The Fourier analysis technique and epsilon-pseudo-eigenvalues

Technical Report · Thu Jul 01 04:00:00 EDT 1993

DOI:https://doi.org/10.2172/10177869· OSTI ID:10177869

Donato, J M

The spectral radii of iteration matrices and the spectra and condition numbers of preconditioned systems are important in forecasting the convergence rates for iterative methods. Unfortunately, the spectra of iteration matrices or preconditioned systems is rarely easily available. The Fourier analysis technique has been shown to be a useful tool in studying the effectiveness of iterative methods by determining approximate expressions for the eigenvalues or condition numbers of matrix systems. For non-symmetric matrices the eigenvalues may be highly sensitive to perturbations. The spectral radii of nonsymmetric iteration matrices may not give a numerically realistic indication of the convergence of the iterative method. Trefethen and others have presented a theory on the use of {epsilon}-pseudo-eigenvalues in the study of matrix equations. For Toeplitz matrices, we show that the theory of c-pseudo-eigenvalues includes the Fourier analysis technique as a limiting case. For non-Toeplitz matrices, the relationship is not clear. We shall examine this relationship for non-Toeplitz matrices that arise when studying preconditioned systems for methods applied to a two-dimensional discretized elliptic differential equation.

• Research Organization: Oak Ridge National Lab., TN (United States)
• Sponsoring Organization: USDOE, Washington, DC (United States)
• DOE Contract Number: AC05-84OR21400
• OSTI ID: 10177869
• Report Number(s): ORNL/TM--12347; ON: DE93018805
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990200
CONVERGENCE
DIFFERENTIAL EQUATIONS
EIGENVALUES
FOURIER ANALYSIS
ITERATIVE METHODS
MATHEMATICS AND COMPUTERS
MATRICES