On The Quadratures Of Gauss-Type

This dissertation is mainly devoted to the Gauss-type quadrature rules-they are Gauss-Radau formulas, Gauss-Lobatto formulas, Gauss-Kronrod formulas and Gauss-Turan formulas with multiple nodes and so on. They are all characterized by approximation to an integral using a linear combination of the integrand values based partially or completely on the zeros of an orthogonal polynomial with respect to some measure, usually a weight function. Gauss-type quadrature rules possess the highest degree of precision compared to Newton-Cotes quadrature rules. This enables us to consider them as the first choice for use. But this advantage is sometimes cancelled by the roundoff error since the Gaussian nodes are generally irrational.However, this does not disable them to find their wide applications in many subjects, such as physics and engineering, etc. Recent applications include the summation of slowly convergent series [61] and moment-preserving spline approximation [60], a problem arising in physics. Thus, a study on Gauss-type quadrature rules is appropriate.In chapter 1, we start from a brief history of quadrature rules followed by the settings and some definition this topic relies on. Given an inner product space, we can then discuss the sequence of orthogonal polynomials relative to some m-distribution which are essential for the construction of Gauss-type quadrature rules. We collect in section 1.3 some basic properties of orthogonal polynomials such as the reality and simplicity of the zeros of them as well as interlacing property that they possess. In addition, if they are the so-called classical orthogonal polynomials(the Jacobi polynomials, the Laguerre polynomials and the Hermite polynomials) which are main interest to us, they further satisfy three-term recurrence. These properties play an important role in theory of Gauss-type quadrature rules briefly discussed in section 1.4. As for the computation of these rules, we mainly refer to the result proposed by Golub and Welsch [42] which says that this can done through the eigenvalues and eigenvectors of the Jacobi matrix derived from the three-term recurrence. Whereas in section 1.2, we survey some basic (Lagrange, Hermite and Newton) interpolation methods as well as some facts concerning the divided differences of a function along some nodes that will be needed in the sequel to derive explicit weights in a Gauss-type quadrature rules.In chapter 2, we are mainly concerned with Gauss-Radau formulas and Gauss-Lobatto formulas for the Jacobi weights and the one in the so-called Gori and Micchelli weight function classwhich is introduced by Gori and Micchelli [45] and consist of all nonnegative integrable functions w on [- 1,1] such thatwhere the prime on the summation indicates that the term corresponding to l = 0 is halved.We show that for any nonnegative measure d, supported on the interval [a,b], the Gauss-Radau formula and Gauss formula for the same weight are connected by the followingTheorem 0.0.1. Let d be any nonnegative measure supported on the interval [a, b] and i= 1,2, ???, n, be the zeros of the n-th degree orthogonal polynomial associated with the modified measure d = (x - a)d,. Suppose further that the measure d assumes the following formula of Gauss-Radau typeWe then have for the weightswhere i = 1,.... ,n, are the Christoffel numbers for the measure dSpecializing this theorem to the case when the weight is , the Jacobi weight, giveswhich are well-known [25]. We also derive the Christoffel numbers for the Laguerre weightwhich can be found in [17] or [25].Similarly, when we consider Gauss-Lobatto formulawe have for the weightsApplying these to the Jacob! weight givesCorollary 0.0.2. is the ith Gauss weight for the Jacobi measure the Lobatto nodes, then the Gauss-Lobatto weight for the same measure can be obtained viaTheorem 0.0.3. For any weight w Wn n N, the Gauss-Lobatto formula takes the following formwhich has algebraic degree of precision 2n- 1.These theorems have the adva...