Generalized Inverse Multivariate Newton-Thiele Matrix Pade Approximation And Its Application In Control Theory

In the actual calculation of the natural sciences and engineering, the basic method to ap-proximate a function is using the part sum of Taylor series. As a special rational approximation, Pade approximation is an extension of Taylor polynomial approximation. The research and de-velopment of Pade approximation are closely linked with analytic functions, approximation theory, the theory of moments, continued fractions and differential equations, etc., and have a wide range of application in other fields like numerical analysis, quantum field theory, control theory of critical phenomena. This thesis is concerned with the generalized inverse multivariate Newton-Thiele type matrix rational interpolation and matrix Pade approximation. The pro-posed algorithms are investigated by comparing with some existing algorithms and by applying the bivariate Newton-Thiele matrix Pade approximation to control theory.This article is divided into five chapters:The first chapter introduces the research background and development, some prior knowl-edge and the main work of this thesis.The second chapter summarizes some existing achievements with additional emphasis on the Newton interpolation polynomial, Thiele-type continued fraction interpolation and one variable matrix rational interpolation.Chapter3consists of three sections. In the first section, we give the generalized inverse Newton-Thiele matrix interpolation formula and its antithetical form. Then we introduce the iterative algorithm with a backward three-term scheme. After that, we obtain the approxima-tion properties including divisibility, characteristic and error of the Newton-Thiele type matrix interpolation. Finally, we conclude experimentally that the Newton-Thiele type interpolation formula is superior to Thiele-type formula. In the second section, we describe the triple Newton-Thiele type rational formula, and verify its effectiveness by a numerical example. In the third section, we obtain the triple Newton-Thiele matrix rational interpolation through extending its scalar prototype introduce in the second section. It is derived by first giving the formula, then introducing its approximation properties, algorithms and error estimate, and finally comparing with other methods for problems with different interpolation nodes.The fourth chapter is divided into three parts. The first section devotes to the definition of the classical matrix Pade approximation, uniqueness and algebraic properties. In the second section, we introduce the definition of generalized inverse Thiele-type matrix Pade approxi-mation, algorithm and approximation properties. In the third section, we first introduce the matrix-valued generalized functionals, and then examine the definition, properties, computa- tional schemes and error analysis of the matrix Pade-type approximation.In the first section of Chapter five, we first give the definitions of blending difference and compounded differences and the relationships. Then we define the (k,l)th compounded deriva-tives of f(x,y) at (x*,y*) and derive its relation with the compounded difference. Based on this, we present Algorithm5.2.4and the matrix expansion of bivariate Newton-Thiele method which further yields the BGMPANt through a (m,n)-th truncation. We also consider the char-acteristic and error estimation, and validate the effectiveness of the formula by some numerical examples. In the last section we show one application of the Newton-Thiele type matrix Pade approximation to control theory.