Differentiation and Integration 4.1 INTRODUCTION

BIT, 1975

The relationships between various numerical methods for obtaining polynomial approximations to the first derivative of a known function are investigated, and their computational advantages discussed. Optimal sequences of interpolation points are then selected with the objective of minimising the relative contribution of rounding errors to the total error, and geometric sequences, though non-optimal in this sense, are considered for computational reasons.

We represent new estimates of errors of quadrature formula, formula of numerical differentiation and approximation using Taylor polynomial. To measure the errors we apply representation of the remainder in Taylor formula by least concave majorant of the modulus of continuity of the n−th derivative of an n−times differentiable function. Our quantitative estimates are special applications of a more general inequality for P n-simple functionals.

Journal of Computational and Applied Mathematics, 2003

A new type of Taylor series based ÿnite di erence approximations of higher-degree derivatives of a function are presented in closed forms, with their coe cients given by explicit formulas for arbitrary orders. Characteristics and accuracies of presented approximations and already presented central di erence higher-degree approximations are investigated by performing example numerical di erentiations. It is shown that the presented approximations are more accurate than the central di erence approximations, especially for odd degrees. However, for even degrees, central di erence approximations become attractive, as the coe cients of the presented approximations of even degrees do not correspond to equispaced input samples.

Journal of Computational and Applied Mathematics, 1999

Numerical di erentiation formulas based on interpolating polynomials, operators and lozenge diagrams can be simpliÿed to one of the ÿnite di erence approximations based on Taylor series. In this paper, we have presented closed-form expressions of these approximations of arbitrary order for ÿrst and higher derivatives. A comparison of the three types of approximations is given with an ideal digital di erentiator by comparing their frequency responses. The comparison reveals that the central di erence approximations can be used as digital di erentiators, because they do not introduce any phase distortion and their amplitude response is closer to that of an ideal di erentiator. It is also observed that central di erence approximations are in fact the same as maximally at digital di erentiators. In the appendix, a computer program, written in MATHEMATICA is presented, which can give the approximation of any order to the derivative of a function at a certain mesh point.

Journal of Informatics and Mathematical Sciences, 2012

In this note, the polynomial interpolation of degree $n$ passing through $n+1$ distinct points is considered. The coefficients of the polynomial interpolation are investigated in terms of finite differences and numerical differentiations. The coefficients are formulated by the use of divided differences and correlated with forward, backward differences and numerical differentiations. It is seen that the coefficients of the polynomial interpolation can be found and computed by using finite differences, numerical differentiations and generating special formulae for equidistant points or not.

Advances in Engineering Software (1978), 1982

Three methods for the numerical evaluation of derivatives from data with errors are compared. These are polynomial fittings, trigonometric series and spline functions. The procedure consists of a number of numerical experiments where a randomly generated error is added to the ordinate of a test function. The effect of this error on the value of the numerical derivatives of three test functions is assessed. It is found that spline functions yield generally better derivatives than the other methods.

BIT, 1980

The first derivative of a real-valued function may be approximated at a certain point by the derivative of a polynomial collocating with the function at this point and a number of other distinct points. The particular points which minimise the magnification of any rounding errors in the function values for any fixed point of truncation error are shown to be closely related to the turning points of a related Chebyshev polynomial.