The Research On Rational Spline Interpolation

There are two sections in this paper.In section one,a kind of rational quartic interpolation function with linear denominator is derived on the base of analysis about the question of rational spline .In fact,this rational quartic spline is the successful generalization of quartic polynomial spline.If we decrease the spline's continuity to C2,it can provide additional freedom degrees,and this is very useful for shape constraint in curve design. In addition, the continuity equations for this C2 rational quartic spline and the method for determing the additional freedom degree are given. And the characteristic of this method is that it can make this rational quartic spline has interpolant precision of cubic polynomial. Meanwhile the continuity equations for this C2 rational quartic spline become tri-diagonal system of equations. At last ,some examples and figures are given.In section two, the algorithms of vector-valued rational interpolants are stated generally. Then a new algorithm of brivate vector -valued rational interpolants by means of complexification of the knots and backward three-term recurrence relations is given. This algorithm avoid using branched continued fractions. Finally, its validity and flexibility are demonstrated by some examples.