Interpolation and Differentiation of Tabular Functions

The monograph outlines the methodology for interpolating tabular functions of one, two, and many independent variables using the nth degree Taylor polynomial. A method for numerical differentiation of tabulated functions of one, two, and many independent variables at arbitrarily located and equidistant interpolation nodes using the nth degree Taylor polynomial, and also a method for solving boundary (edge) problems for one- and two-dimensional regions using difference analogues of differential operators are presented as well. A method for calculating the kth order derivatives of a tabulated function with arbitrarily located interpolation nodes has been formulated. The method uses the product of the nth degree Taylor vector-row for one, two or many independent variables with the kth order matrix of its differentiation and with the vector-column of the values of the nth degree interpolant coefficients. Furthermore, a method for determining the difference analogues of the kth order differential operators for various arrangements of equidistant nodal points of a tabulated function of one and two independent variables has been developed. The method uses the product of the kth order Taylor matrix for a given set of nodal points with the kth order matrix of its differentiation and with the inverse of the kth order Taylor matrix. In addition, a method for calculating the kth order derivatives at equidistant interpolation nodes for one and two independent variables has been devised. The method uses the product of the matrix of difference analogues of the kth order differential operators and the vector-column of interpolation node values with the nth degree Taylor polynomial. Moreover, differential analogues of the kth order differential operators were calculated using the matrix method, based on the nth degree Taylor polynomials for one and two independent variables, as well as various schemes for the arrangement of equidistant nodal points, which makes it possible to calculate derivatives of the kth order for a given set of nodal points. Keywords: boundary (edge) problems in software engineering; the nth degree Taylor polynomial; nodal points of a tabular function; arbitrarily located and equidistant interpolation nodes; kth order derivative of the nth degree Taylor polynomial; difference analogue of kth order differential operator; diagram location of equidistant nodal points; vector-row of values of elements of the nth degree Taylor polynomial; vector-column of values of interpolation nodes of Taylor polynomial of nth degree; Taylor matrix formed by the nth degree polynomial; inverse Taylor matrix.