Thiele-Werner Type Continued Fraction Complex Vector Valued Rational Interpolation Problems And Applications

The vector valued rational interpolation(Denoted as GVRI), based on the Samel-son inverse of vectors, was at first brought forward by Wynn in 1963, which was developed in the modal analysis of vibrating structures by Graves-Morris. In late thirty years, many mathematicians have achieved many results in this field. In this article, we keep on doing this work and stress on the continued fraction vector valued rational interpolation in complex number field. The contents of the article are introduced as follows:In the first chapter, we outline the background and the main results obtained in this article.In the second chapter, we state some basic concepts and properties of GVRI, the main computation methods and results on GVRI and its serval applications.In the third chapter, we study the Thiele-type GVRI under complex variable condition, the main results obtained include the solution approach and computation formation. It should be pointed out that the complex variable GVRI may been achieved with Lagrange determinant method, but it is a new research topic on the continued fraction interpolating approach.In the fourth-fifth chapter, we study the GVRIs with Thiele-Werner continued fraction structure. The results achieved mainly include five aspects:(1) The first to obtain the explicit expression on the degrees of denominator and numerator of the Thiele-Werner type GVRIs, which is of the functional forms (See Theorem 4.3).(2) The first to obtain the types distribution on the GVRIs constructed by Thiele-Werner continued fraction (See Theorem 4.4).(3) The uniqueness characteristic of the Thiele-Werner type GVRIs is obtained (See Theorem 4.5).(4) The first to obtain vector valued osculatory rational interpolations with Thiele-Werner continued fraction structure (See Algorithm 5.2.1).(5) The algebraic properties of osculatory GVRIs are established (See Theorem 5.1 - Theorem 5.5).We put forward the Thiele-Werner type GVRIs and study its algebraic construction and characteristic properties roundly. Using the expression obtained on the degrees of denominator an numerator, we may construct various Thiele-Werner type GVRIs expediently. The previous study on Thiele type GVRI is only its special case(E.g.Corollary 4.3.4). Because the Thiele-Werner type continued fraction include various construction of GVRIs, we may solve GVRI with various forms and choose the GVIR with less rational process among them. At the same time, we further put forward and study the osculatory GVRIs with Thiele-Werner continued fraction structure, we also give its algebraic construction and characteristic properties. So we may unify GVRI in the Thiele-Werner continued fraction structure. The study on Thiele-Werner type GVRI enrich the content of GVRI greatly.In the sixth chapter, we study the recurrence relations on the Thiele and Thiele-werner continued fraction GVRI. The results achieved mainly include three aspects:(1) Obtain the stepwise rational interpolation sequence of complex variable Thiele-type GVRI by forward the recursive approach (See Theorem 6.2, Theorem 6.4, Theorem 6.5).(2) Establish the interrelation between the forward recurrence relation and backward recurrence relation of complex variable Thiele-type GVRI (See Theorem 6.3, Theorem 6.6).(3) Extend the recurrence relations of Thiele type GVRI to the case of complex variable Thiele-Werner type GVRI (See Theorem 6.7-Theorem 6.12).All the discussion of recurrence relations are established in the jth convergence of the continued fractions, which are different from the way of backward recurrence relation without inheritable attribute. The recurrence relations obtained in interpolating case can extend directly to the case of general vector continued fraction.