The Method Of Padé Approximations And Their Applications In Integral Equations

In this paper,two methods of the function-valued Pade-type approximation are constructed——Function-Valued Pade-Type Approximant Via Formal Orthogonal Polynomials(FPTAVOP) and Function-Valued Partial Pade-Type Approximant(FPPTA),and at the same time they are also applied to estimate eigenvalues of Fredholm integral equation of the second kind and obtain the approximating solution of the second kind of Fredholm integral equations,and more,we use the methods of numerical algebra to compute the determinant formulas of the methods of Pade approximant and Pade-type approximant.In Chapter 2,the method of Function-Valued Pade-Type Approximant Via Formal Orthogonal Polynomialsis established.We first define a new linear functional different from previous ones,based on which we give the definition of Function- Valued Formal Orthogonal Polynomials,and then we deduce its recurrence relation.Based on all of our preparation work,and after our strict proof,we create the basic construction of FPTAVOP,i.e.,its denominator polynomial is just a Function-Valued Formal Orthogonal Polynomial,and its numerator polynomial can also be obtained easily.After a effectual algorithm being given, we use a typical numerical example to verify that until now the method of FPTAVOP is the best and the most effectual method to estimate and compute the eigenvalues and the solution of Fredholm integral equation of the second kind.In Chapter 3,the method of Function-Valued Partial Pade-Type Approximant is established. In practice,sometimes we only know partial eigenvalues of an integral equation, and how do we use these known eigenvalues to obtain the remainders? This just is our goal in this chapter.To achieve this objective,we first define the concept of Function-Valued Partial Padé-Type Approximant,and then investigate its algebraic properties,error formulas,and many kinds of existence situation of maximum and minimum of eigenvalues. Subsequently,by means of the definition and the error formulas we construct the determinant formulas of FPPTA,and several numerical examples verify its validity and practicability.In the end of this chapter,two convergence theorems,de Montessus theorem and the convergence theorem established on the error formula in terms of functional are discussed in detail.In Chapter 4,we strive to get help from the methods of numerical algebra to solve the computation problems occurring in Padéapproximation.First of all,we discover that the constructions of the determinant formulas of the methods of Padéapproximation (including function-valued Padé-type Approximation) are resemblant,and so do the determinant formulas of the methods of generalized inverse,(function-valued,vector-valued and matrix-valued) Padéapproximation.So we sum up the determinant formulas of the methods of Padéapproximation in one form,and do the same way with the determinant formulas of the methods of generalized inverse Padéapproximation,and then the methods of numerical algebra to compute these determinants are studied.At last,convergence theorem of some methods are investigated.