on the methods for solving yule walker equations the three well known fast algorithms for the solution of yule walker equations the levinson euclidean and berlekamp massey algorithms are reviewed and the relationship between each of them and the pade approximation problem is shown the algorithms are classified with reference to three criteria namely 1 the path they follow in the pade table 2 the organization of the computations one pass or two pass and 3 the auxiliary variables used this classification shows that the set of known classical algorithms is not complete and the missing variants are proposed with these variants of the berlekamp massey and euclid algorithms both forward and backward predictors can be obtained without additional cost a unified representation of the two pass algorithms is given in such a way that the application of the divide and conquer strategy becomes straightforward a general doubling algorithm which represents all the associated doubling algorithms in an exhaustive way is provided

necessary conditions of asymptotic stability for unilateral dynamical systems in this paper we develop a mathematical tool that can be used to state necessary conditions of asymptotic stability of isolated stationary solutions of a class of unilateral dynamical systems more precisely nonlinear evolution variational inequalities are considered instability criteria are also given applications can be found in mechanics or electrical circuits

differential variational inequalities this paper introduces and studies the class of differential variational inequalities dvis in a finite dimensional euclidean space the dvi provides a powerful modeling paradigm for many applied problems in which dynamics inequalities and discontinuities are present examples of such problems include constrained time dependent physical systems with unilateral constraints differential nash games and hybrid engineering systems with variable structures the dvi unifies several mathematical problem classes that include ordinary differential equations odes with smooth and discontinuous right hand sides differential algebraic equations daes dynamic complementarity systems and evolutionary variational inequalities conditions are presented under which the dvi can be converted either locally or globally to an equivalent ode with a lipschitz continuous right hand function for dvis that cannot be so converted we consider their numerical resolution via an euler time stepping procedure which involves the solution of a sequence of finite dimensional variational inequalities borrowing results from differential inclusions dis with upper semicontinuous closed and convex valued multifunctions we establish the convergence of such a procedure for solving initial value dvis we also present a class of dvis for which the theory of dis is not directly applicable and yet similar convergence can be established finally we extend the method to a boundary value dvi and provide conditions for the convergence of the method the results in this paper pertain exclusively to systems with index not exceeding two and which have absolutely continuous solutions

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

on a new class of differential variational inequalities and a stability result this paper addresses a new class of differential variational inequalities that have recently been introduced and investigated in finite dimensions as a new modeling paradigm of variational analysis to treat many applied problems in engineering operations research and physical sciences this new subclass of general differential inclusions unifies ordinary differential equations with possibly discontinuous right hand sides differential algebraic systems with constraints dynamic complementarity systems and evolutionary variational systems the purpose of this paper is two fold firstly we show that these differential variational inequalities when considering slow solutions and the more general level of a hilbert space contain projected dynamical systems another recent subclass of general differential inclusions this relation follows from a precise geometric description of the directional derivative of the metric projection in hilbert space which is based on the notion of the quasi relative interior secondly we are concerned with stability of the solution set to this class of differential variational inequalities here we present a novel upper set convergence result with respect to perturbations in the data including perturbations of the associated set valued maps and the constraint set here we impose weak convergence assumptions on the perturbed set valued maps use the monotonicity method of browder and minty and employ mosco convergence as set convergence also as a consequence we obtain a stability result for linear complementarity systems

a new algorithm for solving toeplitz systems of equations abstract we present some recurrences that are the basis for an algorithm to invert an n n toeplitz system of linear equations with computational complexity o n log 2 n the recurrences used are closely related to the zohar trench and bareiss algorithms but do not have any obvious connection with other asymptotically fast algorithms for the inversion of toeplitz systems

on the optimality of the discrete karhunen loeve expansion a short proof of the optimality of the discrete karhunen loeve expansion as the best linear approximation in the quadratic mean is presented

a generalization of the levinson algorithm for hermitian toeplitz matrices with any rank profile the paper describes a recursive algorithm for solving hermitian toeplitz systems of linear equations without any restriction on the ranks of their nested toeplitz subsystems such a general algorithm is needed e g to obtain the eigenfilters for signal processing applications or to compute the inverse of a nondefinite toeplitz matrix the regular portion of the algorithm is made of the classical levinson recursion the singular portion requires solving some well defined systems of linear equations with gradient structure the dimension of each of these sytems equals the amplitude of the corresponding singularity

projected gabor deconvolution abstractgabor deconvolution consists of approximating a nonstationary problem as several stationary subproblems via the gabor frame solving each subproblem independently and then recombining projecting the subsolutions into an approximate solution to the original nonstationary problem the approximations however cause inherent instability due to systematic errors that prevent the algorithm from converging to the true solution even for noise free cases furthermore the method will be time consuming when a nonlinear optimization is used for shaping the reflectivity structure an alternative projected gabor deconvolution is considered which is based on the gabor expansion however in contrast to the conventional form in the new method first the problem is projected into the time domain and then an inversion is performed to obtain the final reflectivity compared with the gabor deconvolution the projected alternative 1 exhibits an improved convergence property 2 is more efficient for sparse de

solution dependence on initial conditions in differential variational inequalities in the first part of this paper we establish several sensitivity results of the solution x t to the ordinary differential equation ode initial value problem ivp dx dt f x x 0 as a function of the initial value for a nondifferentiable f x specifically we show that for xi t equiv x t xi 0 0 leq t leq t a if f is b differentiable on xi t then so is the solution operator x t at 0 b if f is semismooth on xi t then so is x t at 0 c if f has a linear newton approximation on xi t then so does x t at 0 moreover the linear newton approximation of the solution operator can be obtained from the solution of a linear differential inclusion in the second part of the paper we apply these ode sensitivity results to a differential variational inequality dvi and discuss a the existence uniqueness and lipschitz dependence of solutions to subclasses of the dvi subject to boundary conditions via an implicit function theorem for semismooth equations and b the convergence of a nonsmooth shooting method for numerically computing such boundary value solutions

solving cone constrained convex programs by differential inclusions a differential inclusion is designed for solving cone constrained convex programs the method is of subgradient projection type it involves projection penalties and lagrangian relaxation nonsmooth data can be accommodated a novelty is that multipliers converge monotonically upwards to equilibrium levels an application to stochastic programming is considered

gabor deconvolution estimating reflectivity by nonstationary deconvolution of seismic data we have extended the method of stationary spiking deconvolution of seismic data to the context of nonstationary signals in which the nonstationarity is due to attenuation processes as in the stationary case we have assumed a statistically white reflectivity and a minimum phase source and attenuation process this extension is based on a nonstationary convolutional model which we have developed and related to the stationary convolutional model to facilitate our method we have devised a simple numerical approach to calculate the discrete gabor transform or complex valued time frequency decomposition of any signal although the fourier transform renders stationary convolution into exact multiplicative factors the gabor transform or windowed fourier transform induces only an approximate factorization of the nonstationary convolutional model this factorization serves as a guide to develop a smoothing process that when applied to the gabor transform of the nonstationary seismic trace estimates the magnitude of the time frequency attenuation function and the source wavelet by assuming that both are minimum phase processes their phases can be determined gabor deconvolution is accomplished by spectral division in the time frequency domain the complex valued gabor transform of the seismic trace is divided by the complex valued estimates of attenuation and source wavelet to estimate the gabor transform of the reflectivity an inverse gabor transform recovers the time domain reflectivity the technique has applications to synthetic data and real data

improved fourier and hartley transform algorithms application to cyclic convolution of real data this paper highlights the possible tradeoffs between arithmetic and structural complexity when computing cyclic convolution of real data in the transform domain both fourier and hartley based schemes are first explained in their usual form and then improved either from the structural point of view or in the number of operations involved namely we first present an algorithm for the in place computation of the discrete fourier transform on real data a decimation in time split radix algorithm more compact than the previously published one second we present a new fast hartley transform algorithm with a reduced number of operations a more regular convolution scheme based on fft s is also proposed finally we show that hartley transforms belong to a larger class of algorithms characterized by their generalized convolution property

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 stability result for differential inclusions in banach spaces on etablit un resultat de stabilite pour les inclusions differentielles definies sur un espace de banach separable

evolutionary problems driven by variational inequalities abstract in this paper we introduce the differential system obtained by mixing an evolution equation and a variational inequality eevi for short first by using kkm theorem and monotonicity arguments we prove the superpositional measurability and upper semicontinuity for the solution set of a general variational inequality then we establish that the solution set of eevi is nonempty and compact our approach is based on the theory of semigroups filippov implicit function lemma and fixed point theory for set valued mappings

implementation of split radix fft algorithms for complex real and real symmetric data a new algorithm is presented for the fast computation of the discrete fourier transform this algorithm belongs to that class of recently proposed 2n fft s which present the same arithmetic complexity the lowest among any previously published one moreover this algorithm has the advantage of being performed in place by repetitive use of a butterfly type structure without any data reordering inside the algorithm furthermore it can easily be applied to real and real symmetric data with reduced arithmetic complexity by removing all redundancy in the algorithm

on the equivalence between complementarity systems projected systems and differential inclusions in this note we prove the equivalence under appropriate conditions between several dynamical formalisms projected dynamical systems two types of differential inclusions and a class of complementarity dynamical systems each of these dynamical systems can also be considered as a hybrid dynamical system this work both generalizes previous results and sheds some new light on the relationship between known formalisms besides it exclusively uses tools from convex analysis

on parametric evolution inclusions of the subdifferential type with applications to optimal control problems in this paper we study parametric evolution inclusions of the subdifferential type and their applications to the sensitivity analysis of nonlinear infinite dimensional optimal control problems the parameter appears in all the data of the problem including the subdifferential operator first we establish several continuity results for the solution multifunction of the subdifferential inclusion then we study how these results can be used to examine the sensitivity properties variational stability of certain broad classes of nonlinear infinite dimensional optimal control problems some examples are worked out in detail illustrating the applicability of our work these include obstacle problems with time varying obstacles optimal control of distributed parameter systems and differential variational inequalities

reconstruction of a sparse spike train from a portion of its spectrum and application to high resolution deconvolution an algorithm is proposed for the reconstruction of a sparse spike train from an incomplete set of its fourier components it is shown that as little as 20 25 percent of the fourier spectrum is sufficient in practice for a high quality reconstruction the method employs linear programming to minimize the l1 norm of the output because minimization of this norm favors solutions with isolated spikes given a wavelet this technique can be used to perform deconvolution of noisy seismograms when the desired output is a sparse spike series relative reliability of the data is assessed in the frequency domain and only the reliable spectral data are included in the calculation of the spike series equations for the unknown spike amplitudes are solved to an accuracy compatible with the uncertainties in the reliable data in examples with 10 percent random noise the output is superior to that obtained using conventional least squares techniques

stochastic sparse spike deconvolution sparse spike deconvolution can be viewed as an inverse problem where the locations and amplitudes of a number of spikes reflectivity are estimated from noisy data seismic traces the main objective is to find the least number of spikes that when convolved with the available band limited seismicwaveletestimate fitthedatawithinagiventolerance errormisfit thedetectionofthespikes timelagsisahighly nonlinearoptimizationproblemthatcanbesolvedusingvery fastsimulatedannealingsa amplitudesareeasilyestimated using linear least squares at each sa iteration at this stage quadratic regularization is used to stabilize the solution to reduce its nonuniqueness and to provide meaningful reflectivitysequences thusavoidingtheneedtoconstrainthe spikes time lags and or amplitudes to force valid solutions impedanceconstraintsalsocanbeincludedatthisstage providing the low frequencies required to recover the acoustic impedance oneadvantageoftheproposedmethodoverother sparse spike deconvolution techniques is that the uncertaintyoftheobtainedsolutionscanbeestimatedstochastically further errorsinthephaseofthewaveletestimatearetolerated foranoptimumconstant phaseshiftisobtainedtocalibratetheeffectivewaveletthatispresentinthedata results using synthetic data including simulated data for the marmousi2 modelandfield 3d data show that physically meaningful high resolution sparse spike sections can be derived fromband limitednoisydata evenwhentheavailablewaveletestimateisinaccurate

seismic absorption compensation a least squares inverse scheme the problem of instability plagues conventional inverse q filtering we formulate the deabsorption problem as an inverse problem in terms of least squares and impose regularization by means of bayes theorem the solution is iterative and nonparametric and returns a reflectivity that has been constrained to be sparse the inverse scheme is tested on both synthetic and real data and the results obtained demonstrate the viability of the approach

existence result for differential variational inequality with relaxing the convexity condition abstract in this paper a class of differential variational inequalities are studied and a new approach is introduced to relax the convexity condition firstly an existence theorem of caratheodory weak solution for the differential variational inequalities is established secondly an algorithm for solving the problem is developed and the convergence analysis for the algorithm is given finally a numerical example is reported to illustrate the proposed algorithm

linear complementarity systems we introduce a new class of dynamical systems called linear complementarity systems the time evolution of these systems consists of a series of continuous phases separated by events which cause a change in dynamics and possibly a jump in the state vector the occurrence of events is governed by certain inequalities similar to those appearing in the linear complementarity problem of mathematical programming the framework we describe is suitable for certain situations in which both differential equations and inequalities play a role for instance in mechanics electrical networks piecewise linear systems and dynamic optimization we present a precise definition of the solution concept of linear complementarity systems and give sufficient conditions for existence and uniqueness of solutions