Numerical differentiation of table-given functions at arbitrarily located interpolation nodes

Abstract

A methodology has been developed for numerically differentiating table-given functions using a Taylor polynomial of degree n, which enables the computation of k-th order derivatives (k £ n) at any point between arbitrarily located interpolation nodes in one, two, or multiple independent variables. Recent research and publications have been analysed, allowing for the assessment of the task complexity of computing derivatives of a function based on the values of independent variables within a certain interval of a table-given function. The formulation of the problem of numerical differentiation of periodic table-given functions using the Taylor polynomial of the nth order from one, two, and multiple independent variables is described. It is established that any tabulated function should be initially smoothed by some function whose analytical expression is a global (local) interpolating polynomial or a polynomial obtained by least squares approximation with some error. The derivative of such a table-given function is understood as the derivative of its interpolant. A method of numerical differentiation of table-given functions is developed, the essence of which is reduced to the product of the Taylor row vector of the n-th degree by the matrix of the k-th order of its differentiation (k £ n) and on the column vector of the coefficients of the corresponding interpolant. Some problem formulations of numerical differentiation of table-given functions using Taylor polynomials of degree n, corresponding solution algorithms, and specific implementation examples are provided. It has been established that to compute the k-th order derivative of a table-given function at a given value of the independent variable, the following steps need to be performed: based on the given table data, form a matrix equation, solve it to obtain the coefficients of the interpolant; substitute into the corresponding matrix expression the obtained interpolant coefficients and the independent variable value, and perform the matrix multiplication operations specified in the expression. The verification of the accuracy of the calculations using the appropriate central difference formulas was made. It was established that the calculated derivatives of the k-th order using the formulas of central finite differences practically coincide with the values ​​obtained using the Taylor polynomial interpolation of the n-th order, that is, the values ​​of the derivatives are calculated correctly.