Osculatory Rational Interpolation On The Hypersphere

It is well-known that rational interpolation method is very important in thestudying of computational mathematics, and the theory of osculatory rationalinterpolation also has some crucial values in practical sense. The summaries ofthis thesis are the researches on the problem of constructing interpolation schemeon the hypersphere, which include traditional osculatory interpolants and the newinterpolation problem on the hypersphere. We base on these theories to constructosculatory rational interpolation scheme on the hypersphere.By the methods of continued fraction, we base on unvaried Thiele-typecontinued fraction and give a recursive algorithm which can judge theinterpolation problem whether have a result or not. Such method can solve manyosculatory problems in some fields, and it’s algorithm also give the procedure ofgetting coefficient i. The coefficient i decide the unique solution curve.In order to solve the interpolation problem, we have created the generalizedinverse vector-valued rational function, which meet those given points andderivative vectors on a unit hyper-sphereSd1. Baseing on those achievements,this paper is inspired by Thierry gensane[5]who constructed Thiele-typevector-valued rational interpolation on the hypersphere. According to the givenalgorithm, we can uniguely get a [2n,2n] type osculatory curve on thehypersphere.Numerical results are given to prove the effectiveness of the method.The numerical results show that when given the same parameters and points anddifferent tangent vectors, we can get different solution curves.