numerical integration of non linear two point boundary value problems using iterated deferred corrections i abstract in several recent papers the idea has been proposed of using implicit runge kutta formulae for the approximate integration of non linear two point boundary value problems these formulae often take a very special form so that computational efficiencies are obtained in certain well defined circumstances in this paper we examine the relative merits of these different classes of formulae and discuss their efficient implementation in a deferred correction framework a theoretical comparison of these formulae is important firstly because they call for different computational tasks to be performed and secondly because they are not always applicable to exactly the same classes of problems the most efficient of these formulae are then compared with methods based on finite differences and with methods based on spline collocation for some simple smooth test functions on the basis of operation counts and numerical experimentation it is concluded that runge kutta methods can be generally competitive with the other methods considered

cubica independent component analysis by simultaneous third and fourth order cumulant diagonalization cubica which is an improved method for independent component analysis ica based on the diagonalization of cumulant tensors is proposed it is based on comon s algorithm but it takes third and fourth order cumulant tensors into account simultaneously the underlying contrast function is also mathematically much simpler and has a more intuitive interpretation it is therefore easier to optimize and approximate a comparison with comon s and three other ica algorithms on different data sets demonstrates its performance

a study of second and third order spectral procedures and maximum likelihood in the identification of a bilinear system an investigation is presented of two spectral moment based identification procedures which are compared to each other and to the method of maximum likelihood for the case of a bilinear system having observed input and corresponding output inputs to the system that are considered are pulse pulse pair and stochastic a principle concern of the analysis is the examination of coupled frequencies relative advantages and disadvantages of the three identification procedures are mentioned advantages of the maximum likelihood approach include the availability of expressions for standard errors and efficiency in an asymptotic sense

an adaptive uniformly convergent numerical method for a semilinear singular perturbation problem a singularly perturbed semilinear two point boundary value problem is considered without any restriction on its turning points a difference scheme is presented for solving this problem on an arbitrary locally quasi uniform mesh it is shown that the solution of the scheme is first order accurate uniformly in the perturbation parameter in a discrete l 1 norm numerical results are presented for an adaptive method based on the scheme

a generalization of joint diagonalization criteria for source separation in the field of blind source separation joint diagonalization based approaches constitute an important framework leading to useful algorithms such as the popular joint approximate diagonalization of eigenmatrices jade and simultaneous third order tensor diagonalization stotd algorithms however they are often restricted to the case of cumulants of order four in this paper we extend the results leading to jade and stotd to cumulants of any order greater than or equal to three by exhibiting a new family of contrast functions that constitutes then a unified framework for the above known results this also leads us to generalize some links between contrast functions and joint diagonalization criteria on which these algorithms are based in turn one contrast of the new family allows us to show that a function previously proposed as a separation criterion is also a contrast moreover for the two generalized jade and stotd contrasts the analytical optimal solution in the case of two sources is derived and shown to keep the same simple expression whatever the cumulant order finally some computer simulations illustrate the potential advantage one can take by considering statistics of different orders for the joint diagonalization of cumulant matrices

statistical tests for presence of cyclostationarity the presence of kth order cyclostationarity is defined in terms of nonvanishing cyclic cumulants or polyspectra relying upon the asymptotic normality and consistency of kth order cyclic statistics asymptotically optimal spl chi sup 2 tests are developed to detect the presence of cycles in the kth order cyclic cumulants or polyspectra without assuming any specific distribution on the data constant false alarm rate tests are derived in both time and frequency domain and yield consistent estimates of possible cycles present in the kth order cyclic statistics explicit algorithms for k spl les 4 are discussed existing approaches are rather empirical and deal only with k spl les 2 case simulation results are presented to confirm the performance of the given tests

numerical patching method for singularly perturbed two point boundary value problems using cubic splines a numerical patching method is presented for solving singularly perturbed two point boundary value problems with the boundary layer at one end left or right point the method is distinguished by the following fact the original singularly perturbed two point boundary value problem is divided into two problems namely inner and outer region problems the terminal boundary condition is obtained from the solution of the reduced problem using general stretching transformation a modified inner region problem is constructed then both inner region problem and outer region problems are solved as two point boundary value problems by employing cubic splines several linear and non linear problems are solved to demonstrate the applicability of the method

a general method for solving singular perturbed impulsive differential equations with two point boundary conditions in this paper a general method is presented for solving singularly perturbed impulsive two point boundary value problems with boundary layer at one end left or right point for the method we divide the defined interval into a series of subintervals and obtain the solution on each subinterval namely on the subinterval of boundary layer we yield an expansion by using matching techniques then by using impulsive and boundary conditions we obtain the solutions on other subintervals finally several linear and nonlinear singular perturbed impulsive problems are provided to be concretely solved according to the new approach

comments on cubica independent component analysis by simultaneous third and fourth order cumulant diagonalization source separation based on joint diagonalization of cumulant matrices or tensors has led now to classical algorithms in a previous paper by blaschke and wiskott an algorithm based on the above principle is proposed it combines third and fourth order cumulants and it is claimed to possess interesting properties this comment has two objectives we point out imprecisions of statement and we give additional references of closely related works some of them presenting identical issues

approximate method for the numerical solution of singular perturbation problems we present an approximate method for the numerical solution of linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval it is motivated by the asymptotic behavior of singular perturbation problems the original problem is divided into inner and outer region problems the reduced problem is solved to obtain the terminal boundary condition then a new inner region problem is created and solved as a two point boundary value problem in turn the outer region problem is also modified and the resulting problem is efficiently treated by employing the trapezoidal formula coupled with discrete invariant imbedding algorithm the proposed method is iterative on the terminal point some numerical experiments have been included to demonstrate its applicability

deconvolution and estimation of transfer function phase and coefficients for nongaussian linear processes nongaussian linear processes are considered it is shown that the phase of the transfer function can be estimated under broad conditions this is not true of gaussian linear processes and in this sense gaussian linear processes are atypical the asymptotic behavior of a phase estimate is determined the phase estimates make use of bispectral estimates these ideas are applied to a problem of deconvolution which is effective even when the transfer function is not minimum phase a number of computational illustrations are given

method of reduction of order for solving singularly perturbed two point boundary value problems in this paper a method of reduction of order is proposed for solving singularly perturbed two point boundary value problems with a boundary layer at one end point it is distinguished by the following fact the original singularly perturbed boundary value problem is replaced by a pair of initial value problems classical fourth order runge kutta method is used to solve these initial value problems several linear and non linear singular perturbation problems have been solved and the numerical results are presented to support the theory

nonparametric cyclic polyspectral analysis of am signals and processes with missing observations by viewing discrete time amplitude modulated signals and processes with missing observations as cyclostationary signals nonparametric mean square sense consistent and asymptotically normal single record estimators are developed for their kth order cumulants and polyspectra along with the asymptotic covariances the proposed estimation schemes use cyclic cumulants and polyspectra and are theoretically insensitive to any additive stationary noise in addition schemes of order k or 3 convey complete phase information and are insensitive to additive cyclostationary gaussian noise of unknown covariance the conventional approaches cannot recover mixed phase linear processes are susceptible to additive noise and are a special case of the proposed schemes simulations demonstrate superior performance of the proposed algorithms

modifications of the euclidean algorithm for isolating periodicities from a sparse set of noisy measurements modifications of the euclidean algorithm are presented for determining the period from a sparse set of noisy measurements the elements of the set are the noisy occurrence times of a periodic event with perhaps very many missing measurements this problem arises in radar pulse repetition interval pri analysis in bit synchronization in communications and in other scenarios the proposed algorithms are computationally straightforward and converge quickly a robust version is developed that is stable despite the presence of arbitrary outliers the euclidean algorithm approach is justified by a theorem that shows that for a set of randomly chosen positive integers the probability that they do not all share a common prime factor approaches one quickly as the cardinality of the set increases in the noise free case this implies that the algorithm produces the correct answer with only 10 data samples independent of the percentage of missing measurements in the case of noisy data simulation results show for example good estimation of the period from 100 data samples with 50 of the measurements missing and 25 of the data samples being arbitrary outliers

a unifying maximum likelihood view of cumulant and polyspectral measures for non gaussian signal classification and estimation classification and estimation of non gaussian signals observed in additive gaussian noise of unknown covariance are addressed using cumulants or polyspectra by integrating ideas from pattern recognition and model identification asymptotically optimum maximum likelihood classifiers and arma autoregressive moving average parameter estimators are derived without knowledge of the data distribution identifiability of noncausal and nonminimum phase arma models is established using a finite number of cumulant or polyspectral lags of any order greater than two a unifying view of cumulant and polyspectral discriminant measures utilizes these lags and provides a common framework for development and performance analysis of novel and existing estimation and classification algorithms tentative order determination and model validation tests for non gaussian arma processes are described briefly illustrative simulations are also presented

a new algorithm for a class of singular boundary value problems abstract in this paper we present a new algorithm to solve a class of singular boundary value problems in the reproducing kernel space w 2 3 0 1 the algorithm is efficiently applied to solving some model problems with the comparison between the numerical solutions and the exact solutions it is demonstrated by the numerical examples that this algorithm is of high precision

nonparametric time series analysis for periodically correlated processes correlation functions of continuous time periodically correlated processes can be represented by a fourier series with coefficient functions it is shown that the usual estimator for stationary covariances formed from a single sample path of the process can be simply modified to provide a consistent in quadratic mean estimator for any of the coefficient functions resulting from the aforementioned representation it is shown that if the process is gaussian and b sub k tau is a fourier integral with respect to a density function g sub k lambda a two dimensional periodogram formed from a single sample function can be smoothed along a line of constant difference frequency to provide a consistent estimator for g sub k lambda this natural extension of the well known procedure for stationary processes provides a method for nonparametric spectral analysis of periodically correlated processes

parameter estimation of cyclostationary am time series with application to missing observations time series with systematic misses occur often in practice and can be modeled as amplitude modulated arma processes with this as a motivating application modeling of cyclostationary amplitude modulated time series is addressed in the paper assuming that the modulating sequence is almost periodic parameter estimation algorithms are developed based on second and higher order cumulants of the resulting cyclostationary observations which may be corrupted by any additive stationary noise of unknown covariance if unknown the modulating sequence can be recovered even in the presence of additive perhaps nonstationary and colored gaussian or any symmetrically distributed noise if the arma process is nongaussian cyclic cumulants of order greater than three can identify non causal and non minimum phase models from partial noisy data simulation experiments corroborate the theoretical results

the cumulant theory of cyclostationary time series ii development and applications for pt i see ibid vol 42 no 12 p 3387 3408 1994 the development of the theory of nonlinear processing of cyclostationary time series that is initiated in part i is continued a new type of cumulant for complex valued variables is introduced and used to generalize the temporal and spectral moments and cumulants for cyclostationary time series from real valued to complex valued time series the relations between the temporal and spectral moments and cumulants at the inputs and outputs of several signal processing operations are determined formulas for the temporal and spectral cumulants of complex valued pulse amplitude modulated time series are derived estimators for the temporal moments and cumulants and for the cyclic polyspectra are presented and their properties are discussed the performance of these estimators is illustrated by several computer simulation examples for pulse amplitude modulated time series the theory is applied to the problems of weak signal detection and interference tolerant time delay estimation

arma spectral estimation of time series with missing observations the problem of estimating the power spectral density of stationary time series when the measurements are not contiguous is considered a new autoregressive moving average arma method is proposed for this problem based on nonlinear optimization of a weighted squared error criterion the method can handle either regularly or randomly missing observations as a special case the method can handle the problem of missing sample covariances the computational complexity is modest compared to exact maximum likelihood estimation of the same parameters the performance of the algorithm is illustrated by some numerical examples and is shown to be statistically efficient in these cases

independent component analysis and simultaneous third order tensor diagonalization comon s 1994 well known scheme for independent component analysis ica is based on the maximal diagonalization in a least squares sense of a higher order cumulant tensor in a previous paper we proved that for fourth order cumulants the computation of an elementary jacobi rotation is equivalent to the computation of the best rank 1 approximation of a fourth order tensor in this paper we show that for third order tensors the computation of an elementary jacobi rotation is again equivalent to a best rank 1 approximation however here it is a matrix that has to be approximated this favorable computational load makes it attractive to do something third order like for fourth order cumulant tensors as well we show that simultaneous optimal diagonalization of third order tensor slices of the fourth order cumulant is a suitable strategy this simultaneous third order tensor diagonalization approach stotd is similar in spirit to the efficient jade algorithm

nonparametric polyspectral estimators for kth order almost cyclostationary processes second and higher order almost cyclostationary processes are random signals with almost periodically time varying statistics the class includes stationary and cyclostationary processes as well as many real life signals of interest cyclic and time varying cumulants and polyspectra are defined for discrete time real kth order cyclostationary processes and their interrelationships are explored smoothed polyperiodograms are proposed for cyclic polyspectral estimation and are shown to be consistent and asymptotically normal asymptotic covariance expressions are derived along with their computable forms higher than second order cyclic cumulants and polyspectra convey time varying phase information and are theoretically insensitive to any stationary for nonzero cycles as well as additive cyclostationary gaussian noise for all cycles

the second order moments of the sample covariances for time series with missing observations the estimation of the covariances of a stationary time series with missing observations is considered general formulas for the asymptotic second order moments of the sample covariances of such a time series are given for either random or deterministic patterns of misses closed form expressions are derived for the random bernoulli pattern and for the deterministic periodic pattern of missing observations and are explicitly evaluated for autoregressive moving average time series these results are useful for constructing and analyzing parameter or spectrum estimation algorithms based on the sample covariances for a stationary time series with missing observations

independent component analysis jacobi like diagonalization of optimized composite order cumulants in the field of blind source separation jacobi like diagonalization based approaches constitute an important tool for independent component analysis ica recently simultaneous diagonalization of cumulant matrices of third and fourth order has been studied by a number of authors in this work we present an optimal parametrized composition of these cumulants that puts two classical contrasts namely the cumulantbased ica and the weighted fourth order contrast in a common framework it is shown that the optimal weight parameter depends on the a priori statistical knowledge of the original mixing sources following the same spirit of the ica algorithm we derive the analytical solution for the case of two sources finally a number of computer simulations have been performed to illustrate the behaviour of the jacobi like iterations for the maximization of the proposed parametrized contrast

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

dynamical analysis and perturbation solution of an seir epidemic model abstract in this paper we present a kind of technique to obtain the asymptotical solution of an seir where s is the susceptible population e is the exposed population i is the infectious population and r is the recovered population epidemic model by employing the method of perturbation at first we investigate the epidemic model by analysing its dynamical behavior then we use the method of perturbation to obtain the analytical solution of the model we assign values for parameters and draw figures to observe the magnitude of error of the perturbation method in a macro view finally we analyse macroscopically the two comparison chart on analytical solution and the exact solution and know that it is feasible to analyse the solution of the epidemic model by using the perturbation method

computational methods for solving two parameter singularly perturbed boundary value problems for second order ordinary differential equations singularly perturbed two point boundary value problems spbvps for second order ordinary differential equations odes with two small parameters multiplying the derivatives are considered computational methods are suggested in this paper to solve this type of problems one of these methods is fitted operator method fom and the other is bondary value technique bvt in fom we suggest an exponentially fitted finite difference effd scheme throughout the domain of definition of the differential equation and the error estimates are derived in bvt the domain of definition of the differential equation is divided into three subintervals called inner and outer regions then we solve the two parameter spbvp tpspbvp over these regions as two point boundary value problems an exponentially fitted finite difference scheme is used in the inner regions and a classical finite difference scheme in the outer region the data for boundary conditions at the transition points are obtained using zero order asymptotic expansion approximation of the solution of the problem the bvt is well suited for parallel computing numerical examples are given to illustrate the method

time and lag recursive computation of cumulants from a state space model time and lag recursive algorithms for the computation of the cumulants of the state vector and the output process of a multiple input multiple output time varying state space model are derived using a kronecker product representation for the cumulants of vector processes the noise processes are not assumed to be stationary symmetry relations for the cumulants of vector processes are discussed computational aspects are examined in detail it is conjectured that the recursive equations derived will be useful in developing state estimators and even optimal controllers based on higher order statistics

the cumulant theory of cyclostationary time series i foundation the problem of characterizing the sine wave components in the output of a polynomial nonlinear system with a cyclostationary random time series input is investigated the concept of a pure nth order sine wave is introduced and it is shown that pure nth order sine wave strengths in the output time series are given by scaled fourier coefficients of the polyperiodic temporal cumulant of the input time series the higher order moments and cumulants of narrowband spectral components of time series are defined and then idealized to the case of infinitesimal bandwidth such spectral moments and cumulants are shown to be characterized by the fourier transforms of the temporal moments and cumulants of the time series it is established that the temporal and spectral cumulants have certain mathematical and practical advantages over their moment counterparts to put the contributions of the paper in perspective a uniquely comprehensive historical survey that traces the development of the ideas of temporal and spectral cumulants from their inception is provided

give your odes a singular perturbation singular perturbations occur when contrary to natural expectations small changes produce substantial effects consequently special methods to approximate solutions both computational and otherwise are needed the topic can provide one way to increase interest motivation and applicability of an ode course or follow up studies this presentation is aimed at those well grounded in the traditional coverage of scalar first order ordinary differential equations and series solutions rather than an applied audience needing to learn singular perturbations per se to asymptotically solve some physical problem they face after figuring out the impact of singular perturbations on these special equations we encourage the consideration of analogous perturbations for more general odes or even pdes

asymptotic theory of mixed time averages and kth order cyclic moment and cumulant statistics we generalize parzen s 1961 analysis of asymptotically stationary processes to mixtures of deterministic stationary nonstationary and generally complex time series under certain mixing conditions expressed in terms of joint cumulant summability we show that time averages of such mixtures converge in the mean square sense to their ensemble averages we additionally show that sample averages of arbitrary orders are jointly complex normal and provide their covariance expressions these conclusions provide us with statistical tools that treat random and deterministic signals on a common framework and are helpful in defining generalized moments and cumulants of mixed processes as an important consequence we develop consistent and asymptotically normal estimators for time varying and cyclic moments and cumulants of kth order cyclostationary processes and provide computable variance expressions some examples are considered to illustrate the salient features of the analysis