54 research outputs found
Quadrature, Interpolation and Observability
Methods of interpolation and quadrature have been used for over 300 years. Improvements in the techniques have been made by many, most notably by Gauss, whose technique applied to polynomials is referred to as Gaussian Quadrature. Stieltjes extended Gauss's method to certain non-polynomial functions as early as 1884. Conditions that guarantee the existence of quadrature formulas for certain collections of functions were studied by Tchebycheff, and his work was extended by others. Today, a class of functions which satisfies these conditions is called a Tchebycheff System. This thesis contains the definition of a Tchebycheff System, along with the theorems, proofs, and definitions necessary to guarantee the existence of quadrature formulas for such systems. Solutions of discretely observable linear control systems are of particular interest, and observability with respect to a given output function is defined. The output function is written as a linear combination of a collection of orthonormal functions. Orthonormal functions are defined, and their properties are discussed. The technique for evaluating the coefficients in the output function involves evaluating the definite integral of functions which can be shown to form a Tchebycheff system. Therefore, quadrature formulas for these integrals exist, and in many cases are known. The technique given is useful in cases where the method of direct calculation is unstable. The condition number of a matrix is defined and shown to be an indication of the the degree to which perturbations in data affect the accuracy of the solution. In special cases, the number of data points required for direct calculation is the same as the number required by the method presented in this thesis. But the method is shown to require more data points in other cases. A lower bound for the number of data points required is given
Quadrature formulas based on rational interpolation
We consider quadrature formulas based on interpolation using the basis
functions on , where are
parameters on the interval . We investigate two types of quadratures:
quadrature formulas of maximum accuracy which correctly integrate as many basis
functions as possible (Gaussian quadrature), and quadrature formulas whose
nodes are the zeros of the orthogonal functions obtained by orthogonalizing the
system of basis functions (orthogonal quadrature). We show that both approaches
involve orthogonal polynomials with modified (or varying) weights which depend
on the number of quadrature nodes. The asymptotic distribution of the nodes is
obtained as well as various interlacing properties and monotonicity results for
the nodes
Collision and re-entry analysis under aleatory and epistemic uncertainty
This paper presents an approach to the design of optimal collision avoidance and re-entry maneuvers considering different types of uncertainty in initial conditions and model parameters. The uncertainty is propagated through the dynamics, with a non-intrusive approach, based on multivariate Tchebycheff series, to form a polynomial representation of the final states. The collision probability, in the cases of precise and imprecise probability measures, is computed considering the intersection between the uncertainty region of the end states of the spacecraft and a reference sphere. The re-entry probability, instead, is computed considering the intersection between the uncertainty region of the end states of the spacecraft and the atmosphere
On some extremalities in the approximate integration
Some extremalities for quadrature operators are proved for convex functions
of higher order. Such results are known in the numerical analysis, however they
are often proved under suitable differentiability assumptions. In our
considerations we do not use any other assumptions apart from higher order
convexity itself. The obtained inequalities refine the inequalities of Hadamard
type. They are applied to give error bounds of quadrature operators under the
assumptions weaker from the commonly used
The impact of Stieltjes' work on continued fractions and orthogonal polynomials
Stieltjes' work on continued fractions and the orthogonal polynomials related
to continued fraction expansions is summarized and an attempt is made to
describe the influence of Stieltjes' ideas and work in research done after his
death, with an emphasis on the theory of orthogonal polynomials
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