On Univariate And Multivariate Padé Approximation

This thesis has done some research on univariate, multivariate Pade approximations and multivariate matrix Pade approximations.Chapter 1 emphasizes the importance and necessity of Pade approximations, summarizes the present situation of the research on some fields of univariate, multivariate Pade approximations and matrix Pade approximations.Chapter 2 considers the cubic, general cubic and quartic Hermite-Pade approx-imatants to e-x on the bases of the work of P.B. Borwein, K. Drive and N.M. Temme on the quadratic Hermite-Pade approximatant to e-x, gets the explicit formulas and differential equations for the coefficient polynomials respectively, obtains the exact asymptotic expressions for the error functions, and shows that these generalized Pade-type approximations can be used to asymptotically minimize the expressions with the same forms on the unit disk.Furthermore, this chapter gives a connection between certain hypergeometric functions and the coefficients of each of the polynomials related with the cubic Hermite-Pade approximatant to e-x, obtains simple expressions for the coefficients, gets the contour integral representations of the polynomials, and by using of the saddle point method, derives the exact asymptotics of the polynomials as the degrees of the polynomials tend to infinity through certain ray sequence.Chapter 3 follows the way of A.W. Mclnnes and R.G. Brooks investigating the quadratic function approximation, considers two kinds of algebraic function approximations, analyses the existence and local behavior of the cubic and general cubic function approximations to a function which has a given power series expansion about the origin, shows that the cubic or general cubic Hermite-Pade form always defines a function that is analytic in a neighbourhood of the origin.Consulting the research on the quadratic function approximation given by A.W. Mclnnes, R.G. Brooks and the definition of Canterbury multivariate algebraic function approximation given by J.S.R. Chisholm, chapter 4 considers three kinds of algebraic function approximations, introduces two kinds of approximations: Karlsson-Wallin type and Lutterodt-type multivariate algebraic function approximations to a real-valued locally analytic multivariate function, analyses the existence and local behavior of the simple off-diagonal Canterbury, diagonal Karlsson-Wallin type and diagonal Lutterodt-type bivariate quadratic algebraic function approximations,and shows that any one of these three kinds of multivariate quadratic Hermite-Pade forms always defines a bivariate quadratic function that is analytic in a neighbourhood of the origin. Numerical examples compare the results obtained by the simple off-diagonal Canterbury approximation with those by the diagonal Chisholm ap-proximant and Taylor polynomial approximant. Finally, this chapter establishes the generation formulae of coefficient polynomials of successive diagonal Canterbury multivariate quadratic approximants and obtains a necessary and sufficient condition for the non-degeneracy ,of Canterbury approximant.Making referrence to the definition of least-squares orthogonal polynomials given by C. Brezinski, chapter 5 introduces two kinds of polynomials: multivariate least-squares orthogonal polynomials from the rectangular and triangular forms respectively, discusses their existence and uniqueness, gives some methods for their recursive computation and constructs two new families of multivariate Pade-type approximants. Numerical examples compare the results obtained by the above multivariate Pade-type approximation from the rectangular form with those of diagonal Chisholm approximant and Taylor polynomial approximant.Following the way of the research on matrix Pade-type approximation given by A. Draux and B. Moalla, chapter 6 introduces a kind of multivariate rectangular matrix Pade-type approximations, discusses their errors and some fundamental properties such as unicity, reciprocal covariance, homographic covariance, obtains the highest order approximations, i.e. matrix Pad...