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[PhD Thesis] A Numerical Investigation of the Rayleigh-Ritz Method for the Solution of Variational Problems
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Dr John Lloyd
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The results of a numerical investigation of the Rayleigh-Ritz method for the approximate solution of two-point boundary value problems in ordinary differential equations are presented. Theoretical results are developed which indicate that the observed behaviour is typical of the method in more general applications. In particular a number of choices of co-ordinate functions for certain second order equations are considered. A new algorithm for the efficient evaluation of an established sequence of .functions related to the Legendre polynomials is described, and the sequence is compared in use with a similar sequence related to the Chebyehev polynomials 0 Algebraic properties of the Rayleigh - Ritz equations for these and other co-ordinate systems are discussed. The Chebyshev system is shown to lead to equations with convenient computational and theoretical properties, and the latter are used to characterize the asymptotic convergence of the approximations for linear equations. These results are subsequently extended to a certain type of non-linear equation. An orthonormalization approach to the solution of the Rayleigh- Ritz equations which has been suggested in the literature is compared in practice with more usual methods and it is shown that the properties of the resulting approximations are not improved. Since it is known that the method requires more work than established ones it cannot be recommended. Quadrature approximations of elements of the Raleigh-Ritz matrices are investigated, and known results for a restricted class of quadrate 'finite element' and 'extended Kantorovich' methods are proposed. A summary of the conclusions discerned from the investigation is given.
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