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Abstract:
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Mathematical modeling of many phenomena which occur in the
natural, technical sciences, economy requires signi cant knowledge of
the theory of numerical integration. In the situations where the integral
cannot be determined analytically, it is necessary to construct the for-
mula which approximates its value with acceptable error. Besides the
traditional formulae, the tendencies in the development of this area refer
to increment of algebraic degree of precision of the quadrature formula
and its error estimation.
The subject of this dissertation is Gaussian quadrature formula with
increased number of nodes which may have double multiplicity (sections
2 and 4) or arbitrary multiplicity (section 3). When we consider the
addition of one node, one of the integration interval ends, the
corresponding quadrature formula is called Gauss-Radau, and in the case
of adding two new nodes, the appropriate formula is of Gauss-Lobatto
type. Beside those formulae, in this dissertation we consider the
Kronrod's extension of Gauss-Tur an quadrature formula, which includes
the derivatives of the function at nodes. Here, the main aim is to deter-
mine efficient error bounds of these formulae with respect to the
different types of Chebyshev weight functions and Gori-Micchelli weight
function which, in one special case, represents the generalization of the
rst Chebyshev weight function.
Introductory theorems and well-known statements that are used in
the rest of the thesis are presented in the rst section. We described basic
properties of orthogonal polynomials participating in the construction
of Gaussian quadrature formula. Furthermore, the different types of
remainder terms of quadrature formula are de ned, whereby in the
following sections we analyze the integral representation of the remainder
term. The recent results relating to the increasing number of the nodes of
arbitrary multiplicity are exposed. The remaining three sections contain
the chapters with original results.
The second section is devoted to the estimation of the error of Gauss-
Lobatto quadrature formulae with a double nodes at the endpoints. Af-
ter detailed analysis of the kernel of the remainder term (see subsection
2.2.1), we proved the Theorems stating the behavior of the maximum
modulus of the kernel. We computed xed values mentioned in the The-
orems and established the real errors and the error bounds of quadrature
formula.
In third section we examine the Kronrod's extension of Gauss-
Tur an quadrature formulae with respect to the special kind of Gori-
Micchelli weight function. This extension, beside the nodes of Gauss-
Tur an quadrature formula, also contains the zeros of Chebyshev polyno-
mial of the second kind. In those new nodes, we consider just the values
of the function (not the values of its derivatives). Previously mentioned
Kronrod's extension is actually Chakalov-Popoviciu quadrature formu-
la, so, by applying the methods for those formulae, we computed the
coefficients and the error of Kronrod's formula.
Gautschi-Li hypotheses are proven in the fourth section. Those
hypotheses concern the Gauss-Radau quadrature formulae with respect
to the Chebyshev weight functions of the second and the third kind.
Subsection 4.4 consists of numerical examples and the tables with
efficient error bounds. |