The Approximation Properties Of Two Kinds Of Rational Cubic Spline Interpolation

In this paper, the author first introduces the origin and the development of Computational Geometry in engineering application. The author also introduces the current situation of the rational cubic spline interpolation, which has important academic meaning and practice value in Computer Aided Geometry Design (CAGD). Secondly, the author introduces the meaning and the importance of the research in the approximation properties of the rational cubic spline interpolation. The author proposes a method which using the Peano-kernel Theory to study the approximation properties of two kinds of rational cubic splines interpolation with linear denominator. One bases on function value and another bases on arithmetic average difference quotient. In this paper, the author mainly studies the approximation properties of both the function value and the derivative of two kinds of splines interpolation mentioned before, and the optimal error constants are obtained finally. At the same time, the author studies the jump of the second derivative at the knots.The paper is arranged as follows.In Chapter 1, the author makes a brief introduction of the development of Computational Geometry. Then the development and some research in approximation properties of spline interpolation, especially the rational cubic spline interpolation is introduced.In Chapter 2, one kind of rational cubic spline interpolation with linear denominator has been proposed, whose derivatives are replaced with accurate derivative values, difference quotient and arithmetic average difference quotientrespectively. The degree of smoothness of these splines interpolation attained is C~1. The latter two kinds of splines are going to be studied in this paper.In Chapter 3 and Chapter 4, the author analyzes the approximation properties ofthe rational cubic spline interpolation based on function value under two conditions where the function interpolated isC1 or C2.In Chapter 5 and Chapter 6, the author analyzes the approximation properties of the rational cubic spline interpolation based on arithmetic average difference quotientunder two conditions where the function interpolated is C' or C2.In Chapter 7, the author summarizes the whole paper and gathers up the main points and results.With the analysis of the approximation properties, the author proposes the date analysis results. Chapter 3, Chapter 4, Chapter 5 and Chapter 6 are the hard cores of this paper.