Multivariate Matrix Pad′e Approximation With Scalar Denominator Polynomial And Its Application In Control Theory

In this thesis, the theory and method of bivariate matrix Pade approximation are im-proved and generalized. A bivariate matrix Pade-type approximation is constructed in the inner product space. Recursive algorithms of its determinantal expression presented is devel-oped to avoid the computation for determinants. A bivariate matrix tensor product Pade-type approximation via formal orthogonal polynomial is proposed and its recurrence formula is presented. A general bivariate matrix Pade approximation is established and several choices of its sets of the indices are given. A bivariate Newton-type matrix rational interpolation is developed. In addition, the function-valued Pade-type approximation for solving Fred-holm integral equations of the second kind is improved and its corresponding algorithms are presented to obtain better numerical accuracy.This thesis consists of seven chapters.In the first chapter, the research background, the existing results and the author's work are introduced.In the second chapter, univariate matrix Pade approximation and its qd-algorithm are introduced. By means of modifying some procedures of the algorithm, modified qd-algorithm(three-term relationship) is given.In the third chapter, by introducing a bivariate matrix linear functional on the scalar polynomial space, a bivariate matrix Pade-type approximation (BMPTA) in the inner prod-uct space is defined. The coefficients of the denominator polynomials of BMPTA are of scalar and can be obtained by making inner product of matrix. An algorithm for the nu-merator polynomials of BMPTA when the denominator of BMPTA is given in advance. A determinantal expression of BMPTA is presented by means of Hankel-like coefficient matrix. Moreover, to avoid the computation of the determinants, two efficient recursive algorithms are proposed. The choice of the sets of indices is discussed. The method of BMPTA is applied to partial realization problems of two dimension linear systems.In the forth chapter, a bivariate tensor product formal orthogonal polynomial(BTPFOP) is defined and its nine-term relationship is proposed. A bivariate matrix tensor product for-mal orthogonal polynomial(BMTPFOP) is accordingly defined and its recurrence formula based on matrix direct inner product is presented. A bivariate matrix Pade-type approxima-tion via formal orthogonal(BMPTAVOP) is consequently defined, its denominator polynomi-als can be computed by making use of nine-term relationship and three-term relationship of univariate matrix Pade-type approximation.In the fifth chapter, a general bivariate matrix Pade approximation(BMPA) in the direct inner product space is defined. Its denominator polynomials are also of scalar. Compared with the rectangular sets of indices of BMPTA, there are more choices of the sets of in-dices of BMPA. The application of range of the method of BMPA is consequently expanded. Analogously, the application are developed.In the sixth chapter, a bivariate Newton-type matrix rational interpolation(BNMRI) is constructed, its existence and recursive algorithm of determinantal expression presented are proposed. In addition, the choice of the index set is discussed to guarantee the duality with respect to two unknowns and the inheritance of the algorithm.In the last chapter, function-valued Pade-type approximation(FPTA) for solving Fred-holm integral equations of the second kind is discussed. By choosing the coefficient in Neu-mann power series of an integral to make the inner product with both sides of a function-valued system of equations on L2 space, a scalar system is yielded. A determinantal solution of FPTA is given. Two algorithms are presented to avoid the computation for determi-nants. The algorithms not only involve most of existing methods, but also overcome their restrictions. Numerical experimentation for a typical integral equation illustrates that the algorithms are simpler and more effective for obtaining the characteristic values and the char-acteristic function than all previous methods. In addition, the algorithms are also applicable to other Fredholm integral equations of the second kind without explicit characteristic values and characteristic functions, which is specified by a corresponding example.