an invariant imbedding method for computation of eigenlengths of singular two point boundary value problems in this article we have described an invariant imbedding method for calculating the smallest eigenlength of a singular tpbvp with the singularity at the origin the invariant imbedding yields a first order nonlinear equation called a riccati equation and also gives the initial conditions at the origin for this equation with the aid of theorem 8 in section 3 we numerically integrate the riccati equation to blowup which gives our computed eigenlength in closing we would like to comment on the numerical merits of the integration to blowup technique on the basis of the examples presented it appears that this technique combined with the available numerical integrators with variable step size is capable of producing accurate results the feature of a variable step size is essential as the value of z approaches the actual eigenlength however it is desirable to have a priori estimate or bounds of the eigenlength similar to those of boland and nelson 2 for the nonsingular case the singular system however presents difficulties due to the lack of sign conditions on the coefficient matrices in obtaining such bounds hopefully an investigation of the matrix r z will yield these results

quasilinear multistep methods and variable step predictor corrector methods for neutral functional differential equations the theory of quasilinear multistep methods and predictor corrector methods for neutral functional differential equations is developed the theorem about the asymptotic behaviour of the global discretization error of quasilinear methods which forms a theoretical basis for step control strategy for predictor corrector methods for neutral equations is given some problems of implementation of these methods are also addressed and the results of numerical experiments are presented

variable step size predictor corrector schemes for second kind volterra integral equations in this paper a family of implicit multistep methods for the solution of volterra integral equations is derived these methods together with an explicit euler predictor permit the use of a variable step size when solving integral equations means of controlling the error and stability by varying the step size and the order of the method are described extensive numerical results are presented

linear conservation laws for odes abstract some physical systems described by odes have quantities that are conserved as the system evolves runge kutta formulas and linear multistep methods preserve linear conservation laws automatically a code for the integration of odes involves a number of algorithms in addition to the basic formula it is shown here that popular codes preserve linear conservation laws if they are used properly an application is made to differential algebraic equations of index 2 arising in the solution of the navier stokes equation by the method of lines

a multi purpose system for the numerical integration of ode s in this paper we describe the development of an experimental multi purpose integration system strongly modular the system allows the use of wide classes of multistep methods for solving ode s singulsr perturbation problems as well as implicit differential algebraic equations we focus our attention on a new family of collocation methods for nonstiff problems while these methods are strongly related to the stiffly stable bdf methods they are particularly well suited for the implementation of a type insensitive integration algorithm compound discretizations in which certain variables are integrated with the first method and others with the second can also be used a stability analysis is given and some implementation issues are discussed

applications of the new scattering formalism the dirichlet boundary condition this is the third in a series of articles concerning the theory of scattering from a rough surface consideration of the scattering problem for a scalar wave obeying dirichlet boundary conditions enables one to investigate the formalism developed in the previous work the advantages of this new representation of the scattering amplitude is discussed this result is supplemented with an improved integral equation for the desired solutions on the surface this new equation not only allows one to reproduce the approximations for the scattering amplitude obtained previously but it can also be used to extend those results to higher orders this is illustrated with a derivation of the second born approximation the correction to the scattering amplitude due to a distribution of point scatters is also obtained to the order under consideration this phenomenon is shown to be equivalent to a perturbation of the surface a similar result is found when the index of refraction is allowed to vary near the scattering surface

a singular nonlinear boundary value problem membrane response of a spherical cap the nonlinear boundary value problem y n x 2 32y 2 lambda 2 8 quad 0 0 nu 0 are constants models the large deflection membrane response of a spherical cap and has been solved numerically by various techniques in the engineering literature we prove existence and uniqueness of a solution of this problem provide qualitative information about the solution and use this information to suggest an approach to numerical computation

stiffness characteristic and the applications of the method of lines on nonuniform grids the method of lines is investigated for the numerical solution of the stream function and vorticity form of the navier stokes equations on nonuniform grids stiffness characteristics of a linear one dimensional model equation are examined to establish the feasibility of applying the method to the vorticity equation in two dimensions the governing equations are transformed from the physical domain with a highly variable grid to a computational domain with a uniform grid the method of lines is used to solve only the vorticity equation and the successive over relaxation technique is used to solve the stream function equation it is observed that the transformed governing equations become stiffer with increased concentration of grid points and also as the number of grid points increases it is also observed that the differencing technique affects the stiffness characteristics the use of forward differencing is not feasible and backward differencing is preferable to central differencing for high reynolds numbers the results of specific applications for the solution of flow in curved wall diffusers and a driven cavity demonstrate that the method of lines under certain circumstances is feasible for the numerical solution of physical problems on domains covered with variable grids

on the exponential fitting of composite multiderivative linear multistep methods exponentially fitted composite multiderivative linear multistep methods of orders up to 5 are derived a numerical investigation of these formulae indicates that they may be a stable for all choices of the fitting parameters an important aspect of the class of integration formulae considered is that each formula contains a built in local error estimate a facility lacking in most other exponentially fitted formulae some preliminary numerical results are given and these show that for problems for which exponential fitting is appropriate the new formulae provide a large saving in computational effort compared with conventional ones

numerical treatment of o d es the theory ofa methods almost all commonly used methods for o d es and their most miscellaneous compositions area methods i e they can be reduced toz o z j az j 1 h x j 1 z j 1 z j h z j s a s s j 1 1 m this paper presents a general theory fora methods and discusses its practical consequences an analysis of local discretization error l d e accumulation results in a general order criterium and reveals which part of the l d e effectively influences the global error this facilitates the comparison of methods and generalizes considerably the concept of error constants it is shown as a consequence that the global error cannot be safely controlled by the size of the l d e and that the conventional error control may fail in important cases furthermore butcher s effective order methods the concept of nordsieck forms and gear s interpretation of lineark step schemes as relaxation methods are generalized the stability of step changing is shortly discussed

variable order adams codes abstract variable step size variable order vsvo adams codes are very effective for solving initial value problems for first order systems of ordinary differential equations the theory of fixed order codes is classical but when the order is varied there is no theory explaining fundamental issues with realistic assumptions about order and step size selection we prove convergence approximate locally the behavior of the error and justify standard error estimators

type insensitive ode codes based on implicit a stable formulas a special concept of stiffness is appropriate for implict a stable formulas it is possible to recognize this kind of stiffness economically and reliably using information readily available during the integration of an ode using this development a variety of effective ode solvers could be made insensitive to the type of problem i e the code would automatically recognize and alter automatically its algorithm at any step depending on whether the problem is stiff there

automatic differentiation of numerical integration algorithms automatic differentiation ad is a technique for automatically augmenting computer programs with statements for the computation of derivatives this article discusses the application of automatic differentiation to numerical integration algorithms for ordinary differential equations odes in particular the ramifications of the fact that ad is applied not only to the solution of such an algorithm but to the solution procedure itself this subtle issue can lead to surprising results when ad tools are applied to variable stepsize variable order ode integrators the computation of the final time step plays a special role in determining the computed derivatives we investigate these issues using various integrators and suggest constructive approaches for obtaining the desired derivatives

hp adaptive time integration based on the bdf for viscous flows this paper presents a procedure based on the backward differentiation formulas of order 1 to 5 to obtain efficient time integration of the incompressible navier stokes equations the adaptive algorithm performs both stepsize and order selections to control respectively the solution accuracy and the computational efficiency of the time integration process the stepsize selection h adaptivity is based on a local error estimate and an error controller to guarantee that the numerical solution accuracy is within a user prescribed tolerance the order selection p adaptivity relies on the idea that low accuracy solutions can be computed efficiently by low order time integrators while accurate solutions require high order time integrators to keep computational time low the selection is based on a stability test that detects growing numerical noise and deems a method of order p stable if there is no method of lower order that delivers the same solution accuracy for a larger stepsize hence it guarantees both that 1 the used method of integration operates inside of its stability region and 2 the time integration procedure is computationally efficient the proposed time integration procedure also features a time step rejection and quarantine mechanisms a modified newton method with a predictor and dense output techniques to compute solution at off step points

diagonally implicit runge kutta formulae for the numerical integration of nonlinear two point boundary value problems abstract fully implicit runge kutta formulae based on interpolatory quadrature schemes for the approximate numerical solution of nonlinear two point boundary value problems have been investigated by weiss such formulae are not used very often in practice however because they generally require such a large computational effort as to make them uncompetitive with for example integration schemes based on the trapezoidal or implicit mid point rules in this paper we consider an alternative class of runge kutta formulae namely diagonally implicit runge kutta dirk formulae which can be implemented more efficiently than the fully implicit formulae considered by weiss we also consider how these dirk formulae can be implemented in a defect or deferred correction framework and we give some numerical results to illustrate the algorithms derived one particular formula belonging to the dirk class is the implicit mid point rule in this paper we derive an efficient implementation of this formula which is applicable when the given boundary conditions are non separated

sensitivity analysis of flexible multibody systems using composite materials components in this paper a general formulation for the computation of the first order analytical sensitivities based on the direct method using automatic differentiation of flexible multibody systems is presented the direct method for sensitivity calculation is obtained by differentiating the equations that define the response of the flexible multibody systems of composite materials with respect to the design variables which are the ply orientations of the laminated in order to appraise the benefits of the approach suggested and to highlight the risks of the procedure the analytical sensitivities are compared with the numerical results obtained by using the finite difference method for the beam composite material elements the section properties and their sensitivities are found using an asymptotic procedure that involves a two dimensional 2 d finite element analysis of their cross section the equations of the sensitivities are obtained by automatic differentiation and integrated in time simultaneously with the equations of motion of the multibody systems the equations of motion and the sensitivities of the flexible multibody system are solved and the accelerations velocities and the sensitivities of accelerations and velocities are integrated in time using a multi step multi order integration algorithm through the application of the methodology to two simple flexible multibody systems the difficulties and benefits of the procedure with respect to finite difference approaches or to the direct implementation of the analytic sensitivities are discussed copyright 2008 john wiley sons ltd

continuous parallel block methods and their applications continuous numerical methods have many applications in the numerical solution of discontinuous ordinary differential equations delay differential equations delay differential algebraic equations neutral delay differential equations integro differential equations etc this paper deals with a continuous extension for the discrete approximate solution of ordinary differential equations generated by a class of parallel block methods existence and uniqueness for the continuous extension are discussed convergence and absolute stability of the continuous parallel block methods for ordinary differential equations are studied as applications we adopt the continuous parallel block methods to solve delay differential equations neutral delay differential equations and delay differential algebraic equations and obtain sufficient and necessary conditions for the continuous parallel block methods to be asymptotically stable several numerical experiments are given to illustrate the performance of the continuous parallel block methods

run time estimation of the spectral radius of jacobians it is useful for ordinary differential equation ode solvers to include an estimator of the spectral radius of the jacobian matrix of the system of ode s since this determines the numerical stability of the method hence a knowledge of spectral radius enables the run time selection of a more efficient integrator some techniques for estimating spectral radius are described and compared they include methods suitable for use with any ode solvers but which require additional computation other methods are described which are suitable with runge kutta or rosenbrock methods and which require little extra computation

an algorithm for starting multistep methods the present paper deals with the problem of starting multistep methods we take into consideration an adams bashforth moulton pece pair with the predictor of order q and the corrector of order q 1 to start this method q 1 starting values are necessary in addition to y0 a well known result from theory says that the order of convergence of the whole integration is q 1 if all those starting values are accurate of that same order present production codes start with a predictor of order 1 and a corrector of order 2 at the first step and then proceed step by step each time raising the order by 1 until all the necessary starting values have been obtained but in this manner all the starting errors keep of order 3 and so the whole integration converges no faster than that order this drawback is normally compensated for by taking very small step sizes in the starting phase the general algorithm we propose furnishes at a reasonably low cost the necessary number q 1 of starting values each of the appropriate order q 1 whatever q might be it is independent of the particular multistep formula considered and is mainly designed to be used for high values of q q 10 where the alternative strategies are too expensive or do not exist at all the numerical results reported show the validity of our approach 2003 elsevier science ltd all rights reserved

a new theory for scattering from a surface a new formal approach to the problem of scattering from a rough surface is derived the first ingredient of this approach is a formal statement of the composite model this consists of an approximate expression for the change in the scattering amplitude corresponding to a perturbation of a reference surface it is good to first order in the perturbation and exact with respect to the reference surface the derivation of this result makes use of the general asymptotic form of the scattering solution the expression is simplified using the boundary conditions when the formal composite model is applied to an unfinitesimal translation a new representation of the scattering amplitude is obtained unlike the traditional representation this expression depends on two full solutions to the wave equation and it manifestly exhibits reciprocity this article is the first in a series application of the results will be considered in future papers

a note on the relative efficiency of adams methods abstract in several pieces of the most effective mathematical software for non stiff and mildly stiff initial value problems in ordinary differential equations the classical adams methods are implemented in a particular mode one reason for the choice of mode is sometimes explained in terms of the associated scaled absolute stability regions for the methods in this note it is shown that the same choice of mode remains appropriate from this point of view when certain families of generalized adams type methods are considered

approximation of weak adjoints by reverse automatic differentiation of bdf methods we shed light on the relation between the discrete adjoints of multistep backward differentiation formula bdf methods and the solution of the adjoint differential equation to this end we develop a functional analytic framework based on a constrained variational problem and introduce the notion of weak adjoint solutions of ordinary differential equations we devise a petrov galerkin finite element fe interpretation of the bdf method and its discrete adjoint scheme obtained by reverse internal numerical differentiation we show how the fe approximation of the weak adjoint is computed by the discrete adjoint scheme and prove its convergence in the space of normalized functions of bounded variation we also show convergence of the discrete adjoints to the classical adjoints on the inner time interval finally we give numerical results for non adaptive and fully adaptive bdf schemes the presented framework opens the way to carry over techniques on global error estimation from fe methods to bdf methods

nonlinear normal modes for the toda chain abstract the toda chain is a nonlinear mass spring chain which can in principle be integrated analytically by the action angle theory of classical mechanics we show that the transformation from physical variables to action variables can be implemented very simply on a computer we study the correspondence between action variables and the motions of the chain and find the role of the action variables to be very similar to the role of normal mode amplitudes in the harmonic chain this similarity leads to a partly rigorous partly heuristic normal mode analysis for the toda chain new results are obtained when this normal mode analysis is applied to certain perturbations of the toda chain such as the fermi pasta ulam chain or the toda chain with a mass impurity

robust function on function regression functional linear regression is a widely used approach to model functional responses with respect to functional inputs however classical functional linear regression models can be severely affect

an examination of the composite roughness scattering model a perturbation technique is applied to derive a composite roughness theory for acoustic scattering from the sea surface the leading term in the expansion obtained is the well known result obtainable using ad hoc arguments higher order terms are evaluated to assess their contribution to the high frequency monostatic backscattering strength of the sea it is concluded that the leading term in the perturbation expansion provides an excellent approximation

functional linear regression analysis for longitudinal data we propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data where both the predictor and response are functions of a covariate such as time predictor and response processes have smooth random trajectories and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects in longitudinal studies the number of repeated measurements per subject is often small and may be modeled as a discrete random number and accordingly only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit we propose a functional regression approach for this situation using functional principal component analysis where we estimate the functional principal component scores through conditional expectations this allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory the resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories asymptotic properties for a sample of n subjects are investigated under mild conditions as n oo and we obtain consistent estimation for the regression function besides convergence results for the components of functional linear regression such as the regression parameter function we construct asymptotic pointwise confidence bands for the predicted trajectories a functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced extending the standard r 2 to the functional case the proposed methods are illustrated with a simulation study longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index

gasturbnlab a multidisciplinary problem solving environment for gas turbine engine design on a network of nonhomogeneous machines abstract gas turbine engines are very complex with 20 40 000 parts and have extreme operating conditions the important physical phenomena take place on scales from 10 100 microns to meters a complete and accurate dynamic simulation of an entire engine is enormously demanding designing a complex system like a gas turbine engine will require fast accurate simulations of computational models from multiple engineering disciplines along with sophisticated optimization techniques to help guide the design process in this paper we describe the architecture of an agent based software framework for the simulation of various aspects of a gas turbine engine utilizing a network of collaborating numerical objects through a set of interfaces among the engine parts moreover we present its implementation using the grasshopper agent middleware and provide simulation results that show the feasibility of the computational paradigm implemented

block runge kutta methods for the numerical integration of initial value problems in ordinary differential equations i the nonstiff case block runge kutta formulae suitable for the approximate numerical integration of initial value problems for first order systems of ordinary differential equations are derived considered in detail are the problems of varying both order and stepsize automatically this leads to a class of variable order block explicit runge kutta formulae for the integration of nonstiff problems and a class of variable order block implicit formulae suitable for stiff problems the central idea is similar to one due to c w gear in developing runge kutta starters for linear multistep methods some numerical results are given to illustrate the algorithms developed for both the stiff and nonstiff cases and comparisons with standard runge kutta methods are made

varying the componentwise order of the multistep methods in solving odes and its absolute stability most variable order variable stepsize codes for solving odes vary the order along the interval of integration but keep the order constant across the system however some codes that do partitioning of the odes into stiff and nonstiff subsystems vary the order for each equation in the system in this paper the reasons for varying the order componentwise are given and some of its salient features are illustrated using a numerical example also given is the direct proof of the existence of a region of absolute stability when different order adams explicit methods are used to solve a system of odes

conservation laws and the numerical solution of odes ii abstract many physical systems described by an initial value problem for a system of ordinary differential equations odes conserve physical quantities such as the net charge or total energy as the system evolves typical codes for the numerical solution of the odes will not conserve these quantities and this can lead to solutions which are not even qualitatively correct one way to impose conservation laws is to perturb the numerical solution at each step of the integration a simple theory is developed in this paper which tells how this should be done so as to guarantee convergence of codes based on one step methods it is also easy to interpret the effect on the accuracy of the perturbations

some tools for functional data analysis multivariate data analysis permits the study of observations which are finite sets of numbers but modern data collection situations can involve data or the processes giving rise to them which are functions functional data analysis involves infinite dimensional processes and or data the paper shows how the theory of l splines can support generalizations of linear modelling and principal components analysis to samples drawn from random functions spline smoothing rests on a partition of a function space into two orthogonal subspaces one of which contains the obvious or structural components of variation among a set of observed functions and the other of which contains residual components this partitioning is achieved through the use of a linear differential operator and we show how the theory of polynomial splines can be applied more generally with an arbitrary operator and associated boundary constraints these data analysis tools are illustrated by a study of variation in temperature precipitation patterns among some canadian weather stations

strategies for solving index one dae with non negative constraints liquid liquid extraction modeling leads to solve an index one dae system for the sake of robustness it is desirable to account for non negative constraints based on the dassl architecture a classical index one dae solver we propose and compare three different strategies to implement these bound constraints each of these strategies corresponds to a different newton modification clipping damping or interior point method the comparisons are made on two test cases the robertson ode problem and an example from liquid liquid extraction modeling

efficient solution of nonlinear ordinary differential equations expressed in s system canonical form ordinary differential equations can be recast into a nonlinear canonical form called an s system evidence for the generality of this class comes from extensive empirical examples that have been recast and from the discovery that sets of differential equations and functions recognized as among the most general are special cases of s systems identification of this nonlinear canonical form suggests a radically different approach to numerical solution of ordinary differential equations by capitalizing on the regular structure of s systems efficient formulas for a variable order variable step taylor series method are developed the minimum improvement in efficiency over conventional runge kutta methods is shown to be more than 20 to 60 percent for equations expressed in s system form tests with implemented methods demonstrate that 10 to 100 fold reductions in time are actually realized for solution of s systems to examine the potential benefit of recasting ordinary differential equations and then sol

using nonconvergence of iteration to partition odes given is a variation of partitioning odes when using the multistep method essentially the nonconvergence property is made use of for partitioning the criteria used are explained by using a test system of two linear equations and generalisation is made to a general system of first order odes finally some numerical results are given that include systems greater than two

rkn type parallel block pc methods with lagrange type predictors abstract this paper describes the construction of block predictor corrector methods based on runge kutta nystrom correctors our approach is to apply the predictor corrector method not only at step points but also at off step points block points so that in each step a whole block of approximations to the exact solution at off step points is computed in the next step these approximations are used to obtain a high order predictor formula using lagrange interpolation by suitable choice of the abscissas of the off step points a much more accurately predicted value is obtained than by predictor formulas based on last step values since the block of approximations at the off step points can be computed in parallel the sequential costs of these block predictor corrector methods are comparable with those of a conventional predictor corrector method furthermore by using runge kutta nystrom corrector methods the computation of the approximation at each off step point is also highly parallel application of the resulting block predictor corrector methods to a few widely used test problems reveals that the sequential costs are reduced by a factor ranging from 4 to 50 when compared with the best sequential methods from the literature

construction of variable stepsize multistep formulas a systematic way of extending a general fixed stepsize multistep formula to a minimum storage variable stepsize formula has been discovered that encompasses fixed coefficient interpolatory variable coefficient variable step and fixed leading coefficient as special cases in particular it is shown that the interpolatory stepsize changing technique of nordsieck leads to a truly variable stepsize multistep formula which has implications for local error estimation and formula changing and it is shown that the variable step stepsize changing technique applicable to the adams and backward differentiation formulas has a reasonable generalization to the general multistep formula in fact it is shown how to construct a variable order family of variable coefficient formulas finally it is observed that the first dahlquist barrier does not apply to adaptable multistep methods if storage rather than stepnumber is the key consideration

implicit schemes for differential equations abstract implicit high order accurate temporal integration schemes are developed to reduce partial differential equations of evolution type to a sequence of ordinary differential equations involving the spatial directions only the ordinary differential equations are solved using various accurate integration methods the schemes are free from extraneous boundary and or initial conditions and are stable for large courant number

the projected gradient methods for least squares matrix approximations with spectral constraints the problems of computing least squares approximations for various types of real and symmetric matrices subject to spectral constraints share a common structure this paper describes a general procedure in using the projected gradient method it is shown that the projected gradient of the objective function on the manifold of constraints usually can be formulated explicitly this gives rise to the construction of a descent flow that can be followed numerically the explicit form also facilitates the computation of the second order optimality conditions examples of applications are discussed with slight modifications the procedure can be extended to solve least squares problems for general matrices subject to singular value constraints

efficient calculation of chemically reacting flow a semi implicit finite volume formulation is used to jrvstudy flow with chemical reactions in this formulation the source terms resulting from the chemical reaction are treated implicitly and the resulting system of partial differential equations is sovled using two time stepping schemes the first is based on the runge kutta method and the second on an adams predictor corrector method results show that improvements in computational efficiency depend to a large extent on the manner in which the source term is treated furthermore analysis and computation indicate that the rungekutta method is more efficient than the adams methods for these systems of differential equations finally an adaptive time stepping scheme is developed to study problems involving shock ignition calculations for a hydrogen air system agree well with other methods

on noncharacteristic boundary conditions for discrete hyperbolic initial boundary value problems abstract the stability properties of some discrete hyperbolic initial boundary value problems involving two dependent variables and noncharacteristic boundary conditions are examined approximations based on the lax wendroff leap frog and crank nicolson methods are considered the most accurate approximations are found to be the least stable the results for the model problems can be used in the stability analysis of larger problems where characteristics occur singly or in pairs in particular discretizations of the shallow water and gas dynamics equations which involve boundary conditions written in natural variables are considered some computations demonstrating the stability results are discussed an appendix discussing numerical stability verification techniques is included

an efficient approach for the numerical simulation of multibody systems the field of kinematics and dynamics of mechanical systems has progressed from a manual graphics art to a highly developed discipline in analytical geometry and dynamics various general purpose formulations for the dynamic analysis of constrained mechanical systems cms lead to mixed differential algebraic equations daes called the euler lagrange equations during the past fifteen years many contributions have been made to the theory of computational kinematics and dynamics of cms also called mutibody dynamics the recent advances in computer hardware and software have tremendously revolutionized the analysis of cms there are various numerical approaches for solving general vector fields in the previous paper appl math comput 92 1998 153 193 a complete and detailed analysis of various approaches for the numerical solution of vector fields is given in this paper we extend the algorithms presented in the above mentioned reference to solve the euler lagrange equations of motion for cms the numerical experiments suggest that perturbation approach performs better than the other approaches

a two loop sparse matrix numerical integration procedure for the solution of differential algebraic equations application to multibody systems abstract in this paper a two loop implicit sparse matrix numerical integration tlismni procedure for the solution of constrained rigid and flexible multibody system differential and algebraic equations is proposed the proposed method ensures that the kinematic constraint equations are satisfied at the position velocity and acceleration levels in this method a sparse lagrangian augmented form of the equations of motion that ensures that the constraints are satisfied at the acceleration level is first used to solve for all the accelerations and lagrange multipliers the independent coordinates and velocities are then identified and integrated using htt or newmark formulas expressed in this paper in terms of the independent accelerations only the constraint equations at the position level are then used within an iterative newton raphson procedure to determine the dependent coordinates the dependent velocities are determined by solving a linear system of algebraic equations in order to effectively exploit efficient sparse matrix techniques and have minimum storage requirements a two loop iterative method is proposed equally important the proposed method avoids the use of numerical differentiation which is commonly associated with the use of implicit integration methods in multibody system algorithms numerical examples are presented in order to demonstrate the use of the new integration procedure

automatic solution of sturm liouville problems using the pruess method abstract we develop algorithms based on coefficient approximation for the automatic solution of regular and singular sturm liouville problems

smoothed functional principal components analysis by choice of norm the principal components analysis of functional data is often enhanced by the use of smoothing it is shown that an attractive method of incorporating smoothing is to replace the usual l 2 orthonormality constraint on the principal components by orthonormality with respect to an inner product that takes account of the roughness of the functions the method is easily implemented in practice by making use of appropriate function transforms fourier transforms for periodic data and standard principal components analysis programs several alternative possible interpretations of the smoothed principal components as obtained by the method are presented some theoretical properties of the method are discussed the estimates are shown to be consistent under appropriate conditions and asymptotic expansion techniques are used to investigate their bias and variance properties these indicate that the form of smoothing proposed is advantageous under mild conditions indeed milder than those for existing methods of smoothed functional principal components analysis the choice of smoothing parameter by cross validation is discussed the methodology of the paper is illustrated by an application to a biomechanical data set obtained in the study of the behaviour of the human thumb forefinger system

error estimates and adaptive time step control for a class of one step methods for stiff ordinary differential equations we prove new optimal a priori error estimates for a class of implicit one step methods for stiff ordinary differential equations obtained by using the discontinuous galerkin method with piecewise polynomials of degree zero and one starting from these estimates we propose a new algorithm for automatic time step control and we discuss the relation between this algorithm and earlier algorithms implemented in packages for the numerical solution of stiff ordinary differential equations

approximate representations of the scattering amplitude this is the second in a series of articles about the theory of scattering from a rough surface a symmetric representation of the scattering amplitude and a formal statement of the composite model both derived in the previous article are used to approximate the scattering amplitude in terms of the known results for a reference surface when the reference surface is a plane the small slope approximation follows while the traditional composite model is obtained if the curvature of the reference surface is neglected the medium is assumed to be homogeneous and the calculation is performed for scalar waves obeying dirichlet or neumann boundary conditions electromagnetic waves obeying perfect conductor boundary conditions and the interface between two media for both types of waves finally the requirement that the formal composite model must reproduce the traditional composite model in the appropriate limit is used to obtain a new approximation to the scattering amplitude this is done for the dirichlet neumann and perfect conductor problems the error goes as the square of the curvature

robust functional principal components a projection pursuit approach in many situations data are recorded over a period of time and may be regarded as realizations of a stochastic process in this paper robust estimators for the principal components are considered by adapting the projection pursuit approach to the functional data setting our approach combines robust projection pursuit with different smoothing methods consistency of the estimators are shown under mild assumptions the performance of the classical and robust procedures are compared in a simulation study under different contamination schemes 1 introduction analogous to classical principal components analysis pca the projection pursuit approach to robust pca is based on finding projections of the data which have maximal dispersion instead of using the variance as a measure of dispersion a robust scale estimator sn is used for the maximization problem this approach was introduced by li and chen 1985 who proposed estimators based on maximizing or minimizing a robust scale in this way given a sample xi r d 1 i n the first robust principal component vector is defined as