NUMERICAL DIFFERENTIATION VIA THE INTERPOLATION METHOD OF HURWITZ-RADON MATRICES

Abstract

Mathematics and computer science need suitable method for numerical calculation of derivative. Classical methods, based on polynomial interpolation, have some negative features: they are useless to interpolate the function that fails to be differentiable at one point or differs from the shape of polynomial considerably, also the Runge’s phenomenon cannot be forgotten. To deal with numerical interpolation and differentiation dedicated methods should be constructed. One of them, called by author the method of Hurwitz-Radon Matrices (MHR), can be used in reconstruction and interpolation of curves in the plane. This novel method is based on a family of Hurwitz-Radon (HR) matrices. The matrices are skew-symmetric and possess columns composed of orthogonal vectors. The operator of Hurwitz-Radon (OHR), built from that matrices, is described. It is shown how to create the orthogonal and discrete OHR and how to use it in a process of function interpolation and numerical differentiation. Created from the family of N-1 HR matrices and completed with the identical matrix, system of matrices is orthogonal only for dimensions N = 2, 4 or 8. Orthogonality of columns and rows is very significant for stability and high precision of calculations. MHR method is interpolating the function point by point without using any formula of function. Main features of MHR method are: accuracy of curve reconstruction depending on number of nodes and method of choosing nodes, interpolation of L points of the curve is connected with the computational cost of rank O(L), MHR interpolation is not a linear interpolation.