Differentiation via Logarithmic Expansions

In this note, we introduce a new finite difference approximation called the Black-Box Logarithmic Expansion Numerical Derivative (BLEND) algorithm, which is based on a formal logarithmic expansion of the differentiation operator. BLEND capitalizes on parallelization and provides derivative approximations of arbitrary precision, i.e., our analysis can be used to determine the number of terms in the series expansion to guarantee a specified number of decimal places of accuracy. Furthermore, in the vector setting, the complexity of the resulting directional derivative is independent of the dimension of the parameter.

Application of Analogue Derivators for Obtaining Direct Non-Measurable Components of the State Vector in Crane Control

This paper presents a simple-to-use system for estimating non-measurable components of crane state vector considering parameter changes. To obtain them, it is possible to use a numerical derivative, where the measurement noise causes great inaccuracies, or the Luenberger observer and Kalman filter, which require knowledge of the dynamics of the controlled system, which is constantly changing with the crane.

A second-order finite difference scheme for singularly perturbed initial value problem on layer-adapted meshes

In this paper, a second-order singularly perturbed initial value problem is considered. A hybrid scheme which is a combination of a cubic spline and a modified midpoint upwind scheme is proposed on various types of layer-adapted meshes. The error bounds are established for the numerical solution and for the scaled numerical derivative in the discrete maximum norm. It is observed that the numerical solution and the scaled numerical derivative are of second-order convergence on the layer-adapted meshes irrespective of the perturbation parameter. To show the performance of the proposed method, it is applied on few test examples which are in agreement with the theoretical results. Furthermore, existing results are also compared to show the robustness of the proposed scheme.

Linear Regression to Minimize the Total Error of the Numerical Differentiation

It is well known that numerical derivative contains two types of errors. One is truncation error and the other is rounding error. By evaluating variables with rounding error, together with step size and the unknown coefficient of the truncation error, the total error can be determined. We also know that the step size affects the truncation error very much, especially when the step size is large. On the other hand, rounding error will dominate numerical error when the step size is too small. Thus, to choose a suitable step size is an important task in computing the numerical differentiation. If we want to reach an accuracy result of the numerical difference, we had better estimate the best step size. We can use Taylor Expression to analyze the order of truncation error, which is usually expressed by the big O notation, that is, E(h) = Chk. Since the leading coefficient C contains the factor f(k)(ζ) for high order k and unknown ζ, the truncation error is often estimated by a roughly upper bound. If we try to estimate the high order difference f(k)(ζ), this term usually contains larger error. Hence, the uncertainty of ζ and the rounding errors hinder a possible accurate numerical derivative.We will introduce the statistical process into the traditional numerical difference. The new method estimates truncation error and rounding error at the same time for a given step size. When we estimate these two types of error successfully, we can reach much better modified results. We also propose a genetic approach to reach a confident numerical derivative.