boundary value methods the third way between linear multistep and runge kutta methods it is well known that the approximation of the solutions of odes by means of k step methods transforms a first order continuous problem in a kth order discrete one such transformation has the undesired effect of introducing spurious or parasitic solutions to be kept under control it is such control which is responsible of the main drawbacks e g the two dahlquist barriers of the classical lmf with respect to runge kutta methods it is however less known that the control of the parasitic solutions is much easier if the problem is transformed into an almost equivalent boundary value problem starting from such an idea a new class of multistep methods called boundary value methods bvms has been proposed and analyzed in the last few years of course they are free of barriers moreover a block version of such methods presents some similarity with runge kutta schemes although still maintaining the advantages of being linear methods in this paper the recent results on the subject are reviewed

parallel implementation of block boundary value methods for odes the parallel solution of initial value problems for odes has been the subject of much research in the last thirty years and different approaches to the problem have been devised in this paper we examine the parallel methods derived by block boundary value methods bvms recently introduced for approximating hamiltonian problems here we restrict the analysis of the methods when applied to linear problems since their nonlinear parallel implementation deserves further study however for linear problems the methods can reach a high parallel efficiency some of these solvers can also be adapted for approximating continuous two point boundary value problems numerical tests carried out on a distributed memory parallel computer are reported

vector arma estimation a reliable subspace approach a parameter estimation method for finite dimensional multivariate linear stochastic systems which is guaranteed to produce valid models approximating the true underlying system in a computational time of a polynomial order in the system dimension is presented this is achieved by combining the main features of certain stochastic subspace identification techniques with sound matrix schur restabilizing procedures and multivariate covariance fitting both of which are formulated as linear matrix inequality problems all aspects of the identification method are discussed with an emphasis on the two issues mentioned above and examples of the overall performance are provided for two different systems

moment problems and orthogonal functions abstract the caratheodory coefficient problem for an infinite sequence c n can be formulated as follows find a caratheodory function f z i e an analytic function mapping the unit disk d z z into the right half plane p w re w 0 whose taylor coefficients at a fixed points in d for convenience the origin are the c n this problem is equivalent to the trigonometric moment problem find a measure on such that e i n d c n for n 0 1 2 these problems are closely related to the theory of szego polynomials i e orthogonal polynomials on the unit circle t z z 1 let n be an arbitrary given sequence of not necessarily distinct points in d and let w n be a given sequence of numbers the nevanlinna pick problem for this situation is to find a caratheodory function f z with the interpolation property f n w n for all n when points n are repeated the interpolation property involves the appropriate number of taylor coefficients also this problem is connected with the problem of finding a measure generating moments the nevanlinna pick problem is related to a theory of rational functions which are orthogonal on t with poles at the points 1 n this relationship is analogous to and generalizes the relationship between the caratheodory coefficient problem or trigonometric moment problem and polynomials orthogonal on t the situation when points n are given on the boundary t or d is also briefly discussed

boundary value methods for volterra integral and integro differential equations abstract new and effective quadrature rules generated by boundary value methods are introduced we employ the introduced quadrature rules to construct quadrature methods for the second kind volterra integral equations and volterra integro differential equations these methods are shown to be effective and possess excellent convergence properties the nonlinear multigrid method is applied to solve the discrete systems derived from the introduced numerical scheme numerical simulations are presented and confirm the efficiency and accuracy of the methods

szego functions and multipoint pade approximation abstract rational functions analogous to szego polynomials are introduced and their application in multipoint pade approximation is discussed

computing stochastic continuous time models from arma models some algorithms for computing the underlying continuous time stochastic model from a sampled arma model are presented three algorithms are given all having a modest computational complexity the properties of the algorithms are analysed and also illustrated by means of numerical examples

on the kalman yakubovich popov lemma the purpose of this note is to present a new elementary proof for the multivariable k y p lemma a minimum of linear algebra and finite dimensional convexity theory is used

brief paper on the indirect approaches for carma model identification in this paper we study the problem of reconstructing a continuous time ct model from an identified discrete time dt model for a continuous time stochastic process we present a new necessary and sufficient condition for the existence of the solution we also show that the solution is unique if it exists our results are useful in modeling multivariable processes as well these results are then used to develop an algorithm where the intermediate discrete time model estimation is not necessary the performance of our algorithm is illustrated using numerical simulations

convergence and stability of boundary value methods for ordinary differential equations a usual way to approximate the solution of initial value problems for ordinary differential equations is the use of a linear multistep formula if the formula has k steps k values are needed to obtain the discrete solution the continuous problem provides only the initial value it is customary to impose the additional k 1 conditions at the successive k 1 initial points however the class of methods obtained in this way suffers from heavy limitations summarized by the two dahlquist barriers it is also possible to impose the additional conditions at different grid points for example some conditions can be imposed at the initial points and the remaining ones at the final points the obtained methods called boundary value methods bvms do not have barriers whatsoever in this paper the question of convergence of bvms is discussed along with the linear stability theory some numerical examples on stiff test problems are also presented

optimal instrumental variable estimates of the ar parameters of an arma process the modified yule walker myw equations for estimating the ar parameters of an arma process are presented as a special case of an instrumental variable iv method the consistency and accuracy of the ar parameter estimates are studied it is shown that estimation accuracy increases monotonically with the number of myw equations for an optimal choice of the weighting matrix used in the least squares solution of these equations the asymptotic error covariance of the optimal iv method equals that of the prediction error method the results of this paper verify experimental results reported in the literature regarding the performance of the myw method and provide the necessary accuracy analysis furthermore they suggest several simple asymptotically efficient multistep algorithms for estimating the ar parameters which are presented in a companion paper

boundary value techniques for initial value problems in ordinary differential equations the numerical solution of initial value problems in ordinary differential equations by means of boundary value techniques is considered we discuss a finite difference method which was already investigated by fox in 1954 and fox and mitchell in 1957 hereby we concentrate on explaining the fundamentals of the method because for initial value problems the boundary value method seems to be fairly unknown we further propose and discuss new jalerkin methods for initial value problems along the lines of the boundary value approach

block boundary value methods for linear hamiltonian systems the problem of characterizing multistep methods suitable to efficiently approximate the solutions of linear hamiltonian systems is discussed showing that the appropriate methods should belong to the class of discrete boundary value methods bvms three families of such methods are proposed the presented methods have infinite regions of absolute stability and can be of any order in fact for every odd k there are k step methods of order up to 2k which is the maximum order reachable by a k step formula

on zero locations for sampled stochastic systems zero locations of stochastic models have significant influence on optimal prediction algorithms as well as on the convergence speed of identification method the possible zero locations for a sampled stochastic system are examined in particular it is shown and numerically illustrated that the sampled zeros can appear only in a restricted set within the unit circle a complete analysis of second order systems is given and all possible zero locations are characterized it is shown how the zero location in this case depends on the continuous time model parameters and the sampling interval

a modified schur algorithm and an extended hamburger moment problem an algorithm for a pick nevanlinna problem where the interpola tion points coalesce into a finite set of points on the real line is introduced its connection with certain multipoint pade approximation problems is discussed and the results are used to obtain the solutions of an extended hamburger mo ment problem the pick nevanlinna problem is the following let za a a be a set of distinct points in the open upper half plane h z imz 0 and let wa a a be a set of points in c find a function f z which is analytic in such that imf z 0 for z e j and f z wa for all a a a function f z which is analytic for z hq with imf z 0 is called a nevanlinna function the problem was solved by pick 27 28 in the case that a is finite by nevanlinna 15 16 in the case that a is countable and by krein and rekhtman 14 in the general case a variant of the problem for finite or countable a arises when all points za coalesce to a single point a and given values wa at the points are replaced by the taylor coefficients at a a problem closely related to this is caratheodory s coefficient problem see 2 3 33 given a finite sequence y0 ym or an infinite sequence yn n n find a function f z which is analytic in the open unit disc d z z 0 for z d and f z ezu 2 ezm okm zk or f z er o z for historical remarks on this problem see 12 by introducing the linear fractional transformation 7 j we can refor mulate the requirement re7 z 0 to read 7 z 0 for z d or a schur function if 7 z 1 for z e 7j schur 31 invented an algorithm called the schur algorithm to deal with this problem the tech nique was adapted by nevanlinna to deal with the pick nevanlinna problem

subspace algorithms for the stochastic identification problem abstract in this paper we derive a new subspace algorithm to consistently identify stochastic state space models from given output data without forming the covariance matrix and using only semi infinite block hankel matrices the algorithm is based on the concept of principal angles and directions we describe how they can be calculated with qr and quotient singular value decomposition we also provide an interpretation of the principal directions as states of a non steady state kalman filter bank

limit results for sampled systems properties of discrete time systems obtained by sampling continuous time systems are described by introducing prefilters we can treat different ways of sampling within one framework results on the convergence of poles and zeros of transfer functions and noise filters as the sampling interval tends to zero are given these results are generalizations of the results of astrom hagander and sternby 1984 on the convergence of poles and zeros for zero order hold sampled transfer functions sampled noise models are also analysed knowledge of these properties is very important in for example discrete time simulations of continuous time systems and identification of continuous time models based on discrete time measurements

the use of compact boundary value method for the solution of two dimensional schrodinger equation in this paper a high order and accurate method is proposed for solving the unsteady two dimensional schrodinger equation we apply a compact finite difference approximation of fourth order for discretizing spatial derivatives and a boundary value method of fourth order for the time integration of the resulting linear system of ordinary differential equations the proposed method has fourth order accuracy in both space and time variables moreover this method is unconditionally stable due to the favorable stability property of boundary value methods the results of numerical experiments are compared with analytical solutions and with those provided by other methods in the literature these results show that the combination of a compact finite difference approximation of fourth order and a fourth order boundary value method gives an efficient algorithm for solving the two dimensional schrodinger equation

comparison of some recent numerical methods for initial value problems for stiff ordinary differential equations we consider the combustion equation as one of the candidates from the class of stiff ordinary differential equations a solution over a length of time that is inversely proportional to d 0 where d 0 is a small disturbance of the pre ignition state is sought this problem has a transient at the midpoint of the integration interval the solution changes from being non stiff to stiff and afterwards becomes non stiff again we provide its asymptotic and numerical solution obtained via a variety of methods comparisons are made for the numerical results which we obtain with the matlab ode solvers ode45 ode15s and ode23s and some nonstandard finite difference methods results corresponding to standard finite difference method are also presented furthermore the discussion on these approaches along with the others provides several open problems for new and young researchers

brief performance evaluation of methods for identifying continuous time autoregressive processes identification of continuous time autoregressive processes from discrete time data by replacing the differentiation operator by an approximation is considered a linear regression model can then be formulated the least squares method and the instrumental variables method must be used with some care to get parameter estimates of good quality the bias is studied explicitly in the paper together with the asymptotic distribution and expressions are presented for the covariance matrix of the estimated parameters it turns out that there are small differences in the dominating bias term for the different methods whereas the statistical properties are comparable overall the performance is similar to that of a prediction error method for short sampling intervals

brief identification of continuous time ar processes from unevenly sampled data when identifying a continuous time ar process from discrete time data an obvious approach is to replace the derivative operator in the continuous time model by an approximation in some cases a linear regression model can then be formulated the well known least squares method would be very desirable to apply since it enjoy good numerical properties and low computational complexity in particular for fast or nonuniform sampling the focus of this paper is the latter i e nonuniform sampling two consistent least squares schemes for the case of unevenly sampled data are presented the precise choice of derivative approximation turns out to be crucial the obtained results are compared to a prediction error method

relative zero location for second order sampled systems it is shown that the pole zero configuration of relevant families of second order continuous time transfer functions is preserved after sampling the results are valid for any sampling period and give tighter bounds for the location of the sample zero than those reported in earlier literature these results complement the analysis made by t soderstrom ibid vol 35 p 1249 53 1990 on the zero location of second order sampled systems

modeling continuous time processes via input to state filters a direct algorithm to estimate continuous time arma carma models is proposed in this paper in this approach we first pass the observed data through an input to state filter and compute the state covariance matrix the properties of the state covariance matrix are then exploited to estimate the half spectrum of the observed data at a set of user defined points on the right half plane finally the continuous time parameters are obtained from the half spectrum estimates by solving an analytic interpolation problem with a positive real constraint as shown by simulations the proposed algorithm delivers much more reliable estimates than indirect modeling approaches which rely on estimating an intermediate discrete time model

solutions of the strong hamburger moment problem abstract the strong hamburger moment problem for a bi infinite sequence c n n 0 1 2 can be described as follows 1 find conditions for the existence of a positive measure on such that c n t n d t for all n 2 when there is a solution find conditions for uniqueness of the solution 3 when there is more than one solution describe the family of all solutions in this paper a theory concerning question 3 is developed in particular an analog to the nevanlinna parametrization describing the solutions of the classical hamburger moment problem is given

strang type preconditioners for solving fractional diffusion equations by boundary value methods the finite difference scheme with the shifted grunwald formula is employed to semi discrete the fractional diffusion equations this spatial discretization can reduce to the large system of ordinary differential equations odes with initial values recently the boundary value method bvm was developed as a popular algorithm for solving the large systems of odes this method requires the solutions of one or more nonsymmetric and large scale linear systems in this paper the gmres method with the block circulant preconditioner is proposed to solve relevant linear systems some conclusions about the convergence analysis and spectrum of the preconditioned matrices are also drawn if the diffusion coefficients are constant finally extensive numerical experiments are reported to show the performance of our method for solving the fractional diffusion equations

orthogonal laurent polynomials and the strong hamburger moment problem abstract this paper is concerned with the strong hamburger moment problem shmp for a given double sequence of real numbers c c n does there exist a real valued bounded non decreasing function on with infinitely many points of increase such that for every integer n c n t n d t necessary and sufficient conditions for the existence of such a function are given in terms of the positivity of certain hankel determinants associated with c our approach is made through the study of orthogonal and quasi orthogonal laurent polynomials referred to here as l polynomials and closely related gaussian type quadrature formulas in the proof of sufficiency an inner product for l polynomials is defined in terms of the given double sequence c since orthogonal l polynomials are believed to be of interest in themselves some examples of specific systems are considered

sensor array processing based on subspace fitting algorithms for estimating unknown signal parameters from the measured output of a sensor array are considered in connection with the subspace fitting problem the methods considered are the deterministic maximum likelihood method ml esprit and a recently proposed multidimensional signal subspace method these methods are formulated in a subspace fitting based framework which provides insight into their algebraic and asymptotic relations it is shown that by introducing a specific weighting matrix the multidimensional signal subspace method can achieve the same asymptotic properties as the ml method the asymptotic distribution of the estimation error is derived for a general subspace weighting and the weighting that provides minimum variance estimates is identified the resulting optimal technique is termed the weighted subspace fitting wsf method numerical examples indicate that the asymptotic variance of the wsf estimates coincides with the cramer rao bound the performance improvement compared to the other techniques is found to be most prominent for highly correlated signals

sampling zeros and the euler frobenius polynomials we show that the zeros of sampled data systems resulting from rapid sampling of continuous time systems preceded by a zero order hold zoh are the roots of the euler frobenius polynomials using known properties of these polynomials we prove two conjectures of hagiwara et al 1993 the first of which concerns the simplicity negative realness and interlacing properties of the sampling zeros of zoh and first order hold foh sampled systems to prove the second conjecture we show that in the fast sampling limit and as the continuous time relative degree increases the largest sampling zero for foh sampled systems approaches 1 e where e is the base of the natural logarithm

unique solvability of an extended hamburger moment problem abstract r functions are rational functions with no poles in the extended complex plane outside a given set a1 ap of points on the real axis methods from the theory of orthogonal polynomials can be extended to r functions by this means the author solved an extended hamburger moment problem given sequences of real numbers cn i n 1 2 i 1 p find conditions for a distribution function to exist such that d t 1 d t t a i m c m i m 1 2 i 1 p in this paper these methods are extended to treat conditions for the moment problem to have a unique solution the results are related to the classical limit point limit circle situation

energy conserving hamiltonian boundary value methods for the numerical solution of the korteweg de vries equation abstract in this paper we study the efficient solution of the well known korteweg de vries equation equipped with periodic boundary conditions a fourier galerkin space semi discretization at first provides a large size hamiltonian ode problem whose solution in time is then carried out by means of energy conserving methods in the hbvm class hamiltonian boundary value methods the efficient implementation of the methods for the resulting problem is also considered and several numerical examples are reported

a variational integrators approach to second order modeling and identification of linear mechanical systems the theory of variational integration provides a systematic procedure to discretize the equations of motion of a mechanical system preserving key properties of the continuous time flow the discrete time model obtained by variational integration theory inherits structural conditions which in general are not guaranteed under general discretization procedures we discuss a simple class of variational integrators for linear second order mechanical systems and propose a constrained identification technique which employs simple linear transformation formulas to recover the continuous time parameters of the system from the discrete time identified model we test this approach on a simulated eight degrees of freedom system and show that the new procedure leads to an accurate identification of the continuous time parameters of second order mechanical systems starting from discrete measured data