Method Of Constructing Bivariate Matrix-valued Rational Interpolation Functions

Rational approximation, one type of nonlinear approximation, is drawing more and more attentions recently. Compared to polynomial interpolations, it's more flexible and can describe physical character of functions more accurately although it is complex. In the past few years, the development of science and technology and the prevalence of computer become the powerful tools of the research of rational interpolation. The research of rational interpolation is going further and it shows some special advantages in applications.In first chapter, we introduce the background of rational interpolation and the main content of this paper.The second chapter introduces continued fraction's basic theories and properties, because continued fraction is very important in constructing matrix-valued rational interpolation, and then introduces the definition and properties of univariate matrix-valued rational interpolation functions.The third chapter introduces the definition and properties of bivariate matrix-valued rational interpolation, also introduces the common method of constructing bivariate matrix-valued rational interpolation functions.In last chapter, we apply polynomial method in constructing bivariate matrix-valued rational interpolation functions. For the calculation of bivariate matrix-valued rational interpolations, multi-parameters are introduced and a group of polynomials with two elements, that is an algebraic polynomial and matrix-valued polynomials, are defined. By using the necessary and sufficient conditions for polynomials identity, linear equations are solved to determine the parameters and the formula of the matrix-valued rational interpolation is given.