Numerical differentiation by integration (Q2871184)

While there are various methods which have been developed for numerical differentiation, the estimation of the derivative of a function is often problematic when one has only noisy values of the function itself. In this instance it is important to employ a method which is able to calculate \(f'(x)\) in a stable manner.NEWLINENEWLINEThis article specifically focuses on the Lanczos method, as interpreted by \textit{C. W. Groetsch} [Am. Math. Mon. 105, No. 4, 320--326 (1998; Zbl 0927.26003)] as a regularization method. Since calculating an approximation of the derivative at the end points and points near the end points of the interval in question cannot be done, this article proposes an integral method capable of doing so. The integral method developed to approximate derivatives of approximately specified functions is applicable for any point on a finite closed interval.NEWLINENEWLINEFirstly the authors propose an integral operator \(D_h f\) such that \((D_h) f(x)\) can be used to approximate \(f(x)\) as \(h \rightarrow 0\). When \(f'_{+}\) and \(f'_{-}\) exist, the convergence behaviour of \(D_h f\) is shown which in fact generalises a result of Groetsch [loc. cit.]. Convergence estimates in \(C[a,b]\) are also provided: if \(f'(x) \in C^{k,\alpha} [a,b]\) (\(k = 0,1\)), then the corresponding convergence rate is \(O\left(\delta^{\frac{\alpha+k}{\alpha+k+1}}\right)\) by an a priori choice of the regularization parameter \(h\). Lastly, convergence estimates in \(L^p[a,b]\) are given: if \(f(x) \in AC[a,b] \bigcap H^{k,p} (a,b)\) (\(k=2,3\)), then the corresponding convergence rate is \(O\left(\delta^{\frac{k-1}{k}}\right)\) by an a priori choice of the regularization parameter \(h\).NEWLINENEWLINETo test the method, essentially a generalised ``Lanczos method'', the authors consider the derivative of several functions on a closed interval. Their numerical examples show that the method is simple and applicable. Stable approximations for the derivatives are obtained for approximately specified functions, and the method is able to calculate these results for any point on the closed interval. The method allows for the calculation of a ``pseudo-derivative'' at points where the function is not differentiable; the significance of which is in the ``properties the derivative shares with other mathematical subjects'' [quoted from article].