight function is well understood for positive weight distributions here we extend this theory and the basic algorithms to the case of an indefinite weight function while the generic indefinite case is formally not much different from the positive definite case there exist nongeneric degenerate situations and these require a different more complicated treatment the understanding of these degenerate situations makes it possible to construct a stable approximate solution of an ill conditioned problem the application to adaptive iterative methods for linear systems of equations is anticipated

generalized biorthogonal bases and tridiagonalisation of matrices in connection with the breakdown problem of the lanczos algorithm a theory of generalized biorthogonal bases is developed the connection between the generalized biorthogonal bases of krylov chains and look ahead lanczos recursions is worked out in detail it is shown how generalized biorthogonal bases with antidiagonal blocks can be constructed with small computational effort finally a special look ahead lanczos algorithm is derived which requires minimal computational effort and storage

differentiable structure of realizable markov parameters abstract the geometry of the finite sequences of p m matrices that are realizable by systems of minimal order d is investigated they can be stratified according to their partial row and column kronecker indices the case when the sum of the number of non zero row and column indices is smaller than the number of elements in the sequence of parameters is considered it is shown that in this case the whole set of markov parameters and each stratum can be endowed with structures of differentiable manifolds and their dimensions are computed

inversion components of block hankel like matrices abstract the inversion problem for square matrices having the structure of a block hankel like matrix is studied examples of such matrices include hankel striped hankel layered and vector hankel matrices it is shown that the components that both determine nonsingularity and construct the inverse of such matrices are closely related to certain matrix polynomials these matrix polynomials are multidimensional generalizations of pade hermite and simultaneous pade approximants the notions of matrix pade hermite and matrix simultaneous pade systems are also introduced these are shown to provide a second set of inverse components for block hankel like matrices a recurrence relation is presented that allows for efficient computation of matrix pade hermite and matrix simultaneous pade systems as a result it is shown that the inverse components can be computed via either the matrix euclidean algorithm or a matrix berlekamp massey algorithm applied to an associated matrix power series an alternative algorithm based on this recurrence relation is also presented for a block hankel like matrix of type n 0 n 1 n k this algorithm is shown to compute the inverse components with a complexity of o k n 0 n k 2 block matrix operations although this can be higher in some pathological cases this is the same complexity as with existing algorithms this algorithm has the significant advantage however that no extra conditions are required on the input matrix other block algorithms require that certain submatrices be nonsingular similar results hold in the case of block toeplitz like matrices

lyapunov lanczos and inertia abstract we present a new proof of the inertia result associated with lyapunov equations furthermore we present a connection between the lyapunov equation and the lanczos process which is closely related to the schwarz form of a matrix we provide a method for reducing a general matrix to schwarz form in a finite number of steps o n 3 hence we provide a finite method for computing inertia without computing eigenvalues this scheme is unstable numerically and hence is primarily of theoretical interest

a new order estimation technique for time series modeling a new approach to estimating the order of the autoregressive moving average model is proposed which is based on the approximate stochastic realization introduced in davis and fotopoulos 1991 the present approach is attractive because overparameterization a very common problem in order determination is avoided successfully simulation results are included to illustrate the effectiveness of the proposed order estimation approach

combined sequence of markov parameters and moments in linear systems the combined sequence of markov parameters and moments in linear systems is discussed this sequence is shown to be generated by a linear feedback shift register application to the minimal degree partial realization from a mixture of any number of markov parameters and moments is considered central in this particular realization problem is the extension of a toeplitz partial behavior matrix this is accomplished on the basis of the algebraic theory of iohvidov indexes for toeplitz matrices

a lanczos type method for multiple starting vectors given a square matrix and single right and left starting vectors the classical nonsymmetric lanczos process generates two sequences of bior thogonal basis vectors for the right and left krylov subspaces induced by the given matrix and vectors in this paper we propose a lanczos type algorithm that extends the classical lanczos process for single starting vectors to mul tiple starting vectors given a square matrix and two blocks of right and left starting vectors the algorithm generates two sequences of biorthogonal basis vectors for the right and left block krylov subspaces induced by the given data the algorithm can handle the most general case of right and left start ing blocks of arbitrary sizes while all previously proposed extensions of the lanczos process are restricted to right and left starting blocks of identical sizes other features of our algorithm include a built in deation procedure to detect and delete linearly dependent vectors in the block krylov sequences and the option to employ look ahead to remedy the potential breakdowns that may occur in nonsymmetric lanczos type methods

continued fraction decomposition of linear systems in the state space abstract various results pertaining to a continued fraction decomposition of matrix formal power series and in particular of multi input multi output linear constant systems are presented from a state space point of view

recursive partial realization from the combined sequence of markov parameters and moments abstract in this paper we develop the recursive structure of partial realization from a two sided sequence of markov parameters and moments the specific feature of this recursive structure is that it accommodates sequences growing in two directions like in the two point pade approximation we present a recursive algorithm for the minimal partial realization from a sequence of markov parameters and moments that amounts to recursive lu factorization of a toeplitz behavior matrix leaping over nongeneric sections it turns out that this algorithm reveals a new continued fraction expansion which extends the classical cauer third form as a corollary various results related to toeplitz matrices two point pade approximations continued fractions etc are unified within this partial realization setting

a look ahead algorithm for the solution of general hankel systems the solution of systems of linear equations with hankel coefficient matrices can be computed with onlyo n 2 arithmetic operations as compared too n 3 operations for the general cases however the classical hankel solvers require the nonsingularity of all leading principal submatrices of the hankel matrix the known extensions of these algorithms to general hankel systems can handle only exactly singular submatrices but not ill conditioned ones and hence they are numerically unstable in this paper a stable procedure for solving general nonsingular hankel systems is presented using a look ahead technique to skip over singular or ill conditioned submatrices the proposed approach is based on a look ahead variant of the nonsymmetric lanczos process that was recently developed by freund gutknecht and nachtigal we first derive a somewhat more general formulation of this look ahead lanczos algorithm in terms of formally orthogonal polynomials which then yields the look ahead hankel solver as a special case we prove some general properties of the resulting look ahead algorithm for formally orthogonal polynomials these results are then utilized in the implementation of the hankel solver we report some numerical experiments for hankel systems with ill conditioned submatrices

direct block tridiagonalization of single input single output systems abstract in this paper we derive a direct method for block tridiagonalizing a single input single output system triple a b c the method is connected to the nonsymmetric lanczos procedure developed in 10 2 1 and also leads to canonical representations of such triples

parallel and superfast algorithms for hankel systems of equations utilizing kernel structure properties a unified construction of hankel matrix inversion algorithms is presented three types of algorithms are obtained 1 o n 2 complexity levinson type 2 o n parallel complexity schur type and 3 o n log2 n complexity asymptotically fast ones all algorithms work without additional assumption like strong nonsingularity

a least squares approach to reduce stable discrete linear systems preserving their stability abstract a new stability preserving model reduction algorithm for discrete linear siso systems based on a least squares approach is proposed similar to the pade approximation an equation system for the markov parameters involving a high dimensional hankel matrix is considered it is proved that approximate solutions computed via the moore penrose pseudo inverse give rise to a stability preserving reduction scheme furthermore the proposed algorithm is compared to the balanced truncation method showing comparable performance of the reduced systems

nonlinear moment matching based model order reduction in this paper we present a time domain notion of moments for a class of single input single output nonlinear systems in terms of the evolution of the output of a generalized signal generator driven by the nonlinear system we also define a new notion of moment matching and present a family of nonlinear parametrized reduced order models that achieve moment matching we establish relations with existing notions of moment for nonlinear systems showing that the newly derived and the existing families of reduced order models that achieve nonlinear moment matching respectively are equivalent furthermore we compute the reduced order model that matches the moments at two chosen signal generators one exciting the input of the system and another driven by the system simultaneously we also present a family of models computed on the basis of a nonlinear extension of the petrov galerkin projection that achieve moment matching finally we specialize the results to the case of nonlinear input affine systems

a fast method to diagonalize a hankel matrix abstract we consider a vandermonde factorization of a hankel matrix and propose a new approach to compute the full decomposition in o n 2 operations the method is based on the use of a variant of the lanczos method to compute a tridiagonal matrix whose eigenvalues are the modes generating the entries in the hankel matrix by adapting existing methods to solve for these eigenvalues and then for the coefficients one arrives at a method to compute the entire decomposition in o n 2 operations the method is illustrated with a simple numerical example

exponential interpolation theory and numerical algorithms we consider the problem of exponential interpolation and show that in some cases it can be solved with effective numerical procedures we show that the problem is closely related to gaussian quadrature and to the problem of partial realization in linear control theory

on recursiveness and related topics in linear systems a decomposition of matrix formal power series linear systems is developed the interconnections between formal power series realization pade approximation recursiveness berlekamp massey algorithm feedback interconnections matrix linear fractional representations matrix continued fractions and the matrix euclidean algorithm are thereby revealed

families of moment matching based low order approximations for linear systems the problem of finding a framework for linear moment matching based approximation of linear systems is discussed it is shown that it is possible to define five equivalent notions of moments i e one from a complex s domain point of view two using krylov projections and two from a time domain perspective based on these notions classes of parameterized reduced order models that achieve moment matching are obtained we analyze the controllability and observability properties of the models belonging to each of the classes of reduced order models we find the subclasses of models of orders lower than the number of matched moments i e we find the sets of parameters that allow for pole zero cancellations to occur furthermore we compute the subclass of minimal order model s the minimal order is equal to half of the number of matched moments i e generically the largest number of possible pole zero cancellations is half of the number of matched moments finally we present classes of reduced order models that match larger numbers of moments based on interconnections between reduced order models

rational approximation in linear systems and control abstract in this paper we want to describe some examples of the active interaction that takes place at the border of rational approximation theory and linear system theory these examples are mainly taken from the period 1950 1999 and are described only at a skindeep level in the simplest possible scalar case we give comments on generalizations of these problems and how they opened up new ranges of research that after a while lived their own lives we also describe some open problems and future work that will probably continue for some years after 2000

an algebraic method to solve the minimal partial realization problem for scalar sequences abstract we give an algebraic method to construct all minimal partial realizations of a finite sequence of scalar numbers we use the minimal number of parameters to characterize all these minimal partial realizations three examples are worked out to illustrate the method

near aggregation the dual ghr and pole zero cancellation it is known that a linear system will aggregate if and only if it is unobservable an equivalent condition is that the transfer function has a pole zero cancellation in this paper we consider siso systems that nearly aggregate and we show that these systems have almost pole zero cancellations defining near unobservability as roughly the existence of an invariant subspace near the null space of c we also show that nearly unobservable systems exhibit almost pole zero cancellations the main tool in this analysis is the dual generalized hessenberg representation dual ghr the dual ghr extends the ghr by incorporating the input structure with the output structure of the ghr the result is a canonical form the dual ghr that relates the internal states to the input output properties of the system by exploiting this structure the main results are obtained

a complete parameterization of all positive rational extensions of a covariance sequence in this paper we formalize the observation that filtering and interpolation induce complementary or dual decompositions of the space of positive real rational functions of degree less than or equal to n from this basic result about the geometry of the space of positive real functions we are able to deduce two complementary sets of conclusions about positive rational extensions of a given partial covariance sequence on the one hand by viewing a certain fast filtering algorithm as a nonlinear dynamical system defined on this space we are able to develop estimates on the asymptotic behavior of the schur parameters 1918 of positive rational extensions on the other hand we are also able to provide a characterization of all positive rational extensions of a given partial covariance sequence indeed motivated by its application to signal processing speech processing and stochastic realization theory this characterization is in terms of a complete parameterization using familiar objects from systems theory and proves a conjecture made by georgiou 1983 1987 our basic result however also enables us to analyze the robustness of this parameterization with respect to variations in the problem data the methodology employed is a combination of complex analysis geometry linear systems and nonlinear dynamics

on the minimal partial realization of 2d discrete linear shift invariant systems the partial realization problem for 2d discrete linear shift invariant systems is discussed the authors attempt to establish a reasonable definition of minimality and propose a method for finding the degree and the system function of a minimal realization of a 2d linear system characterized by a given 2d impulse response array the method used is based on the 2d berlekamp massey algorithm which has a close connection with the 2d hankel matrix derived from the 2d array while it compares in efficiency with other methods for identification and approximation of 2d linear systems it gives a novel approach to the problem in the sense that one does not need any knowledge or assumptions concerning the system degree

model order reduction for large systems in computational electromagnetics abstract this paper examines classical model order reduction mor strategies in view of the particular properties and needs of computational electromagnetism hereby reduced models are mainly needed for two reasons the fast calculation of certain characteristics such as the transfer behavior over a range of excitation frequencies especially for highly resonant systems and the generation of macromodels such as equivalent circuits simplifying coupled simulations in the first case the computational cost of the method is of main interest while in the second one the size of the reduced model as well as the conservation of stability and passivity plays a major role the considered methods partial realization moment matching and modal extraction are well known and have been investigated for about two decades now however their efficiency appears in a different light if the number of unknowns reaches hundreds of thousands or even millions this paper compares the suitability and efficiency of the mentioned methods for lossless or weakly lossy structures discretized by the finite integration theory fit close relations and even transitions between the algorithms are shown finally some specific properties of fit enable the application of a method called two step lanczos tsl a successive application of partial realization and moment matching which is highly efficient in both computation time and model size while preserving the passivity of the reduced models tsl allows to compute the broadband transfer behavior of systems with hundreds of thousands of unknowns within minutes on a standard pc additionally the resulting model can easily be implemented as a physical electric equivalent circuit

rational interpolation via orthogonal polynomials abstract relations between rational interpolants and hankel matrices are investigated a modification of a jacobi like algorithm for rational interpolation proposed in 1 is developed the proposed algorithm eliminates all the auxiliary assumptions required to implement the algorithm of 1 in this way it can be used to construct all existing rational interpolants with the cost of o n 2 arithmetic operations where n 1 is the number of data points

a unified approach to loewner and hankel matrices abstract we introduce so called generalized loewner matrices the definition is almost the same as antoulas and anderson s our class of generalized loewner matrices contains both classical loewner and hankel matrices as special cases the properties of such matrices are studied together with the associated interpolation problem the detailed characterization of its nonsolvable cases is given by means of the s c minimal generalized solution by means of rank properties of the loewner matrix and by means of the s c l polynomial introduced by fiedler

fast fraction free triangularization of bezoutians with applications to sub resultant chain computation an algorithm for the computation of the lu factorization over the integers of an n times n bezoutian b is presented the algorithm requires o n 2 arithmetic operations and involves integers having at most o n log nc bits where c is an upper bound of the moduli of the integer entries of b as an application by using the correlations between bezoutians and the euclidean scheme we devise a new division free algorithm for the computation of the polynomial pseudo remainder sequence of two polynomials of degree at most n in o n 2 arithmetic operations the growth of the length of the integers involved in the computation is kept at the minimum value i e o n log nc bits no matter if the sequence is normal or not where c is an upper bound of the moduli of the input coefficients the algorithms that rely on the bareiss technique and on the properties of the schur complements of bezoutians improve the previous ones

functional models in linear algebra abstract this paper examines the role of functional models in linear algebra and some related areas of application notably linear systems theory this approach adopts an intermidiate level of abstraction somewhere between abstract module theory on the one hand and matrix theory on the other it is a powerful method which simplifies many proofs provides insights and illuminates the less obvious connections between linear algebra and operator theory approximation theory moment problems interpolation and control theory

computing the polynomial remainder sequence via bezout matrices in this paper we address the task of computing the polynomial remainder sequence appearing in the euclidean algorithm applied to two polynomials u x and v x of degree n and m respectively m n via renewed approach based on the approximate block diagonalization of a b ezout matrix b u v more specifically this algorithm consists on using the e cient schur complementation method given in terms of a bmatrix b u v associated with the input polynomials we also compare the proposed approach to the approximate block diagonalization of a hankel matrix h u v in which the successive transformation matrices are upper triangular toeplitz matrices all algorithms have been implemented in matlab numerical experiments performed with a wide variety of test problems show the e ectiveness of this algorithm in terms of e cience stability and robustness be two polynomials with deg u x n deg v x m and m n the classical euclidean algorithm applied to u x and v x returns a sequence of quotients qk x and remainders rk x such that r 1 x u x r0 x v x rk 2 x rk 1 x qk x rk x k 1 k 2 where rk x is the polynomial remainder of the division of rk 2 x and rk 1 x and rk x is the greatest common divisor gcd of u x and v x the question of computing the quotients and the remainders is beginning to be studied by collins and brown see 11 9 10 via e cient euclidean based methods naturally the classical algorithm turns out to be unstable when applied to approximate polynomials one might think of designing stabilized numerical versions of the euclidean algorithm to the approximate case see 25 23 several papers have been devoted to describe the natural relation between the euclidean algorithm and the block lu factorization of the hankel and b

the matrix partial realization problem an algebraic approach this paper introduces a new approach to the multivariable partial realization problem it formulates the problem in algebraic terminology which sheds new light on the nature of the problem and our existing knowledge of it the structure of the state space is analysed in terms of polynomial models the main idea is analysis of the algebraic structure of the kernels of truncated hankel maps the elements of such kernels are directly related to the partial realization problem in the sense that the columns of the denominator matrix of a partial realization are elements of such kernels the numerator matrix is determined by an appropriate denominator matrix

uniqueness properties of minimal partial realizations abstract we characterize quantities poles and pole chains of the transfer functions of the minimal partial realizations of a given sequence of matrices s s0 s1 sn 1 which are uniquely determined by s furthermore the set of all points at which the transfer functions of all minimal partial realizations are analytic is described

a look ahead lanczos algorithm for unsymmetric matrices abstract the two sided lanczos algorithm is known to suffer from serious breakdowns these occur when the associated moment matrix does not permit triangular factorization we modify the algorithm slightly so that it corresponds to using a 2 x 2 pivot in triangular factorization whenever a 1 x 1 pivot would be dangerous the incidence of breakdown is greatly reduced the price paid is that the tridiagonal matrix produced by the algorithm now has bumps whenever a 2 x 2 pivot is used author

bruhat canonical form for linear systems based on a generalization of the classical bruhat factorization of nonsingular matrices to arbitrary rectangular matrices a new canonical form for state space equivalence of controllable and observable linear systems is introduced the proposed canonical form is shown to be closely related to a canonical form due to bosgra and van der weiden moreover in the single input single output case and up to minor details the proposed normal form is equivalent to the continued fraction canonical form introduced by kalman connections to a cell decomposition by fuhrmann and krishnaprasad are discussed as well discrete invariants appearing in the bruhat canonical form are shown to be invariants for restricted output feedback equivalence

experimental evidence showing that stochastic subspace identification methods may fail it is known that certain popular stochastic subspace identification methods may fail for theoretical reasons related to positive realness in fact these algorithms are implicitly based on the assumption that the positive and algebraic degrees of a certain estimated covariance sequence coincide in this paper we describe how to generate data with the property that this condition is not satisfied using these data we show through simulations that several subspace identification algorithms exhibit massive failure

on minimality in the partial realization problem abstract a compression algorithm is constructed which allows one to reduce an arbitrary realization of a finite sequence of matrices m 1 m r to a minimal partial realization of m 1 m r furthermore a state space criterion of minimality of a partial realization and a formula for the minimal state space dimension are obtained

toda flows inverse spectral transform and realization theory abstract in this paper we establish some new relationships between the solution techniques for completely integrable nonlinear differential equations of the toda type and topics in realization theory

canonical correlation analysis approximate covariance extension and identification of stationary time series abstract in this paper we analyze a class of state space identification algorithms for time series based on canonical correlation analysis in the light of recent results on stochastic systems theory in principle these so called subspace methods can be described as covariance estimation followed by stochastic realization the methods offer the major advantage of converting the nonlinear parameter estimation phase in traditional arma models identification into the solution of a riccati equation but introduce at the same time some nontrivial mathematical problems related to positivity the reason for this is that an essential part of the problem is equivalent to the well known rational covariance extension problem therefore the usual deterministic arguments based on factorization of a hankel matrix are not valid for generic data something that is habitually overlooked in the literature we demonstrate that there is no guarantee that several popular identification procedures based on the same principle will not fail to produce a positive extension unless some rather stringent assumptions are made which in general are not explicitly reported in this paper the statistical problem of stochastic modeling from estimated covariances is phrased in the geometric language of stochastic realization theory we review the basic ideas of stochastic realization theory in the context of identification discuss the concept of stochastic balancing and of stochastic model reduction by principal subsystem truncation the model reduction method of desai and pal 1982 a realization approach to stochastic model reduction proc 1st decision and control conf pp 1105 1112 based on truncated balanced stochastic realizations is partially justified showing that the reduced system structure has a positive covariance sequence but is in general not balanced as a byproduct of this analysis we obtain a theorem prescribing conditions under which the subspace identification methods produce bona fide stochastic systems

a hybrid approach to the computation of the inertia of a parametric family of bezoutians with application to some stability problems for bivariate polynomials abstract given two polynomials with coefficients over k k the associated bezout matrix b k with entries over z k defines a parametric family of bezout matrices with entries over k it is intended in this paper to propose a hybrid approach for determining the inertia of b k for any value of k in some real interval this yields an efficient solution to certain root location problems for bivariate polynomials we first develop a fast fraction free method for computing an inverse triangular factorization of the bezout matrix b k over the integral domain z k in this way we may easily compute the sequence i k of the training principal minors of b k for almost any value k of k the associated sign sequence sign i k specifies the inertia of b k the function sign i k is final obtained by numerically computing rational approximations of the real zeros of i k z k

matrix pade approximation definitions and properties abstract some proposals are made to give a general definition of matrix pade approximants depending on the normalization of the denominator we define type i constant term is the unit matrix and type ii by conditions on the leading coefficient approximants existence and uniqueness are considered determinant expressions are given and relations among types i ii and left right approximants are considered

partial realization of symmetric matrix rational functions abstract the partial realization problem is treated for the special case in which the given data are a sequence of symmetric matrices the induced algebraic structure of the state space for this special case is analyzed and implemented to construct rational matrix functions whose coefficients in the power series expansion match with the given sequence up to a certain finite number the interrelation between the mcmillan degree and the number of coefficients that match is also treated

connectivity properties of spaces of partial realizations abstract this paper addresses the problem of determining the number of connected components of the space of real scalar m n hankel matrices of rank n for n

linear feedback shift registers and the minimal realization problem the first author is partially supported by grants mineco mtm2017 83624 p mtm2017 90682 redt and upv ehu giu16 42 the second author is partially supported by grants mineco mtm2017 83624 p and mtm2017 90682 redt

block diagonalization and lu equivalence of hankel matrices this article presents a new algorithm for obtaining a block diagonalization of hankel matrices by means of truncated polynomial divisions such that every block is a lower hankel matrix in fact the algorithm generates a block lu factorization of the matrix two applications of this algorithm are also presented by the one hand this algorithm yields an algebraic proof of frobenius theorem which gives the signature of a real regular hankel matrix by using the signs of its principal leading minors on the other hand the close relationship between hankel matrices and linearly recurrent sequences leads to a comparison with the berlekamp massey algorithm

structure preserving model reduction of port hamiltonian systems by moment matching at infinity model reduction of port hamiltonian systems by means of the krylov methods is considered aiming at port hamiltonian structure preservation it is shown how to employ the arnoldi method for model reduction in a particular coordinate system in order to preserve not only a specific number of the markov parameters but also the port hamiltonian structure for the reduced order model furthermore it is shown how the lanczos method can be applied in a structure preserving manner to a subclass of port hamiltonian systems which is characterized by an algebraic condition in fact for the same subclass of port hamiltonian systems the arnoldi method and the lanczos method turn out to be equivalent in the sense of producing reduced order port hamiltonian models with the same transfer function

an algorithm for constructing a minimal partial realization in the multivariable case it is known that the berlekamp massey algorithm from information theory can be used to compute scalar minimal partial realizations recently it has been interpreted in terms of exact modeling of behaviors in this paper we extend these results to the multivariable case we put the behavioral theory of exact modeling to work in deriving an extension of the berlekamp massey algorithm for computing multivariable minimal partial realizations

the sylvester equation and approximate balanced reduction abstract the purpose of this paper is to investigate the problem of iterative computation of approximately balanced reduced order systems the resulting approach is completely automatic once an error tolerance is specified and also yields an error bound this is to be contrasted with existing projection methods namely pade via lanczos pvl and rational krylov which do not satisfy these properties our approach is based on the computation and approximation of the cross gramian of the system the cross gramian is the solution of a sylvester equation and therefore some effort is dedicated to the study of this equation leading to some new insights our method produces a low rank approximation to this gramian in factored form and thus directly provides a reduced order model and a reduced basis for the original system it is well suited to large scale problems because there are no matrix factorizations of the large sparse system matrix only matrix vector products are required

a canonical matrix continued fraction solution of the minimal partial realization problem abstract a canonical algorithm for minimal partial realization of mimo systems is given the paper starts from the work of dickinson et al and anderson et al but reformulates it in the style of antoulas so that the elements to construct a matrix continued fraction are obtained it gives a detailed explicit algorithm to compute a uniquely defined canonical form of this continued fraction a number of remarkable properties of this canonical form are proved an example is included

partial realization of descriptor systems abstract we address the partial realization problem for linear descriptor systems a general solution to this problem using the markov parameters of the system defined via its laurent series is provided for proper descriptor systems we also discuss numerically feasible algorithms for computing a partial minimal realization based on the unsymmetric lanczos and arnoldi processes applications to model reduction for examples from computational fluid dynamics and mechanical systems with holonomic constraints are also given

on the geometry of realizable markov parameters by simo and miso systems abstract let p m n d be positive integers and let l n d denote the set of sequences l l 1 l n of p m real or complex matrices which are realizable by systems of minimal order d it was shown in 5 14 that l n d can be endowed with a structure of differentiable manifold when p m 1 that is when the sequences are realizable by single input single output siso systems in this paper a similar result is obtained for more general sequences specifically we will consider the set l n r s of sequences l which are realizable by systems of minimal order d and having r and s as brunovsky indices of controllability and observability respectively it is shown in this paper that when one of the two collections of indices r or s is constant then l n r s can be provided with a structure of differentiable manifold and a formula of its dimension is given the special cases r 1 1 or s 1 1 correspond to sequences realizable respectively by single input multi output simo or multi input single output miso systems

a class of single input single output systems stabilizable by reduced order controllers abstract let the euclidian algorithm be applied to the numerator and denominator polynomial of an n th order system transfer function it isshown that there exists an n 1 k nt order stabilizing controller provided one of the remainder polynomials occuring in the algorithm is a k th order hurwitz polunomial the controller is constructed using almost disturbance decoupling techniques the result specifies an algebraic condition under which a reduced order controller can be determined using a reduced order model obtained by minimal partial realization or a certain continued fraction expansion

identifiability and well posedness of shaping filter parameterizations a global analysis approach in this paper we study the well posedness of the problems of determining shaping filters from combinations of finite windows of cepstral coefficients covariance lags or markov parameters for example we determine whether there exists a shaping filter with a prescribed window of markov parameters and a prescribed window of covariance lags we show that several such problems are well posed in the sense of hadamard that is one can prove existence uniqueness identifiability and continuous dependence of the model on the measurements our starting point is the global analysis of linear systems where one studies an entire class of systems or models as a whole and where one views measurements such as covariance lags and cepstral coefficients or markov parameters from data as functions on the entire class this enables one to pose such problems in a way that tools from calculus optimization geometry and modern nonlinear analysis can be used to give a rigorous answer to such problems in an algorithm independent fashion in this language we prove that a window of cepstral coefficients and a window of covariance coefficients yield a bona fide coordinate system on the space of shaping filters thereby establishing existence uniqueness and smooth dependence of the model parameters on the measurements from data

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

recursive independent state space and matrix fraction realizations a recursive multivariate partial realization algorithm is proposed the novel features are that it shows the explicit link between state space and matrix fraction solutions and that each solution is obtained not from the other but directly from data this is achieved by means of a certain factorization of the generalized hankel matrix which yields the entries of both realizations updated with a low computational effort

an algebraic approach to the partial realization problem abstract the partial realization problem is treated algebraically using polynomial models the main idea is the kernels of certain projections of the hankel map which give rise to orthogonal direct sum decomposition of the state space minimality of partial realizations is based on minimal elements of these subspaces which arise as a natural consequence of the structural properties

partial realization and the euclidean algorithm the authors show how the euclidean algorithm fits into the behavioral framework of exact modeling and how it computes solutions of the scalar minimal partial realization problem it turns out that the euclidean algorithm can be considered as a special instance of wolovich s procedure 1974 to achieve row reducedness for a given polynomial 2 spl times 2 matrix the authors show in detail how this approach yields a parameterization of all minimal solutions in terms of polynomials that are sequentially produced by the euclidean algorithm

on constructing a shortest linear recurrence relation it has been shown in the literature that a formulation of the minimal partial realization problem in terms of exact modeling of a behavior lends itself to an iterative polynomial solution for the scalar case we explicitly present such a solution in full detail unlike classical solution methods based on hankel matrices the algorithm is constructive it iteratively constructs a partial realization of minimal mcmillian degree the algorithm is known in information theory as the berlekamp massey algorithm and is used for constructing a shortest linear recurrence relation for a finite sequence of numbers

a simple hankel interpretation of the berlekamp massey algorithm abstract a simple interpretation of the berlekamp massey algorithm in the light of the hankel matrix is presented the salient result is that the jump of the linear feedback shift register lfsr length is derived almost trivially from the so called iohvidov index of the hankel matrix prior to making any reference to the berlekamp massey algorithm itself next the hankel system of equations that yields the updated connection polynomial is solved via the natural lu factorization of the hankel matrix which itself leads to the berlekamp massey algorithm in a simple and transparent manner