Analyses of the Lanczos Algorithm and of the Approximation Problem in Richardson's Method
Grcar, Joseph Frank
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https://hdl.handle.net/2142/68189
Description
Title
Analyses of the Lanczos Algorithm and of the Approximation Problem in Richardson's Method
Author(s)
Grcar, Joseph Frank
Issue Date
1981
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
Two algorithms of use in sparse matrix computation are studied. The rounding errors of the computational Lanczos algorithm are examined in order to account for the differences between the ideal and the machine-operator quantities. The observed behavior of these errors is explained by means of a formal error analysis which relates the errors to properties of the matrix tridiagonalization problem solved by the algorithm. An investigation of the orthogonal polynomials associated with the algorithm partially explains the observed phenomena.
For Richardson's method of solving systems of linear equations, the associated uniform approximation problem is taken as a particular case of a more general problem. The solutions to the latter are characterized. A version of the algorithm of Remez is shown to solve these problems.
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