correction of numerov s eigenvalue estimates the error in the estimate of thekth eigenvalue of a regular sturm liouville problem obtained by numerov s method with mesh lengthh iso k6h4 we show that a simple correction technique of paine de hoog and anderssen reduces the error to one ofo k3h4 numerical examples demonstrate the usefulness of this correction even for low values ofk

trigonometric polynomial or exponential fitting approach the application of a trigonometric polynomial and an exponential fitting approach is compared for a three point formula for second order derivatives for simpson s quadrature rule and for numerov s scheme for second order differential equations the expressions for the occurring parameters are constructed in both the approaches and the behaviour of these parameters with respect to the introduced frequency is studied the errors for specific problems obtained in both the approaches as a function of the frequency are compared

families of backward differentiation methods based on a new type of mixed interpolation abstract backward differentiation methods based on a new type of mixed interpolation for the first order initial value problems whose solutions are known to be periodic are constructed the angular frequency k is calculated by minimizing the local truncation error within each integration interval the resulting methods depend on a parameter hk h the step length and reduce to classical backward methods if k 0 numerical examples are provided to illustrate the algorithms

approximating grammar probabilities solution of a conjecture it is proved that the production probabilities of a probabilistic context free grammar may be obtained as the limit of the estimates inferred from an increasing sequence of randomly drawn samples from the language generated by the grammar

numerical solutions of the spectral problem for arbitrary self adjoint extensions of the one dimensional schrodinger equation a numerical algorithm to solve the spectral problem for arbitrary self adjoint extensions of one dimensional regular schrodinger operators is presented it is shown that the set of all self adjoint extensions of one dimensional regular schrodinger operators is in one to one correspondence with the group of unitary operators on the finite dimensional hilbert space of boundary data and they are characterized by a generalized class of boundary conditions that include the well known dirichlet neumann robin and quasi periodic boundary conditions the numerical algorithm is based on a nonlocal boundary extension of the finite element method and their convergence is proved an appropriate basis of boundary functions must be introduced to deal with arbitrary boundary conditions and the conditioning of its computation is analyzed some significant numerical experiments are also discussed as well as the comparison with some standard algorithms in particular it is shown that appropriate perturbations of stand

computing the eigenvalues of the generalized sturm liouville problems based on the lie group sl 2 r for the generalized sturm liouville problems we can construct an sl 2 r lie group shooting method to find eigenvalues by using the closure property of the lie group a one step lie group transformation between the boundary values at two ends of the considered interval is established hence we can theoretically derive an analytical characteristic equation to determine the eigenvalues for the generalized sturm liouville problems because the closed form formulas are derived to calculate the unknown left boundary values in terms of l the present method provides an easy numerical implementation and has a cheap computational cost numerical examples are examined to show that the present sl 2 r lie group shooting method is effective

solution of an open problem on probabilistic grammars it has been proved that when the production probabilities of an unambiguous context free grammar g are estimated by the relative frequencies of the corresponding productions in a sample s from the language l g generated by g the expected derivation length and the expected word length of the words in l g are precisely equal to the mean derivation length and the mean world length of the words in the same s respectively

a parallel approach to the modified numerov like eigenvalue determination for sturm liouville problems abstract a modified numerov like eigenvalue algorithm previously introduced is parallelized an inplementation of this algorithm on a helios based parallel processing transputer system is discussed time savings with respect to a sequential approach are commented

probabilistic hyperedge replacement grammars abstract in a probabilistic graph grammar each production has a probability attached to it this induces a probability assigned to each derivation tree and to each derived graph conditions for this probability function to be a probabilistic measure are discussed the statistical properties of the generated language are investigated we show how to compute the average size of an inductive function and the probability of an inductive graph predicate a relationship can be established between production probabilities and some statistical information of the generated language this way probabilities can be assigned to productions so that the generated graphs satisfy some statistical conditions

accurate computation of higher sturm liouville eigenvalues the computation of eigenvalues of regular sturm liouville problems is considered it is shown that a simple step dependent linear multistep method can be used to reduce the error of the orderk 4 h 2 of the centered finite difference estimate of thek th eigenvalue with uniform step lengthh to an error of orderkh 2 by an appropriate minimization of the local error term of the method one can obtain even more accurate results a comparison of the simple correction techniques of paine de hoog and anderssen and of andrew and paine is given numerical examples demonstrate the usefulness of this correction even for low values ofk

frequency domain uncertainty and the graph topology a new metric on linear time invariant systems is defined this metric is no greater than the gap metric and is in fact the smallest metric for which a certain robust stabilization result holds unlike other known metrics which induce the graph topology it has a clear frequency response interpretation this allows questions regarding robustness in the face of parametric uncertainty to be considered in terms of this metric

cp methods for the schro dinger equation revisited abstract on constructing cpm propagators with an abundant number of terms by mathematica we have shown that the cpm n q where n is the number of polynomial terms by which the potential is approximated in each interval and q the number of corrections introduced is a method of order 2n 2 at low energies if q 2 3 n 1 and of order n at high energies if q 1 we have also proved that in the last case the error damps out as 1 e for both initial and boundary value problems we have written a program for boundary value problems which is of order 12 10 at low and high energies respectively and have found out that it is far more efficient than the well established codes sl02f sledge and sleign

numerov method maximally adapted to the schro dinger equation abstract we construct and investigate the two step scheme which is maximally specialized for the schrodinger equation the new scheme is found to preserve all attractive features of the standard numerov method ease in programming flexibility in application speed in run etc with the important advantage of being more efficient for problems involving high values of energy its efficiency is actually found to be close to that of the piecewise perturbation methods a fact which strongly recommends it for further applications

approximations of sturm liouville eigenvalues using differential quadrature dq method the polynomial based differential quadrature pdq and the fourier expansion based differential quadrature fdq methods are applied in this work to compute eigenvalues of the sturm liouville problem it is demonstrated through some examples that accurate results for the first kth eigenvalues of the problem where k 1 2 3 can be obtained by using at least 2k mesh points the results of this work are compared with other published results in the literature

inference of k testable languages in the strict sense and application to syntactic pattern recognition the inductive inference of the class of k testable languages in the strict sense k tssl is considered a k tssl is essentially defined by a finite set of substrings of length k that are permitted to appear in the strings of the language given a positive sample r of strings of an unknown language a deterministic finite state automation that recognizes the smallest k tssl containing r is obtained the inferred automation is shown to have a number of transitions bounded by o m where m is the number of substrings defining this k tssl and the inference algorithm works in o kn log m where n is the sum of the lengths of all the strings in r the proposed methods are illustrated through syntactic pattern recognition experiments in which a number of strings generated by ten given source non k tssl grammars are used to infer ten k tssl stochastic automata which are further used to classify new strings generated by the same source grammars the results of these experiments are consistent with the theory and show the ability of stochastic k tssls to approach other classes of regular languages

design variables for bias distribution in transfer function estimation estimation of transfer functions of linear systems is one of the most common system identification problems several different design variables chosen by the user for the identification procedure affect the properties of the resulting estimate in this paper it is investigated how the choices of prefilters noise models sampling interval and prediction horizon i e the use of k step ahead prediction methods influence the estimate an important aspect is thai the true system is not assumed to be exactly represented within the chosen model set the estimate will thus be biased it is shown how the distribution of bias in the frequency domain is governed by a weighting function which emphasizes different frequency bands the weighting function in turn is a result of the previously listed design variables it is shown e g thai the common least squares method has a tendency to emphasize high frequencies and that this can be counteracted by prefiltering it is also shown that asymptotically it is only the prediction horizon itself and not how it is split up into sampling interval times number of predicted sampling instants that affects this weighting function

comparison of the closed loop identfication methods in terms of the bias distribution several schemes for plant model identification in closed loop operation including classical direct method two step identification and closed loop output error algorithms are considered these methods are analyzed and compared in terms of the bias distribution of the estimates for the case that the noise model is estimated as well as the case that a fixed model of noise is considered output error structure the problems concerning the filtered direct method which is often used in the iterative identification and control scheme are mentioned it is shown that these problems may be solved by the closed loop output error identification method

recursive algorithms for identification in closed loop a unified approach and evaluation a unified presentation of recursive algorithms for plant model identification in closed loop is given from the basic formulation of the problem of finding the plant model that gives the best predictor for the closed loop system two families of algorithms using either a reparameterized predictor for the closed loop or a plant predictor operating on filtered data are presented and their asymptotic properties are examined validation tests for the models identified in closed loop are proposed a comparative evaluation of the various algorithms in simulation and on real time experiments concludes the paper

approximate computation of eigenvalues with chebyshev collocation method in this study chebyshev collocation method is investigated for the approximate computation of higher sturm liouville eigenvalues by a truncated chebyshev series using the chebyshev collocation points this method transform the sturm liouville problems and given boundary conditions to matrix equation by solving the algebraic equation system the approximate eigenvalues can be computed hence by using asymptotic correction technique corrected eigenvalues can be obtained

a finite element estimate with trigonometric hat functions for sturm liouville eigenvalues abstract the computation of eigenvalues of regular sturm liouville problems is considered numerical results show that the error of the k th eigenvalue obtained by the finite element method with trigonometric hat functions and mesh length h is of the same order as the one obtained with the linear hat functions together with a simple asymptotic correction technique of paine de hoog and anderssen 1981

correction of finite element estimates for stum liouville eigenvalues it is shown that a simple asymptotic correction technique of paine de hoog and anderssen reduces the error in the estimate of thekth eigenvalue of a regular sturm liouville problem obtained by the finite element method with linear hat functions and mesh lengthh fromo k 4 h 2 too k h 2 the result still holds when the matrix elements are evaluated by simpson s rule but if the trapezoidal rule is used the error iso k 2 h 2 numerical results demonstrate the usefulness of the correction even for low values ofk

spectral corrections for sturm liouville problems abstract the numerical solution of the sturm liouville problem can be achieved using shooting to obtain an eigenvalue approximation as a solution of a suitable nonlinear equation and then computing the corresponding eigenfunction in this paper we use the shooting method both for eigenvalues and eigenfunctions in integrating the corresponding initial value problems we resort to the boundary value method the technique proposed seems to be well suited to supplying a general formula for the global discretization error of the eigenfunctions depending on the discretization errors arising from the numerical integration of the initial value problems a technique to estimate the eigenvalue errors is also suggested and seems to be particularly effective for the higher index eigenvalues numerical experiments on some classical sturm liouville problems are presented

approximate calculation of eigenvalues with the method of weighted residuals collocation method in this study the collocation method of the weight residual methods are investigated for the approximate computation of higher sturm liouville eigenvalues where trial solution is accepted as the chebyshev series the obtained approximate eigenvalues are compared with the previous computational results anziam j 42 2000 c96 numer math 47 1985 289 numer math 59 1991 243 appl numer math 23 1997 311 computing 26 1981 123

some remarks on the bias distribution analysis of discrete time identification algorithms based on pseudo linear regressions abstract in 1998 a karimi and i d landau published in this journal an article entitled comparison of the closed loop identication methods in terms of bias distribution one of its main purposes was to provide a bias distribution analysis in the frequency domain of closed loop output error identication algorithms that had been recently developed the expressions provided in that paper are only valid for prediction error identification methods pem not for pseudo linear regression plr ones for which we give the correct frequency domain bias analysis both in open and closed loop although plr was initially and is still considered as an approximation of pem we show that it gives better results at high frequencies