University of Illinois Urbana-Champaign

Optimum Semi-Iterative Methods for the Solution of Any Linear Algebraic System With a Square Matrix

Smolarski, Dennis Chester

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Description

Title
Optimum Semi-Iterative Methods for the Solution of Any Linear Algebraic System With a Square Matrix
Author(s)
Smolarski, Dennis Chester
Issue Date
1982
Department of Study
Computer Science
Discipline
Computer Science
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Computer Science
Abstract
  • In 1975, T. A. Manteuffel developed a method for the iterative solution of a non-symmetric linear system, Ax = b, when the eigenvalues have positive real parts. The iterative parameters are reciprocals of the roots of a scaled and translated Chebyshev polynomial and depend upon an ellipse enclosing the spectrum of the system matrix.
  • In some applications, a matrix will occur whose spectrum is not well-approximated by an ellipse. In this thesis, a method is developed to determine optimal iteration parameters for use in a complex version of Richardson's iteration for a spectrum contained in any simply-connected bounded open set (in practice, a polygon symmetric with respect to the real axis).
  • The proposed method is a generalization of algorithms found in a 1958 paper of E. L. Stiefel in which real orthogonal polynomials were used. In this thesis, Stiefel's work is extended to complex orthogonal and bi-orthogonal polynomials. In addition, numerical methods are developed to obtain the desired iteration parameters by means of least squares theory.
  • Results are presented which show that the methods of this thesis compare favorably to Manteuffel's method and the Lanczos algorithm for test matrices whose spectra had pre-determined shapes.
Graduation Semester
1982
Type of Resource
text
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http://hdl.handle.net/2142/69508

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