ANALYSIS OF ERROR IN INTERPOLATION OF CONTINUOUS FUNCTIONS WITH UNKNOWN CONTINUOUS DERIVATIVES USING CUBIC SPLINES

Abstract

Methodology of determination of error of s_k^(d) interpolation of function is worked out by a cube spline in case of unknown fourth derivative. It is shown that value of s_k^(d) on an interval under a number k equal h|d_k|/6, where h is a size of interval, d_k are coefficients of polynomials of spline at a variable in the third degree. The algorithm of calculation of coefficients of spline supposes implementation of next terms: a function is continuous and possesses continuous the first and second derivatives, on the borders of function the first derivatives are set, second derivative on a right border is equal to the zero. In case of the unknown first derivatives on borders possibility of their calculation is envisaged with the use of values of function in key points. The proposed method was tested on monotone functions sin(πх/2), (1-exp(-x))/(1-exp(-1)), log(1+х)/log(2), (2x+x2+x3+x4)/5, which do not have inflections and extreme points on the segment [0,1]. The standard deviation of the values of the reconstructed function sin(πх/2) from the values of a spline of three polynomials, equal to s=0.92·10^-2, is in good agreement with the error calculated with the proposed method and equal to s_k^(d)=1.3·10^-2. Increase of amount of knots to 10³ and more provides exactness of interpolation to 10^-7 and higher. In all cases the size of s_k^(d) well comports with the exact meaning of s. An analogical situation is observed for many droningly decreasing or increasing functions, not having bends and extreme points. At renewal of any functional dependences it is recommended to break up the investigated segment on such intervals, wherever functions have bends and extreme points.

Keywords: renewal of function, interpolation polynomial, interpolation, cube spline, error of interpolation, function.