the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

on the generation of mono implicit runge kutta nystro m methods by mono implicit runge kutta methods mono implicit runge kutta methods can be used to generate implicit runge kutta nystrom irkn methods for the numerical solution of systems of second order differential equations the paper is concerned with the investigation of the conditions to be fulfilled by the mono implicit runge kutta mirk method in order to generate a mono implicit runge kutta nystrom method mirkn that is p stable one of the main theoretical results is the property that mirk methods in standard form cannot generate mirkn methods in standard form of order greater than 4 many examples of mirkn methods generated by mirk methods are presented

reduced order generalized h_ infty filtering for linear discrete time systems with application to channel equalization this paper investigates the problem of reduced order generalized h infty filtering for linear discrete time systems generalized h infty filtering covers standard h infty filtering as a special case and can make use of the frequency domain characteristics of practical signals moreover reduced order filters are more attractive than full order ones regarding implementability by virtue of the generalized bounded real lemma a new necessary and sufficient condition is proposed for analyzing the generalized h infty performance of the filtering error system to compute reduced order generalized h infty filters a necessary and sufficient condition is then derived in terms of matrix inequalities to solve which iterative linear matrix inequality algorithms are further developed in addition the proposed filter design method is applied to cope with the channel equalization problem examples are finally presented for illustrating the proposed method

highly localized solutions of the wave equation simple explicit solutions of the linear wave equation in three dimensions are presented which describe wave packets exponentially localized near a point moving with the wave speed for large values of a certain free parameter these new solutions are localized in gaussian manner with respect to longitudinal and transverse variables and time this agrees with considerations by babich ulin and ralston who have presented an asymptotic description of solutions exhibiting such local behavior global estimates and large time asymptotics of these solutions are given

fault detection filter design for markovian jump singular systems with intermittent measurements this paper addresses the problem of fault detection filter design for discrete time markovian jump singular systems with intermittent measurements the measurement transmission from the plant to the fault detection filter is assumed to be imperfect and a stochastic variable is utilized to model the phenomenon of data missing our attention is focused on the design of a fault detection filter such that the residual system is stochastically markovian jump admissible and satisfies some expected performances a new necessary and sufficient condition for a class of discrete time markovian jump singular systems to be stochastically markovian jump admissible is proposed in the form of strict linear matrix inequalities sufficient conditions are established for the existence of the fault detection filter finally a numerical example is provided to demonstrate the usefulness and applicability of the developed theoretical results

adaptive reliable h_ infty filtering against sensor failures this paper is concerned with the adaptive reliable hinfin filtering problem against sensor failures for continuous time linear systems an adaptive hinfin performance index is defined to describe the disturbance attenuation performance of systems with time varying parameter estimations by combining the linear matrix inequality lmi approach for hinfin filter design and adaptive method a method of designing adaptive reliable hinfin filters is proposed where the filter parameter matrices are updated automatically to compensate the sensor failures effects on systems based on the online estimations of eventual faults it is shown that the design condition for the newly proposed adaptive hinfin filtering is more relaxed than that for the traditional fixed gain filter design based on lmi approach an illustrative example and its simulation results are given to illustrate the effectiveness of the proposed design method

robust h sub 2 h sub spl infin filtering for linear systems with error variance constraints in this correspondence we consider the robust h sub 2 h sub spl infin filtering problem for linear perturbed systems with steady state error variance constraints the purpose of this multiobjective problem is to design a linear filter that does not depend on the parameter perturbations such that the following three performance requirements are simultaneously satisfied 1 the filtering process is asymptotically stable 2 the steady state variance of the estimation error of each state is not more than the individual prespecified value 3 the transfer function from exogenous noise inputs to error state outputs meets the prespecified h sub spl infin norm upper bound constraint we show that in both continuous and discrete time cases the addressed filtering problem can effectively be solved in terms of the solutions of a couple of algebraic riccati like equations inequalities we present both the existence conditions and the explicit expression of desired robust filters an illustrative numerical example is provided to demonstrate the flexibility of the proposed design approach

brief paper robust filtering with stochastic nonlinearities and multiple missing measurements this paper is concerned with the filtering problem for a class of discrete time uncertain stochastic nonlinear time delay systems with both the probabilistic missing measurements and external stochastic disturbances the measurement missing phenomenon is assumed to occur in a random way and the missing probability for each sensor is governed by an individual random variable satisfying a certain probabilistic distribution over the interval 01 such a probabilistic distribution could be any commonly used discrete distribution over the interval 01 the multiplicative stochastic disturbances are in the form of a scalar gaussian white noise with unit variance the purpose of the addressed filtering problem is to design a filter such that for the admissible random measurement missing stochastic disturbances norm bounded uncertainties as well as stochastic nonlinearities the error dynamics of the filtering process is exponentially mean square stable by using the linear matrix inequality lmi method sufficient conditions are established that ensure the exponential mean square stability of the filtering error and then the filter parameters are characterized by the solution to a set of lmis illustrative examples are exploited to show the effectiveness of the proposed design procedures

on lmi formulations of some problems arising in nonlinear stochastic system analysis this work involves formulation of some problems arising in the analysis of a class of discrete time nonlinear stochastic systems in terms of linear matrix inequalities this allows one to utilize a wide variety of numerical techniques available for tackling both the feasibility problem and the actual numerical solution in an efficient manner

variance constrained cal h _ infty filtering for a class of nonlinear time varying systems with multiple missing measurements the finite horizon case this paper is concerned with the robust h finite horizon filtering problem for a class of uncertain nonlinear discrete time varying stochastic systems with multiple missing measurements and error variance constraints all the system parameters are time varying and the uncertainty enters into the state matrix the measurement missing phenomenon occurs in a random way and the missing probability for each sensor is governed by an individual random variable satisfying a certain probabilistic distribution in the interval the stochastic nonlinearities under consideration here are described by statistical means which can cover several classes of well studied nonlinearities sufficient conditions are derived for a finite horizon filter to satisfy both the estimation error variance constraints and the prescribed h performance requirement these conditions are expressed in terms of the feasibility of a series of recursive linear matrix inequalities rlmis simulation results demonstrate the effectiveness of the developed filter design scheme

finite horizon h_ infty filtering with missing measurements and quantization effects in this paper a new h filtering approach is developed for a class of discrete time varying systems subject to missing measurements and quantization effects the missing measurements are modeled via a diagonal matrix consisting of a series of mutually independent random variables satisfying certain probabilistic distributions on the interval 0 1 the measured output is quantized by a logarithmic quantizer attention is focused on the design of a stochastic h filter such that the h estimation performance is guaranteed over a given finite horizon in the simultaneous presence of probabilistic missing measurements quantization effects as well as external non gaussian disturbances a necessary and sufficient condition is first established for the existence of the desired time varying filters in virtue of the solvability of certain coupled recursive riccati difference equations rdes owing to its recursive nature the proposed rde approach is shown to be suitable for online application without the need of increasing the problem size the simulation experiment is carried out for the mobile robot localization problem with non gaussian disturbances missing measurements and quantization effects the effectiveness of the proposed method is demonstrated in the numerical example

fixed order robust h sub spl infin filter design for markovian jump systems with uncertain switching probabilities this paper discusses the fixed order robust h sub spl infin filtering problem for a class of markovian jump linear systems with uncertain switching probabilities the uncertainties under consideration are assumed to be norm bounded in the system matrices and to be elementwise bounded in the mode transition rate matrix respectively first a criterion based on linear matrix inequalities is provided for testing the h sub spl infin filtering level of a filter over all the admissible uncertainties then a sufficient condition for the existence of the fixed order robust h sub spl infin filters is established in terms of the solvability of a set of linear matrix inequalities with equality constraints to determine the filter a globally convergent algorithm involving convex optimization is suggested finally a numerical example is used to illustrate that the developed theory is more effective than the existing results

extended double stride l stable methods for the numerical solution of odes abstract usmani and agarwal 1 had proposed construction of extended one step higher than two order a stable methods by coupling classical linear multistep methods lmms following on their idea in the present paper we describe a class of extended double stride methods based on simpson s rule which are fourth order and l stable

gaussian beam methods for the schrodinger equation with discontinuous potentials we propose eulerian and lagrangian gaussian beam methods for the schrodinger equation with discontinuous potentials at the quantum barriers where the potential is discontinuous we derive suitable interface conditions to account for quantum scattering information these scattering interface conditions are then built into the numerical fluxes in the eulerian level set formulation of the gaussian beam methods and are also used in the lagrangian formulation including an interface condition for the hessian matrix we carry both 1d and 2d numerical examples to verify the accuracy of the method

h filtering for descriptor systems with strict lmi conditions in this paper the problem of h filtering for descriptor systems with strict linear matrix inequalities lmis is investigated the necessary and sufficient conditions for the solvability and the expression of the solution are obtained for both continuous time and discrete time descriptor systems the proposed continuous time filters extend the results presented by xu and lam 2006 by removing semi definite conditions and introducing new formulation for the filter parameters which does not require determination of intermediate variables for discrete time descriptors differing from existing methods that provide sufficient conditions only two necessary and sufficient conditions are introduced new proposed design methods propose less conservative results comparing with existing works two numerical examples are presented to demonstrate the applicability of the proposed methods

fast multiscale gaussian beam methods for wave equations in bounded convex domains motivated by fast multiscale gaussian wavepacket transforms and multiscale gaussian beam methods which were originally designed for pure initial value problems of wave equations we develop fast multiscale gaussian beam methods for initial boundary value problems of wave equations in bounded convex domains in the high frequency regime to compute the wave propagation in bounded convex domains we have to take into account reflecting multiscale gaussian beams which are accomplished by enforcing reflecting boundary conditions during beam propagation and carrying out suitable reflecting beam summation to propagate multiscale beams efficiently we prove that the ratio of the squared magnitude of beam amplitude and the beam width is roughly conserved and accordingly we propose an effective indicator to identify significant beams we also prove that the resulting multiscale gaussian beam methods converge asymptotically numerical examples demonstrate the accuracy and efficiency of the method

robust h sub spl infin filtering for stochastic time delay systems with missing measurements in this paper the robust hinfin filtering problem is studied for stochastic uncertain discrete time delay systems with missing measurements the missing measurements are described by a binary switching sequence satisfying a conditional probability distribution we aim to design filters such that for all possible missing observations and all admissible parameter uncertainties the filtering error system is exponentially mean square stable and the prescribed hinfin performance constraint is met in terms of certain linear matrix inequalities lmis sufficient conditions for the solvability of the addressed problem are obtained when these lmis are feasible an explicit expression of a desired robust hinfin filter is also given an optimization problem is subsequently formulated by optimizing the hinfin filtering performances finally a numerical example is provided to demonstrate the effectiveness and applicability of the proposed design approach

variance constrained state estimation for complex networks with randomly varying topologies this paper investigates the variance constrained h infty state estimation problem for a class of nonlinear time varying complex networks with randomly varying topologies stochastic inner coupling and measurement quantization a kronecker delta function and markovian jumping parameters are utilized to describe the random changes of network topologies a gaussian random variable is introduced to model the stochastic disturbances in the inner coupling of complex networks as a kind of incomplete measurements measurement quantization is taken into consideration so as to account for the signal distortion phenomenon in the transmission process stochastic nonlinearities with known statistical characteristics are utilized to describe the stochastic evolution of the complex networks we aim to design a finite horizon estimator such that in the simultaneous presence of quantized measurements and stochastic inner coupling the prescribed variance constraints on the estimation error and the desired h infty performance requirements are guaranteed over a finite horizon sufficient conditions are established by means of a series of recursive linear matrix inequalities and subsequently the estimator gain parameters are derived a simulation example is presented to illustrate the effectiveness and applicability of the proposed estimator design algorithm

robust kalman filtering for uncertain discrete time systems this paper is concerned with the problem of a kalman filter design for uncertain discrete time systems the system under consideration is subjected to time varying norm bounded parameter uncertainty in both the state and output matrices the problem addressed is the design of a linear filter such that the variance of the filtering error is guaranteed to be within a certain bound for all admissible uncertainties furthermore the guaranteed cost can be optimized by appropriately searching a scaling design parameter

eulerian gaussian beams for high frequency wave propagation we design an eulerian gaussian beam summation method for solving helmholtz equations in the high frequency regime the traditional gaussian beam summation method is based on lagrangian ray tracing and local ray centered coordinates we propose a new eulerian formulation of gaussian beam theory which adopts global cartesian coordinates level sets and liouville equations yielding uniformly distributed eulerian traveltimes and amplitudes in phase space simultaneously for multiple sources the time harmonic wavefield can be constructed by summing up gaussian beams with ingredients provided by the new eulerian formulation the conventional gaussian beam summation method can be derived from the proposed method there are three advantages of the new method 1 we have uniform resolution of ray distribution 2 we can obtain wavefields excited at different sources by varying only source locations in the summation formula 3 we can obtain wavefields excited at different frequencies by varying only frequencies in the summation formula numerical experiments indicate that the gaussian beam summation method yields accurate asymptotic wavefields even at caustics the new method may be used for seismic modeling and migration

distributed filtering for a class of time varying systems over sensor networks with quantization errors and successive packet dropouts this paper is concerned with the distributed finite horizon filtering problem for a class of time varying systems over lossy sensor networks the time varying system target plant is subject to randomly varying nonlinearities rvns caused by environmental circumstances the lossy sensor network suffers from quantization errors and successive packet dropouts that are described in a unified framework two mutually independent sets of bernoulli distributed white sequences are introduced to govern the random occurrences of the rvns and successive packet dropouts through available output measurements from not only the individual sensor but also its neighboring sensors according to the given topology a sufficient condition is established for the desired distributed finite horizon filter to ensure that the prescribed average filtering performance constraint is satisfied the solution of the distributed filter gains is characterized by solving a set of recursive linear matrix inequalities a simulation example is provided to show the effectiveness of the proposed filtering scheme

multistride l stable fourth order methods for the numerical solution of odes recently chawla et al 1 have obtained a one parameter family of double stride l stable methods of fourth order by the coupling of three linear multistep methods lmms here we present an alternative derivation technique based on mono implicit runge kutta mirk methods the mirk method provides a framework within which all families of double and triple stride l stable methods of fourth order can be expressed the well established theory for runge kutta methods can then be used to derive these families in a straightforward manner

h_ bm infty fuzzy filtering of nonlinear systems with intermittent measurements this paper is concerned with the problem of h infin fuzzy filtering of nonlinear systems with intermittent measurements the nonlinear plant is represented by a takagi sugeno t s fuzzy model the measurements transmission from the plant to the filter is assumed to be imperfect and a stochastic variable satisfying the bernoulli random binary distribution is utilized to model the phenomenon of the missing measurements attention is focused on the design of an h infin filter such that the filter error system is stochastically stable and preserves a guaranteed h infin performance a basis dependent lyapunov function approach is developed to design the h infin filter by introducing some slack matrix variables the coupling between the lyapunov matrix and the system matrices is eliminated which greatly facilitates the filter design procedure the developed theoretical results are in the form of linear matrix inequalities lmis finally an illustrative example is provided to show the effectiveness of the proposed approach

a class of stabilized extended one step methods for the numerical solution of odes abstract to overcome the order barrier imposed by a stability on linear multistep methods lmms usmani and agarwal 1 had constructed a third order a stabilized extended one step method by coupling two lmms in the present paper we introduce a class of extended one step methods generalizing the method of usmani and agarwal methods of this class of orders three and four which are a and or l stable have been given previously in 2 3 it is natural to ask the maximum attainable order for methods of this class which are also a stable or l stable the purpose of this paper is to show that the maximum attainable order of a method in this class is five we derive fifth order extended one step methods and show the existence of sub families of these methods which are a stable or l stable these methods are illustrated by considering two numerical examples

order results for mono implicit runge kutta methods the mono implicit runge kutta methods are a subclass of the well known implicit runge kutta methods and have application in the efficient numerical solution of systems of initial and boundary value ordinary differential equations although the efficiency and stability properties of this class of methods have been studied in a number of papers the specific question of determining the maximum order of an s stage mono implicit runge kutta method has not been dealt with in addition to the complete characterization of some subclasses of these methods having a number of stages s leq 5 a main result of this paper is a proof that the order of an s stage mono implicit runge kutta method is at most s 1

optimal filtering with random sensor delay multiple packet dropout and uncertain observations this paper studies the problem of optimal filtering of discrete time systems with random sensor delay multiple packet dropout and uncertain observation the random sensor delay multiple packet dropout or uncertainty in observation is transformed to a stochastic parameter in the system representation a new formulation enables us to design an optimal filter for a system with multiple packet dropout in sensor data based on a stochastic definition of the norm of a system with a stochastic parameter new relations for stochastic norm are derived the stochastic norm of the estimation error is used as a criterion for the filter design the relations derived for the new norm definition are used to obtain a set of linear matrix inequalities lmis to solve the filter design problems simulation examples show the effectiveness of the proposed method

dissipative control for state saturated discrete time varying systems with randomly occurring nonlinearities and missing measurements in this paper the dissipative control problem is investigated for a class of discrete time varying systems with simultaneous presence of state saturations randomly occurring nonlinearities as well as multiple missing measurements in order to render more practical significance of the system model some bernoulli distributed white sequences with known conditional probabilities are adopted to describe the phenomena of the randomly occurring nonlinearities and the multiple missing measurements the purpose of the addressed problem is to design a time varying output feedback controller such that the dissipativity performance index is guaranteed over a given finite horizon by introducing a free matrix with its infinity norm less than or equal to 1 the system state is bounded by a convex hull so that some sufficient conditions can be obtained in the form of recursive nonlinear matrix inequalities a novel controller design algorithm is then developed to deal with the recursive nonlinear matrix inequalities

robust h filtering for networked systems with multiple state delays in this paper a new robust h filter design problem is studied for a class of networked systems with multiple state delays two kinds of incomplete measurements namely measurements with random delays and measurements with stochastic missing phenomenon are simultaneously considered such incomplete measurements are induced by the limited bandwidth of communication networks and are modelled as a linear function of a certain set of indicator functions that depend on the same stochastic variable attention is focused on the analysis and design problems of a full order robust h filter such that for all admissible parameter uncertainties and all possible incomplete measurements the filtering error dynamics is exponentially mean square stable and a prescribed h attenuation level is guaranteed some recently reported methodologies such as delay dependent and parameter dependent stability analysis approaches are employed to obtain less conservative results sufficient conditions which are dependent

optimal full order and reduced order estimators for discrete time systems with multiple packet dropouts this paper is concerned with the estimation problem for discrete time stochastic linear systems with multiple packet dropouts based on a recently developed model for multiple packet dropouts the original system is transferred to a stochastic parameter system by augmentation of the state and measurement the optimal full order linear filter of the form of employing the received outputs at the current and last time instants is investigated the solution to the optimal linear filter is given in terms of a riccati difference equation governed by packet arrival rate the optimal filter is reduced to the standard kalman filter when there are no packet dropouts the steady state filter is also studied a sufficient condition for the existence of the steady state filter is given and the asymptotic stability of the optimal filter is analyzed at last a reduced order filter is investigated

robust cal h _ infty filtering for markovian jump systems with randomly occurring nonlinearities and sensor saturation the finite horizon case this paper addresses the robust h filtering problem for a class of discrete time varying markovian jump systems with randomly occurring nonlinearities and sensor saturation two kinds of transition probability matrices for the markovian process are considered namely the one with polytopic uncertainties and the one with partially unknown entries the nonlinear disturbances are assumed to occur randomly according to stochastic variables satisfying the bernoulli distributions the main purpose of this paper is to design a robust filter over a given finite horizon such that the h disturbance attenuation level is guaranteed for the time varying markovian jump systems in the presence of both the randomly occurring nonlinearities and the sensor saturation sufficient conditions are established for the existence of the desired filter satisfying the h performance constraint in terms of a set of recursive linear matrix inequalities simulation results demonstrate the effectiveness of the developed filter design scheme

h sub spl infin filtering for fuzzy singularly perturbed systems with pole placement constraints an lmi approach this paper addresses the problem of designing an h sub spl infin filter for a class of nonlinear singularly perturbed systems described by a takagi sugeno ts fuzzy model based on a linear matrix inequality lmi approach we develop a fuzzy h sub spl infin controller that guarantees that i the l sub 2 gain from an exogenous input to a filter error is less than or equal to a prescribed value and ii the poles of each local filter are within a prespecified region in order to alleviate the ill conditioning resulting from the interaction of slow and fast dynamic modes solutions to the problem are given in terms of linear matrix inequalities which are independent of the singular perturbation spl epsiv when spl epsiv is sufficiently small the proposed approach does not involve the separation of states into slow and fast ones and it can be applied not only to standard but also to nonstandard singularly perturbed nonlinear systems a numerical example is provided to illustrate the design developed in this paper

induced l sub 2 and generalized h sub 2 filtering for systems with repeated scalar nonlinearities this paper provides complete results on the filtering problem for a class of nonlinear systems described by a discrete time state equation containing a repeated scalar nonlinearity as in recurrent neural networks both induced l sub 2 and generalized h sub 2 indexes are introduced to evaluate the filtering performance for a given stable discrete time systems with repeated scalar nonlinearities our purpose is to design a stable full order or reduced order filter with the same repeated scalar nonlinearities such that the filtering error system is asymptotically stable and has a guaranteed induced l sub 2 or generalized h sub 2 performance sufficient conditions are obtained for the existence of admissible filters since these conditions involve matrix equalities the cone complementarity linearization procedure is employed to cast the nonconvex feasibility problem into a sequential minimization problem subject to linear matrix inequalities which can be readily solved by using standard numerical software if these conditions are feasible a desired filter can be easily constructed these filtering results are further extended to discrete time systems with both state delay and repeated scalar nonlinearities the techniques used in this paper are very different from those used for previous controller synthesis problems which enable us to circumvent the difficulty of dilating a positive diagonally dominant matrix a numerical example is provided to show the applicability of the proposed theories

gaussian wave packets in inhomogeneous media with curved interfaces time dependent particle like pulses are considered as asymptotic solutions of the classical wave equation the wave packets are localized in space with gaussian envelopes the pulse centres propagate along the rays of the wave equation and the envelope parameters satisfy evolution equations very similar to the ray equations for time harmonic disturb ances however the present theory contains an extra degree of freedom not found in the time harmonic theory explicit results are presented for media with constant velocity gradients and interesting new phenomena are identified for example a pulse that is initially long in the direction of propagation and comparatively narrow in the orthogonal direction maintains its initial spatial orientation even as the propagation direction rotates the reflection and transmission of a pulse incident upon an interface are also discussed the various theoretical results are illustrated by numerical simulations this method of solution could be very useful for fast forward modelling in large scale structures it is formulated explicitly in the time domain and does not suffer from unphysical singularities at caustics

brief robust h filtering with error variance constraints for discrete time varying systems with uncertainty the robust h filtering problem with error variance constraints is considered for discrete time varying systems subject to norm bounded parameter uncertainties in both the state and the output matrices of the state space model sufficient conditions for a finite horizon filter to satisfy state estimation error variance constraints as well as prescribed h performance for all admissible perturbations are given in terms of two discrete riccati difference equations which are of a form suitable for recursive computation the results are extended to cover the case of stationary filtering over an infinite horizon for uncertain time invariant systems

brief paper optimal linear estimation for systems with multiple packet dropouts this paper is concerned with the optimal linear estimation problem for linear discrete time stochastic systems with multiple packet dropouts based on a packet dropout model the optimal linear estimators including filter predictor and smoother are developed via an innovation analysis approach the estimators are computed recursively in terms of the solution of a riccati difference equation of dimension equal to the order of the system state plus that of the measurement output the steady state estimators are also investigated a sufficient condition for the convergence of the optimal linear estimators is given simulation results show the effectiveness of the proposed optimal linear estimators

h sub spl infin filtering for multiple time delay measurements the aim of this paper is to study the h sub spl infin filtering problem for linear continuous time systems with both current and multiple time delay measurements the key technique applied for deriving the filter is the reorganized innovation analysis approach in krein space a necessary and sufficient condition for the existence of an h sub spl infin filter is derived and the solution to the h sub spl infin filter is given in terms of solutions of riccati and matrix differential equations an example is finally presented to show the effectiveness of the proposed approach

analysis and synthesis of randomly switched systems with known sojourn probabilities in this paper a new approach is proposed and investigated for the stability analysis and stabilizing controller design of randomly switched linear discrete systems the approach is based on sojourn probabilities and it is assumed that these probabilities are known a prior a new lyapunov functional is constructed and two main theorems are proved in this paper theorem 1 gives a sufficient condition for a switched system with known sojourn probabilities to be mean square stable theorem 2 gives a sufficient condition for the design of a stabilizing controller the applications of these theorems and the corresponding corollary and lemma are demonstrated by three numerical examples finally some future research is proposed

an a stable extended trapezoidal rule for the integration of ordinary differential equations abstract we examine a single step implicit integration algorithm which is obtained by a modification of the well known trapezoidal rule the obtained new method is a third order numerical process and preserves the property of a stability of the trapezoidal rule numerical examples involving stiff linear systems of first order differential equations are also included to demonstrate the practical usefulness of this new integration procedure

brief paper optimal estimation of linear discrete time systems with stochastic parameters this paper considers optimal linear state estimation in the general case of linear discrete time systems with stochastic parameters which are statistically independent with respect to time the estimator is derived by transforming the system to one with deterministic parameters and state dependent additive system and observation noise it is shown that mean square stability of the system is a sufficient and almost necessary condition for the existence uniqueness and stability of the time invariant estimator