improved weighted least squares algorithm for the design of quadrature mirror filters this paper presents an improved weighted least squares wls algorithm for the design of quadrature mirror filters qmfs first a new term is incorporated into the objective function that effectively prevents an optimization algorithm from producing suboptimal qmfs these suboptimal qmfs exhibit a transition band anomaly the frequency responses of the filters have large oscillatory components in the transition band the new term can be applied to the wls design of any fir filter to prevent a similar transition band anomaly next we present an algorithm to obtain the qmf coefficients that minimize the objective function incorporating the new term the computational requirement of this algorithm is also briefly discussed last we include a set of practical design rules for use with our algorithm these rules simplify the design process by providing good estimation of the design parameters such as the minimum filter length to meet a given set of qmf specifications

on fiedler and parter vertices of acyclic matrices let a be a real symmetric matrix and let be a real number the algebraic multiplicity of as an eigenvalue of a is denoted by ma and the principal submatrix of a obtained by deleting row and column i from a is denoted by a i if ma i ma resp ma i ma then index i is said to be a fiedler vertex resp a parter vertex of a for in this paper we provide geometric characterizations of fiedler and parter vertices of acyclic matrices and give a geometric proof for the parter wiener theorem in c r johnson a leal duarte c m saiago the parter wiener theorem refinement and generalization siam j matrix anal appl 25 2003 352 361 furthermore we describe a structure of an acyclic matrix in terms of fiedler and parter vertices which enables us to construct an acyclic matrix of a desired form according to the locations of fiedler and parter vertices

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

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

tensor sylvester matrices and the fisher information matrix of varmax processes the purpose of this paper is to develop compact expressions for the fisher information matrix fim of a gaussian stationary vector autoregressive and moving average process with exogenous or input variables a vector armax or varmax process we develop a representation of the fim based on multiple sylvester matrices an extension of this representation yields another one but in terms of tensor sylvester matrices in order to obtain the results presented in this paper the approach used in a klein g melard p spreij on the resultant property of the fisher information matrix of a vector arma process linear algebra appl 403 2005 291 313 is extended

numerical solution of integral equations fast algorithms and krein sobolev equation this paper contains a unified rigorous approach for the treatment of fast numerical algorithms for different classes of fredholm integral equations of second kind the krein sobolev functional differential nonlinear equation for the resolvent provides the ground for a unified approach new results concerning the general analysis of the krein sobolev equation and the convergence and stability of related numerical schemes are also presented

computation of theoretical autocovariance matrices of multivariate autoregressive moving average time series matrix expressions relating the theoretical autocovariances of arma processes to their parameters are derived and used to design an efficient procedure for computing autocovariance sequences of multivariate arma processes the method proposed is more efficient than others suggested in the literature and in particular reduces the computational burden associated with exact maximum likelihood estimation of arma models the closed form expressions facilitate the implementation of algorithm for computing multivariate autocovariances

integration with respect to fractal functions and stochastic calculus i the classical lebesgue stieltjes integral b a fdg of real or complex valued functions on a finite interval a b is extended to a large class of integrands f and integrators g of unbounded variation the key is to use composition formulas and integration by part rules for fractional integrals and weyl derivatives in the special case of holder continuous functions f and g of summed order greater than 1 convergence of the corresponding riemann stieltjes sums is proved the results are applied to stochastic integrals where g is replaced by the wiener process and f by adapted as well as anticipating random functions in the anticipating case we work within slobodeckij spaces and introduce a stochastic integral for which the classical ito formula remains valid moreover this approach enables us to derive calculation rules for pathwise defined stochastic integrals with respect to fractional brownian motion

new analogs of split algorithms for arbitrary toeplitz plus hankel matrices fast algorithms for solving arbitrary toeplitz plus hankel systems of equations are presented the algorithms are analogs of the split levinson and schur algorithms although the more general toeplitz plus hankel structure requires that the algorithms be based on a four term recurrence relations with the previous split algorithms are considered the algorithms require roughly half as many multiplications as previous fast algorithms for toeplitz plus hankel systems

all pairs suffice for a p set abstract a p set of a symmetric matrix a is a set of indices such that the nullity of the matrix obtained from a by removing rows and columns indexed by is more than the nullity of a it is known that each subset of a p set is a p set it is also known that a set of indices such that each singleton subset is a p set need not be a p set this note shows that if all pairs of vertices of a set with at least two elements are p sets then the set is a p set

the euclid algorithm and the fast computation of cross covariance and autocovariance sequences a simple linear procedure is given to compute the cross covariance sequence associated with the outputs of two rational digital transfer functions driven by the same white noise sequence such a computation often appears in the study of digital filters in wiener filtering in noise variance estimation in the study of low order approximations and in the study of multichannel systems a fast algorithm based on the euclid algorithm is introduced to solve the linear system of equations involved in the computation and a detailed analysis of the matrix is given the special case of the autocovariance computation is reviewed and the same study is performed alternate polynomial presentations are given and are shown to involve the same matrices and similar fast algorithms

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

generation of cross covariance sequences we propose a linear method for the generation of cross covariance sequences in multichannel systems described by rational transfer functions the method relies on the determination of a linear discrete time filter whose impulse response is the desired cross covariance sequence we also prove existence and uniqueness of the solution

a note on the evaluation of complex integrals using filtering interpretations the evaluation of complex integrals of the form r k frac 1 2 pij oint h z h z 1 z k frac dz z for all k is frequently required in control communications and digital filtering problems the major drawback of all existing methods is that the evaluation of the integral has to be preceded or followed by a check of the stability of the transfer function h z to ensure that r k min infin max infin is a true correlation sequence in this correspondence we modify the method proposed in 1 to yield a simple linear algorithm that does not require a separate cheek for the stability of h z the filtering interpretations proposed in 1 are extended to the evaluation of more general complex integrals with integrands of the form h z g z 1 finally an algorithm using the levinson recursions is presented as a fast method to generate correlation sequences in the arma case

toward a fundamental theory of optimal feature selection part ii implementation and computational complexit certain algorithms and their computational complexity are examined for use in a vlsi implementation of the real time pattern classifier described in part i of this work the most computationally intensive processing is found in the classifier training mode wherein subsets of the largest and smallest eigenvalues and associated eigenvectors of the input data covariance pair must be computed it is shown that if the matrix of interest is centrosymmetric and the method for eigensystem decomposition is operator based the problem architecture assumes a parallel form such a matrix structure is found in a wide variety of pattern recognition and speech and signal processing applications each of the parallel channels requires only two specialized matrix arithmetic modules these modules may be implemented as linear arrays of processing elements having at most o n elements where n is the input data vector dimension the computations may be done in o n time steps this compares favorably to o n3 operations for a conventional or general rotation based eigensystem solver and even the o 2n2 operations using an approach incorporating the fast levinson algorithm for a matrix of toeplitz structure since the underlying matrix in this work does not possess a toeplitz structure some examples are provided on the convergence of a conventional iterative approach and a novel two stage iterative method for eigensystem decomposition

multiband least squares fir filter design this paper gives two methods for the least squares approximation design of fir digital filters they both allow multiple pass and stop bands and arbitrary transition bands all with explicit control of band edges the first method is as simple to use as a window method this design method builds up an optimal multiband frequency response by sequentially adding and subtracting optimal lowpass filters with transition bands however it assumes the ideal response has spline transition functions and does not allow error weighting the second method allows different weights in the multiple passbands stopbands and transition bands but it requires numerical solution of simultaneous equations which may be ill conditioned these methods compliment the parks mcclellan algorithm which minimizes the chebyshev error

linear interpolation lattice for nonstationary signals a ladder algorithm for linear interpolation of nonstationary signals is developed the algorithm is based on the sliding window least squares method and can be implemented using a lattice structure furthermore by assuming that the input signal is stationary the number of parameters required to be calculated is reduced the lattice structure in the case of stationary input is also presented

spectral estimation via minimum energy correlation extension it is well known that the discrete fourier transform of a truncated correlation sequence the blackman tukey estimate can become negative at certain frequencies since the true power spectrum must in fact be positive negative estimates are undesirable in this paper we obtain the positive spectral estimate which is as close to the blaekman tukey estimate as possible in the l 2 norm while still matching the correlation constraints in the time domain this estimate yields the correlation extension of least energy for the specified truncated correlation sequence algorithms and numerical examples will be provided multidimensional minimum energy correlation extension is discussed in multidimensions the minimum energy estimate is shown to exist in certain cases when the autoregressive maximum entropy estimate fails to exist

weighted least squares design of recursive allpass filters a method for the design of allpass filters is described in this method an error reflecting the difference between the desired phase response and the phase response of the practical allpass filter is formulated in a quadratic form the coefficients are obtained by solving a system of linear equations involving the sum of a toeplitz and an hankel matrix

computing theoretical autocovariances of multivariate autoregressive moving average models by using a block levinson method a closed form expression relating the theoretical autocovariances of a multivariate autoregressive moving average arma model to the arma parameters in terms of a system of linear equations has been presented recently using this result and the special hankel plus toeplitz structure of the coefficient matrix obtained after suitable partitioning this paper presents a method for computing multivariate autocovariance functions that is based on a block levinson procedure the method is particularly advantageous for models with high order autoregressive components and or a large number of variables

inversion of certain extensions of toeplitz matrices it is shown how an algorithm for inverting toeplitz matrices using o n2 operations can be modified to deal with a certain type of extension named conjugate toeplitz matrices the block partitioned case is also included

synthesis of extremal wavelet generating filters using gaussian quadrature orthogonal wavelets can be generated from finite impulse response quadrature mirror filters these filters are also used in perfect reconstruction filter banks this paper addresses the problem of efficiently synthesizing such filters a class of extremal filters is defined by the property that their magnitude spectrum maximizes an integral criterion it is found that these filters are characterized by their zeros on the unit circle which frequently can be obtained from a set of orthogonal polynomials a family of filters is constructed that minimize the subband aliasing energy and can generate wavelets with an arbitrary number of vanishing moments the algorithm for generating these filters makes use of the levinson recursions gaussian quadrature and a fast version of euclid s algorithm similar to other methods for constructing quadrature mirror filters the spectral factorization of a polynomial is the computationally expensive part of this algorithm

mixed stochastic differential equations with long range dependence existence uniqueness and convergence of solutions for a mixed stochastic differential equation involving standard brownian motion and an almost surely holder continuous process z with holder exponent c 1 2 we establish a new result on its unique solvability we also establish an estimate for difference of solutions to such equations with different processes z and deduce a corresponding limit theorem as a by product we obtain a result on existence of moments of a solution to a mixed equation under an assumption that z has certain exponential moments

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

a fast algorithm for optimal linear interpolation a fast algorithm for computing the optimal linear interpolation filter is developed the algorithm is based on the sherman morrison inversion formula for symmetric matrices the relationship between the derived algorithm and the levinson algorithm is illustrated it is shown that the new algorithm in comparison with the well known algorithms requires fewer multiplications and hence is of lower complexity

a newton raphson method for moving average spectral factorization using the euclid algorithm an implementation of the newton raphson approach to compute the minimum phase moving average spectral factor of a finite positive definite correlation sequence is presented each step in the successive approximation method involves a system of linear equations that is solved using either the levinson algorithm backwards the jury stability test or a symmetrized version of the euclid algorithm various properties of the newton raphson map are studied the algorithm is generalized to other symmetries other than with respect to the unit circle the special case of the symmetry with respect to the imaginary axis is presented and related to the routh hurwitz stability test for continuous time transfer function

convergence of solutions of mixed stochastic delay differential equations with applications the paper is concerned with a mixed stochastic delay differential equation involving both a wiener process and a holder continuous process with 1 2 e g a fractional brownian motion with hurst parameter greater than 1 2 it is shown that its solution depends continuously on the coefficients and the initial data two applications of this result are given the convergence of solutions to equations with vanishing delay to the solution of equation without delay and the convergence of euler approximations for mixed stochastic differential equations as a side result of independent interest the integrability of solution to mixed stochastic delay differential equations is established

an optimal instrumental variable method for arma spectral estimation a multistep iterative and fast recursive algorithm for autoregressive moving average arma spectral estimation is presented the ar parameters of an arma process are estimated using the extended instrumental variable eiv method the optimal choice of instruments prefilter and weighting matrix is investigated a bootstrapping procedure that has computational convenience is proposed for the algorithm the statistical analysis and experiments show that the optimal iv estimate is unbiased consistent efficient asymptotically normal and equivalent to the maximum likelihood ml estimate and the prediction error pe estimate and the proposed algorithm has the advantages of sharper resolution less frequency bias and better efficiency of convergence

fast o n complexity algorithms for diagonal innovation matrices in general the direct solution of an n dimensional system of linear equations requires o n3 arithmetic operations frequently in high data rate signal processing fast algorithms of complexity lower than o n3 are required to solve a large system of linear equations in several interesting applications the specific structure of the coefficient matrix associated with the linear system may be used to reduce the required number of operations fast algorithms have been developed when the coefficient matrix is a toeplitz matrix a hankel matrix or when it can be represented as a sum of toeplitz and hankel matrices the arithmetic complexity associated with these fast algorithms is o n2 in this paper fast algorithms of o n are presented that can be used for a class of matrices called diagonal innovation matrices dim previous results for this class of matrices require an arithmetic complexity of o n2 a number of results are presented regarding linear systems having dim coefficient matrices and several special cases are examined the effect of recursively increasing the order of the coefficient matrix on the number of operations is studied and some observations are made

on a fast algorithm for the exact information matrix of a gaussian arma time series the paper is devoted to a new algorithm for the computation of the exact fisher information matrix of a gaussian autoregressive moving average time series the number of operations is an order of magnitude smaller than the fastest existing procedure the algorithm is based on a set of new recursions for the covariance matrix of the derivatives of the state vector with respect to the parameters combined with the chandrasekhar recursions used in the evaluation of the likelihood function

an exact recursion for the composite nearest neighbor degeneracy for a 2 n lattice space a set theoretic argument is utilized to develop a recursion relation that yields exactly the composite nearest neighbor degeneracy for simple indistinguishable particles distributed on a 2 n lattice space the associated generating functions as well as the expectation of the resulting statistics are also treated

a superfast solver for sylvester s resultant linear systems generated by a stable and an anti stable polynomial abstract we develop a superfast method for the solution of n m n m sylvester s resultant linear systems associated with two real polynomials a z and c z of degree n and m respectively where a z is a stable polynomial i e all its roots lie inside the unit circle whereas c z is an anti stable polynomial i e z m c z 1 is stable the proposed scheme proceeds by iteratively constructing a sequence of increasing approximations of the solution it is based on a blend of ideas from structured numerical linear algebra computational complex analysis and linear operator theory each iterative step can be performed in o n m log n m arithmetic operations by combining fast polynomial arithmetic based on fft with displacement rank theory for structured matrices in addition the resulting process is shown to be quadratically convergent right from the start since the approximation error at the j th iteration is o r 2 j where r max a i 0 i min c i 0 i denotes the separation ratio between the spectrum of a z and c z finally we report and discuss the results of many numerical experiments which confirm the effectiveness and the robustness of the proposed algorithm

computations with infinite toeplitz matrices and polynomials we relate polynomial computations with operations involving infinite band toeplitz matrices and show applications to the numerical solution of markov chains of nonlinear matrix equations to spectral factorizations and to the solution of finite toeplitz systems in particular two matrix versions of graeffe s iteration are introduced and their convergence properties are analyzed correlations between graeffe s iteration for matrix polynomials and cyclic reduction for block toeplitz matrices are pointed out the paper contains a systematic treatment of known topics and presentation of new results improvements and extensions

noise reduction in recursive digital filters using high order error feedback the problem of solving the optimal minimum noise error feedback coefficients for recursive digital filters is addressed in the general high order case it is shown that when minimum noise variance at the filter output is required the optimization problem leads to set of familiar wiener hopf or yule walker equations demonstrating that the optimal error feedback can be interpreted as a special case of wiener filtering as an alternative to the optimal solution the formulas for suboptimal error feedback with symmetric or antisymmetric coefficients are derived in addition the design of error feedback using power of two coefficients is discussed the efficiency of high order error feedback is examined by test implementations of the set of standard filters it is concluded that error feedback is a very powerful and versatile method for cutting down the quantization noise in any classical infinite impulse response iir filter implemented as a cascade of second order direct form sections the high order schemes are attractive for use with high order direct form sections

on the relative position of multiple eigenvalues in the spectrum of an hermitian matrix with a given graph for hermitian matrices whose graph is a given tree the relationships among vertex degrees multiple eigenvalues and the relative position of the underlying eigenvalue in the ordered spectrum are discussed in detail in the process certain aspects of special vertices whose removal results in an increase in multiplicity are investigated

on the computation of the cramer rao bound for arma parameter estimation the cramer rao lower bound crlb provides a useful tool for evaluating the performance of parameter estimation techniques several techniques for the computation of the asymptotic form of the crlb for arma models are presented it is shown that the asymptotic crlb can be expressed as an explicit function of the model parameters

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

computation of the fisher information matrix for siso models closed form expressions and an algorithm for obtaining the fisher information matrix of gaussian single input single output siso time series models are presented it enables the computation of the asymptotic covariance matrix of maximum likelihood estimators of the parameters the procedure makes use of the autocovariance function of one or more autoregressive processes under certain conditions the siso model can be a special case of a vector autoregressive moving average arma model for which there is a method to evaluate the fisher information matrix that method is compared with the procedure described in the paper

root finding by expansion with independent constraints elimination methods are highly effective for the solution of linear and nonlinear systems of equations but reversal of the elimination principle can be beneficial as well competent incorporation of additional independent constraints and variables or more generally immersion of the original computational problem into a larger task defined by a larger number of independent constraints and variables can improve global convergence of iterative algorithms that is their convergence from the start a well known example is the dual linear and nonlinear programming which enhances the power of optimization algorithms we believe that this is just an ad hoc application of general principle of expansion with independent constraints it should be explored systematically for devising iterative algorithms for the solution of equations and systems of equations and for optimization at the end of this paper we comment on other applications and extensions of this principle presently we show it at work for the approximation of a single zero of a univariate polynomial p of a degree n empirical global convergence of the known algorithms for this task is much weaker than that of the algorithms for all n zeros such as weierstrass durand kerner s root finder which reduces its root finding task to viete s vieta s system of n polynomial equations with n unknowns we adjust this root finder to the approximation of a single zero of p preserve its fast global convergence and decrease the number of arithmetic operations per iteration from quadratic to linear together with computing a zero of a polynomial p the algorithm deflates this polynomial as by product and then could be reapplied to the quotient to approximate the next zero of p alternatively by using m processors that exchange no data one can concurrently approximate up to m zeros of p our tests confirm the efficiency of the proposed algorithms technically our root finding boils down to computations with structured matrices polynomials and partial fraction decompositions our study of these links can be of independent interest e g as by product we express the inverse of a sylvester matrix via its last column thus extending the celebrated result of gohberg and sementsul 1972 22 from toeplitz to sylvester matrix inverses

discrete fast algorithms for two dimensional linear prediction on a polar raster new generalized split levinson and schur algorithms for the two dimensional linear least squares prediction problem on a polar raster are derived the algorithms compute the prediction filter for estimating a random field at the edge of a disk from noisy observations inside the disk the covariance function of the random field is assumed to have a toeplitz plus hankel structure for both its radial part and its transverse angular part this assumption is valid for some types of random fields such as isotropic random fields the algorithms generalize the split levinson and schur algorithms in two ways 1 to two dimensions and 2 to toeplitz plus hankel covariances

cramer rao bounds for deterministic modal analysis the accuracy with which deterministic modes can be identified from a finite record of noisy data is determined by computing the cramer rao bound on the error covariance matrix of any unbiased estimator of mode parameters the bound is computed for many of the standard parametric descriptions of a mode including autoregressive and moving average parameters poles and residues and poles and zeros asymptotic frequency domain versions of the cramer rao bound bring insight into the role played by poles and zeros application of the bound to second and fourth order systems illustrates the influence of mode locations on the ability to identify them application of the bound to the estimation of an energy spectrum illuminates the accuracy of estimators that presume to resolve spectral peaks

fast algorithms for linear least squares smoothing problems in one and two dimensions using generalized discrete bellman siegert krein resolvent identities new fast algorithms for linear least square smoothing problems in one and two dimensions are derived these are discrete and multidimensional generalizations of the bellman siegert krein resolvent identity which has been applied to the continuous one dimensional stationary smoothing problem by t kailath 1969 the new equations relate the linear least squares prediction filters associated with discrete random fields to the smoothing filters for those fields this results in new fast algorithms for deriving the latter from the former in particular used in conjunction with recently developed generalized one two dimensional split levinson and schur algorithms for covariances with block toeplitz plus hankel structure these algorithms can be used to compute smoothing filters for random fields defined on a polar raster using fewer computations than those required by previous algorithms

spectral multiplicity and splitting results for a class of qualitative matrices abstract this paper presents some results concerning the location and multiplicity of eigenvalues of sign symmetric matrices whose associated graphs are trees in particular it extends previous spectral multiplicity and splitting results proved by others

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

correction to comments on a curiosum concerning discrete time convolution plus a remark in the author s comments to an article by hall and wise see ibid vol 38 no 6 p 1059 1990 included a typographic error which is corrected he also points out that the sequences used in the example become l sub 1 norm sequences

fast algorithms for linear prediction and system identification filters with linear phase a general finite impulse response fir filter can be used as a linear prediction filter if given only an input sample sequence or as a system identification model if given the input and output sequences from an unknown system with known correlation the coefficients of the fir filter that minimize the mean square error in both applications are found by solution of a set of normal equations with toeplitz structure using only data samples the coefficients that yield the least squared error in both applications are found by solution of a set of normal equations with near to toeplitz structure computationally efficient fast algorithms have been published to solve for the coefficients from both types of normal equation structures if the fir filter is constrained to have a linear phase then the impulse response must be symmetric this then leads to normal equations with toeplitz plus hankel or near to toeplitz plus hankel structure fast algorithms for solving these normal equations for the filter coefficients are developed in this paper they have computational complexity proportional to m2and parameter storage proportional to m where m is the filter order an application of one of these algorithms for spectral estimation concludes the paper

kalman filtering methods for computing information matrices for time invariant periodic and generally time varying varma models and samples abstract under general conditions the inverse sample information matrix can be used to establish a cramer rao lower bound of the covariance matrix of parameter estimates of a model and the inverse asymptotic information matrix is the asymptotic covariance matrix of the parameter estimates the paper does two things first it derives a recursive kalman filtering method for computing exact sample and asymptotic information matrices for time invariant periodic or generally time varying gaussian vector autoregressive moving average varma models and samples second it specializes the recursive method to a nonrecursive method for computing exact asymptotic information matrices for time invariant or periodic varma models and samples

linear statistical models for stationary sequences and related algorithms for cholesky factorization of toeplitz matrices we review the levinson durbin recursions for computing the cholesky factors of the inverse of a toeplitz correlation matrix r and the schur recursions for computing the cholesky factors of r itself we show that both algorithms may be imbedded in a single vector lattice recursion then depending only on how the initial conditions are set one gets either the levinson durbin or the leroux gueguen recursions for computing reflection coefficients we extend the analysis to treat the toeplitz nonsymmetric case and place the corresponding algorithms into the same framework we also review the relation between the vector lattice recursions and their implementations in a serial lattice structure then we show how the factorizations may be run backwards to compute the cholesky factors and correlations from the reflection coefficients this generalizes a result usually attributed to robinson and treitel one of our main purposes is to emphasize that the levinson durbin recursions for going back and forth between correlations reflection coefficients and autoregressive filter parameters may be replaced with a dual set of recursions for going back and forth between correlations reflection coefficients and moving average filter parameters when the correlation sequence is arma p p then we show how the schur recursions simplify to the morf sidhu kailath algorithm for computing kalman gains finally the msk algorithm may be specialized to the rissanen algorithm for factoring a pure ma correlation sequence

new progress in real and complex polynomial root finding matrix methods are increasingly popular for polynomial root finding the idea is to approximate the roots as the eigenvalues of the companion or generalized companion matrix associated with an input polynomial the algorithms also solve secular equation qr algorithm is the most customary method for eigen solving but we explore the inverse rayleigh quotient iteration instead which turns out to be competitive with the most popular root finders because of its excellence in exploiting matrix structure to advance the iteration we preprocess the matrix and incorporate newton s linearization repeated squaring homotopy continuation techniques and some heuristics the resulting algorithms accelerate the known numerical root finders for univariate polynomial and secular equations and are particularly well suited for the acceleration by using parallel processing furthermore even on serial computers the acceleration is dramatic for numerical approximation of the real roots in the typical case where they are much less numerous than all complex roots

blind deconvolution of symmetric noncausal impulse responses using two sided linear prediction considers the problem of estimating the time symmetric noncausal impulse response of a linear time invariant system from measurements of the response of the system to an unknown input signal which is assumed to be a realization of a white random process the symmetric impulse response is modeled by a two sided ar or arma system model the two sided ar coefficients are estimated using a two step procedure first an estimate of an unconstrained parameter vector is computed by solving a close to toeplitz plus hankel system of equations using previously developed fast algorithms then the polynomial square root of the result is obtained by solving a constrained least squares problem which has a simple solution unlike previous methods this approach requires no iterative procedure however it may lead to an unstable model in some extreme cases simulation results illustrate the performance of the proposed methods

computation of the exact information matrix of gaussian dynamic regression time series models in this paper the computation of the exact fisher information matrix of a large class of gaussian time series models is considered this class which is often called the single input single output siso model includes dynamic regression with autocorrelated errors and the transfer function model with autoregressive moving average errors the method is based on a combination of two computational procedures recursions for the covariance matrix of the derivatives of the state vector with respect to the parameters and the fast kalman filter recursions used in the evaluation of the likelihood function it is much faster than existing procedures an expression for the asymptotic information matrix is also given

fast and stable least squares approach for the design of linear phase fir filters we present a new numerical algorithm for solving the normal equations associated with the least squares design of linear phase fir filters the usual solution methods have a computational complexity of o n sup 3 moreover solving the normal equations with gaussian elimination commonly yields numerical errors especially when the filter is long here we convert a least squares method into the problem of constructing a system of orthonormal functions the proposed design algorithm needs only o n sup 2 computations and numerical errors can be reduced some examples are given to show the performance of the algorithm

maximal p sets of matrices whose graph is a tree abstract the graph of a real n n symmetric matrix a g a is the undirected graph on n vertices whose edges correspond precisely with the nonzero off diagonal entries of a let m a 0 denote the multiplicity of 0 as an eigenvalue of a it is possible that the multiplicity of 0 as an eigenvalue of an n 1 n 1 principal submatrix a n 1 of a is larger than m a 0 when this occurs the vertex in g a corresponding to the row and column of a that was deleted to form a n 1 is called a p vertex of a if there is a list of principal submatrices of a a a n 1 a n 2 a n k each one being obtained from the matrix preceding it in the list by deleting a row and column such that m a 0 m a n 1 0 m a n k 0 the set of vertices in g a corresponding to the rows and columns of a that were deleted to form a n k is a called a p set of a let a be a matrix whose graph is a tree i e a connected graph with no cycles it is known that a set of p vertices of a is not necessarily a p set of a a main result of this paper shows that if every subset of size 2 of a set is a p set of a then the entire set is a p set of a this and other results are used to determine the maximal p sets and thus all of the p sets of a

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

accelerated approximation of the complex roots and factors of a univariate polynomial the known algorithms approximate the roots of a complex univariate polynomial in nearly optimal arithmetic and boolean time they are however quite involved and require a high precision of computing when the degree of the input polynomial is large which causes numerical stability problems we observe that these difficulties do not appear at the initial stages of the algorithms and in our present paper we extend one of these stages analyze it and avoid the cited problems still achieving the solution within a nearly optimal complexity estimates provided that some mild initial isolation of the roots of the input polynomial has been ensured the resulting algorithms promise to be of some practical value for root finding and can be extended to the problem of polynomial factorization which is of interest on its own right we conclude with outlining such an extension which enables us to cover the cases of isolated multiple roots and root clusters

reconstruction of signals from phase efficient algorithms segmentation and generalizations in this paper we develop several new formulations of the phase only reconstruction problem which lead to new efficient algorithms for signal reconstruction we describe a procedure for approximately reconstructing a long duration signal from short segments phase only blind deconvolution for linear phase filters relies on mirror image symmetry in the filter transfer function h z we generalize this deconvolution result to include several other types of symmetries for h z

least squared error fir filter design with transition bands the authors propose the use of transition bands and transition functions in the ideal amplitude frequency response to allow the analytical design of optimal least squared error fir digital filters with an explicit control of the transition band edges design formulas are derived for approximations to ideal frequency responses which use pth order spline transition functions a mixed analytical and numerical method for zero error weight in the transition bands and passband and stopband error weighting functions with an integral squared error approximation are derived a variable order spline transition function is developed and a method for choosing the optimal order to minimize the integral squared approximation error is given

a new autoregressive method for high performance spectrum analysis in this paper a new method of autoregressive ar spectrum estimation is presented it shall be called two sided autoregressive spectrum estimation because an interpolative or smoothing model is postulated as opposed to the predictive one sided model used in ar modeling the matrix equations arising in the estimation procedures proposed in this paper exhibit a special structure the exploitation of these structures leads to fast solutions that reduce the total number of computations by an order of magnitude compared with straightforward approaches also special attention is directed to the constrained two sided ar model simulation examples show higher resolution capability of the proposed method relative to the least squares ar method

a fast algorithm for the exact likelihood of stationary and partially nonstationary vector autoregressive moving average processes summary an expression for the likelihood function of a stationary vector autoregressive moving average process is developed the expression is very efficient numerically and applies to any stationary but not necessarily invertible model in particular when the multivariate process is autoregressive the exact likelihood can be evaluated with a small number of operations depending on the order of the autoregressive operator and the process dimension but not on the size of the observed series the expression also provides an efficient method for the evaluation of the exact likelihood of a partially nonstationary vector autoregressive moving average process for which the determinant of the autoregressive operator has at least one unit root and the remaining roots are outside the unit circle this method does not require differencing the series so that complications caused by over differencing the series such as noninvertibility and parameter identifiability problems are avoided the results for autoregressive models are also applied to testing the stationarity and invertibility of any autoregressive moving average model with given parameter values

a hermitian toeplitz matrix is unitarily similar to a real toeplitz plus hankel matrix two unitary matrices are presented that transform a hermitian toeplitz matrix into a real toeplitz plus hankel matrix and vice versa additional properties and consequences of these unitary transformations are also presented for a certain class of n dimensional toeplitz plus hankel systems of equations an efficient method of solution is presented an example is given to illustrate the similarity transforms toeplitz plus hankel systems arise in a wide variety of applications such as linear filtering theory discrete inverse scattering and discretization of certain integral equations arising in mathematical physics

matrix differential calculus applied to multiple stationary time series and an extended whittle formula for information matrices abstract the purpose of this paper is to set forth easily implementable expressions for the fisher information matrix fim of a gaussian stationary vector autoregressive and moving average process with exogenous or input variables a vector armax or varmax process the entries of the fim are represented as circular integral expressions and can be computed by applying cauchy s residue theorem an extension of the whittle formula for the fim of multiple time series processes is developed for varmax processes it will be shown that the extended whittle formula yields the fim when a bivariate structure consisting of the varmax process and the exogenous input process is considered consequently the equivalence between a frequency and time domain representation of the fim of varmax processes is established in order to obtain the results presented in this paper the differentiation techniques developed and used in a klein p spreij an explicit expression for the fisher information matrix of a multiple time series process linear algebra appl 417 2006 140 149 are applied

the number of p vertices in a matrix with maximum nullity abstract let t be a tree with n 2 vertices set s t for the set of all real symmetric matrices whose graph is t let a s t and i 1 n we denote by a i the principal submatrix of a obtained after deleting the row and column i we set m a 0 for the multiplicity of the eigenvalue zero in a the nullity of a when m a i 0 m a 0 1 we say that i is a p vertex of a as usual m t denotes the maximum nullity occurring of b s t in this paper we determine an upper bound and a lower bound for the number of p vertices in a matrix a s t with nullity m t we also prove that if the integer b is between these two bounds then there is a matrix e s t with b p vertices and maximum nullity