Explicit Nordsieck methods with extended stability regions

We describe the construction of explicit Nordsieck methods of order p and stage order q = p with large regions of absolute stability. We also discuss error propagation and estimation of local discretization errors. The error estimators are derived for examples of general linear methods constructed in this paper. Some numerical experiments are presented which illustrate the effectiveness of proposed methods.     

Convergence analysis of the Jacobi spectral-collocation method for fractional integro-differential equations

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L^∞ norm and weighted L²-norm. The numerical examples are given to illustrate the theoretical results.     

QR methods and error analysis for computing Lyapunov and Sacker�CSell spectral intervals for linear differential-algebraic equations

In this paper, we propose and investigate numerical methods based on QR factorization for computing all or some Lyapunov or Sacker?CSell spectral intervals for linear differential-algebraic equations. Furthermore, a perturbation and error analysis for these methods is presented. We investigate how errors in the data and in the numerical integration affect the accuracy of the approximate spectral intervals. Although we need to integrate numerically some differential-algebraic systems on usually very long time-intervals, under certain assumptions, it is shown that the error of the computed spectral intervals can be controlled by the local error of numerical integration and the error in solving the algebraic constraint. Some numerical examples are presented to illustrate the theoretical results.     

Optimale Fehlerabschätzungen mit dualen Funktionalen bei semilinearen Randwertaufgaben gewöhnlicher Differentialgleichungen

In Shampine [7] it is shown how to obtain variational error bounds for approximate solutions of boundary value problems for semilinear ordinary differential equations. These bounds are depending on a certain constantK, the existence of which is assumed. Our paper aims at practical computation; in order to get applicableL _∞-error bounds,K has to be computed explicitly. Using Gårding's inequality, we obtainK=K(?) depending on a positive parameter ?. In order to make these bounds efficient,K(?) will be optimized. In application only the maximal zeros of three polynomials have to be computed. Some numerical examples are given to compare the error bounds with the actual errors.     

Legendre spectral collocation method for volterra-hammerstein integral equation of the second kind

This paper is concerned with obtaining the approximate solution for VolterraHammerstein integral equation with a regular kernel. We choose the Gauss points associated with the Legendre weight function ω(x) = 1 as the collocation points. The Legendre collocation discretization is proposed for Volterra-Hammerstein integral equation. We provide an error analysis which justifies that the errors of approximate solution decay exponentially in L~2 norm and L~∞ norm. We give two numerical examples in order to illustrate the validity of the proposed Legendre spectral collocation method.     

Adapted Falkner-type methods solving oscillatory second-order differential equations

The classical Falkner methods (Falkner, Phil Mag S 7:621, 1936) are well-known for solving second-order initial-value problems u′′(t)?=?f(t, u(t), u′(t)). In this paper, we propose the adapted Falkner-type methods for the systems of oscillatory second-order differential equations u′′(t)?+?Mu(t)?=?g(t, u(t)) and make a rigorous error analysis. The error bounds for the global errors on the solution and the derivative are presented. In particular, the error bound for the global error of the solution is shown to be independent of ||M||. We also give a stability analysis and plot the regions of stability for our new methods. Numerical examples are included to show that our new methods are very competitive compared with the reformed Falkner methods in the scientific literature.     

A modified Markov model for the estimation of computer software performance

The model proposed by Trivedo and Shooman [8] is extended and modified by assuming that (1) the error occurrence rate when the machine is running is proportional to the number of errors in the system; (2) the error correction rate has two components, either an error is corrected with correction rate μ₀ or an error is corrected but a new error is created with ineffective correction rate μ₁. The solution of the differential equations corresponding to the model is obtained in closed form.     

Optimal nonparametric estimation of the density of regression errors with finite support

Knowledge of the probability distribution of error in a regression problem plays an important role in verification of an assumed regression model, making inference about predictions, finding optimal regression estimates, suggesting confidence bands and goodness of fit tests as well as in many other issues of the regression analysis. This article is devoted to an optimal estimation of the error probability density in a general heteroscedastic regression model with possibly dependent predictors and regression errors. Neither the design density nor regression function nor scale function is assumed to be known, but they are suppose to be differentiable and an estimated error density is suppose to have a finite support and to be at least twice differentiable. Under this assumption the article proves, for the first time in the literature, that it is possible to estimate the regression error density with the accuracy of an oracle that knows “true” underlying regression errors. Real and simulated examples illustrate importance of the error density estimation as well as the suggested oracle methodology and the method of estimation.     

Fehleranalyse für die Gauß-Elimination zur Berechnung der Lösung minimaler Länge

Summary Presented is a realistic, elementwise analysis for the rounding errors of a generalization of Gauss elimination for solving the linear best least squares problem without pivoting. The bounds are suitable to determine the class of well-posed problems to the given method. A mixed error analysis is given and then the effects of errors in the input data are studied. Numerical examples demonstrate the efficiency.

     

Estimation of the regression operator from functional fixed-design with correlated errors

We consider the estimation of the regression operator r in the functional model: Y=r(x)+ε, where the explanatory variable x is of functional fixed-design type, the response Y is a real random variable and the error process ε is a second order stationary process. We construct the kernel type estimate of r from functional data curves and correlated errors. Then we study their performances in terms of the mean square convergence and the convergence in probability. In particular, we consider the cases of short and long range error processes. When the errors are negatively correlated or come from a short memory process, the asymptotic normality of this estimate is derived. Finally, some simulation studies are conducted for a fractional autoregressive integrated moving average and for an Ornstein-Uhlenbeck error processes.     

Error estimates for Miller's algorithm

Summary A rigorous error analysis is given of both truncation and rounding errors in Miller's algorithm for three-term scalar recursions and 2×2 matrix-vector recursions. The error bounds are shown to be very realistic and this will be supported by examples. The results are generalized to recursions of higher order.     

Rounding errors in numerical solutions of two linear equations in two unknowns

New condition numbers and stability constants for the numerical behaviour of Cramer's rule and Gaussian elimination for solving two linear equations in two unknowns under data perturbations and rounding errors of floating-point arithmetic are established. By these means fundamental error estimates and stability theorems are proved. The error estimates are illustrated by a series of numerical examples.     

Quadrature error of the load vector in the finite element method

We analyze the error in thep version of the of the finite element method when the effect of the quadrature error is taken in the load vector. We briefly study some results on theH ¹ norm error and present some new results for the error in theL ₂ norm. We investigate the quadrature error due to the numerical integration of the right hand side We present theoretical and computational examples showing the sharpness of our results.     

Sinc-based computations of eigenvalues of Dirac systems

The celebrated classical sampling theorem is used to compute approximate values of the eigenvalues of Dirac systems with eigenvalue parameter in the boundary conditions. We deal with problems with an eigenparameter in one or two boundary conditions. The error analysis is established considering both truncation and amplitude errors associated with the sampling theorem. We indicate the role of the amplitude error as well as other parameters in the method via illustrative examples. AMS subject classification (2000)  34L16, 65L15, 94A20     

Means that minimize relative error,and an associated integral equation

Determinations are made of the means that minimize various relative errors in the sense that the harmonic mean of a and b minimizes the traditional relative error on [a, b]. The general problem for averaged relative error leads to a nonlinear integral equation for which we prove existence and uniqueness results, as well as constructive solution procedures. The inverse problem of finding the measure of relative error corresponding to a given mean is also analyzed in detail. These studies shed light on both new and previously known inequalities for specific means.     

On the construction of error estimators for implicit Runge-Kutta methods

For implicit Runge-Kutta methods intended for stiff ODEs or DAEs, it is often difficult to embed a local error estimating method which gives realistic error estimates for stiff/algebraic components. If the embedded method's stability function is unbounded at z=∞, stiff error components are grossly overestimated. In practice, some codes ‘improve’ such inadequate error estimates by premultiplying the estimate by a ‘filter’ matrix which damps or removes the large, stiff error components. Although improving computational performance, this technique is somewhat arbitrary and lacks a sound theoretical backing. In this scientific note we resolve this problem by introducing an implicit error estimator. It has the desired properties for stiff/algebraic components without invoking artificial improvements. The error estimator contains a free parameter which determines the magnitude of the error, and we show how this parameter is to be selected on the basis of method properties. The construction principles for the error estimator can be adapted to all implicit Runge-Kutta methods, and a better agreement between actual and estimated errors is achieved, resulting in better performance.     

On a conjecture of D. B. Hunter

We consider errors of positive quadrature formulas applied to Chebyshev polynomials. These errors play an important role in the error analysis for many function classes. Hunter conjectured that the supremum of all errors in Gaussian quadrature of Chebyshev polynomials equals the norm of the quadrature formula. We give examples, for which Hunter's conjecture does not hold. However, we prove that the conjecture is valid for all positive quadratures if the supremum is replaced by the limit superior. Considering a fixed positive quadrature formula and the sequence of all Chebyshev polynomials, we show that large errors are rare.     

Adaptive finite element analysis of structures under transient dynamic loading using modal superposition

In this paper, the error estimation and adaptive strategy developed for the linear elastodynamic problem under transient dynamic loading based on the Z–Z criterion is utilized for 2D and plate bending problems. An automatic mesh generator based on “growth meshing” is utilized effectively for adaptive mesh refinement. Optimal meshes are obtained iteratively corresponding to the prescribed domain discretization error limit and for a chosen number of basis modes satisfying modal truncation errors. Numerous examples show the effectiveness of the integrated approach in achieving the target accuracy in finite element transient dynamic analysis.     

A sequential algorithm and error sensitivity analysis for the inverse heat conduction problems with multiple heat sources

This paper proposes a sequential approach to determine the unknown parameters for inverse heat conduction problems which have multiple time-dependent heat sources. There are two main aims in this study, one is to derive an inverse algorithm that can estimate the unknown conditions effectively, and the other is to bring up a theoretical sensitivity analysis to discuss what causes the growth of errors. This paper has three major achievements with regard to the literature on IHCPs, as follows: (1) proposing an efficient sequential inverse algorithm that can simultaneously determine several unknown time-dependent parameters; (2) exploring why the sequential function specification method can provide a stable but inaccurate estimation when tackling problems with larger measurement errors; and (3) discussing the sensitivity problem and analyzing what factors cause the growth in error sensitivity. Three examples are applied to demonstrate the performance of the proposed method, and the numerical results show that the accurate estimations can be obtained by alleviating the error sensitivity when the measurement error is considered.     

A Certain Class of Weighted Approximations for Integrable Functions and Applications

A certain class of weighted approximations, which extends the results of Masjed-Jamei [6] is introduced for integrable functions and some of upper bounds are obtained for the absolute value of the errors of such approximations in two L¹[a, b] and L^∞[a, b] spaces. As the main motivation for introducing the aforesaid class, it is shown that many new inequalities can be generated from the given error bounds. Some illustrative examples are presented in this sense. Moreover, by using the obtained error bounds, a nonstandard type of three-point weighted quadrature rules is introduced and its error bounds are computed.