Study On Multivariate Function-Valued Padé Approximation

Function-valued Pade approximation is an important part of rational approximation, and has a wide range of applications in many fields. This thesis mainly studies the determinantal representation of one and two function-valued Pade approximation and their applications.This thesis consists of four parts. In the first part we review the background and evolution of function-valued Pade approximation and the Fredholm integral equation of the second kind.The second part introduces basic concepts and preliminaries used in the next two parts.In the third part, a new determinantal representation for function-valued Pade approximation in one variable is constructed by converting the problem of function-valued Pade approximation into the problem of vector-valued Pade approximation with scalar components. Each element of the determinant are obtained by the operation of dot product of two vectors rather than by evaluating a definite integral of a function done by the traditional manner. So the calculation of the determinant introduced in this paper is much less than the traditional one. In addition, this method is also used to find the numerical solution of the Fredholm integral equation of the second kind. Some numerical examples are illustrated the method and show good approximate accuracy.In the last part, we generalize the study to the bivariate case. Firstly, a determinantal representation for bivariate function-valued Pade approximation is given by use of double integrals and some recursive algorithms are deduced. Secondly, another determinantal representation for bivariate function-valued Pade approximation is constructed by converting the problem of bivariate function-valued Pade approximation into the problem of matrix-valued Pade approximation. Again some recursive algorithms are presented. These methods significantly reduce the calculation. Finally, the results of some numerical examples show excellent approximations of these methods.