Further Study On Existence Of (Vector-Valued) Rational Interpolation

This thesis consists of four parts. In the first part, we review the evolution of the research on (vector-valued) rational interpolation, especially the study for existence of (vector-valued) rational interpolation.In the second part, forward and backward methods for testing the existence for a rational interpolation function are given. In case that the rational interpolation function has unattainable point(s), a kind of blending rational interpolation is constructed to change the unattainable point(s) into the attainable one(s). Compared with the previous ones, the degrees of the numerator and denominator of the blending rational interpolation introduced in this part are lower. Consequently, the computation of the method is much less. In addition, the algorithms for the above methods are simple and easy to program. Some numerical examples illustrate these methods.In the third part, a method of constructing the vector-valued rational interpolation is presented by using Samelson inverse and the theory of Thiele-Werner type rational interpolation. The method can also test the degeneration of the constructing vector-valued rational interpolation and unattainable points. If the constructed interpolation has unattainable points, then a modified interpolation, a kind of piecewise vector-valued rational interpolation, is given to fit all the interpolation conditions. The methods in this part are flexible and easy to program. Again some examples on these methods are offered.The forth part briefly introduces the existing main results of the bivariate vector-valued rational interpolation and the study of existence for the bivariate rational interpolation.