On some numerical and algebraic computations with matrices and polynomials.
Item
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Title
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On some numerical and algebraic computations with matrices and polynomials.
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Identifier
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AAI9959171
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identifier
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9959171
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Creator
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Chen, Zhao Qin.
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Contributor
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Adviser: Victor Y. Pan
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Date
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2000
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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Polynomial and matrix computations constitute the bulk of the area of modern computations for sciences, engineering and communication. From mathematical point of view, these are usually algebraic problems but the effectiveness of practical algorithms for their solution can frequently be increased by involving approximation techniques and other methods of numerical computation. We demonstrate this by the examples of some important computations from three areas.;In the first chapter of the dissertation, we follow [PACLS98] and [PACPS98]. We consider Trummer's classical computational problem, having several important applications to celestial mechanics (n-body problem), fluid mechanics, computation of Riemann zeta function, conformal maps, solution of integral equations, polynomial and rational interpolation and evaluation on a fixed node set.;In the second chapter of the dissertation, we follow [PC99]. We consider most effective algebraic and numerical algorithms for polynomial division, which we relate to the computations with structured matrices and polynomials.;In the third chapter of the dissertation, we follow [PC99a]. We consider the matrix eigenproblem, which is a central problem of applied linear algebra. In the unsymmetric case, the known customary algorithms have substantial deficiency, that is, they are lack of guaranteed fast convergence and, indeed, diverge or stumble in practice in the important cases of multiple and clustered eigenvalues. (The latter cases routinely arise when the input matrix with multiple eigenvalues is perturbed by small errors.) To yield solution algorithms that remain fast even on the worst case input, we reduce the computation of the eigenvalues of a matrix to the approximation of the roots (zeros) of its minimum or characteristic polynomials, for which one may apply the recent highly effective polynomial rootfinders (e.g. ones that yield nearly linear solution cost bound [P95]).;Summarizing, in all three parts we demonstrated that combined applications of numerical and algebraic techniques is an effective tool for reducing computational cost for the solution of various algebraic tasks. (Abstract shortened by UMI.).
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
