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Title: Preconditioning for Toeplitz-related systems
Author: Hon, Sean
ISNI:       0000 0004 7653 4447
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2018
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This thesis concerns preconditioning for Toeplitz-related systems. Specifically, we consider functions of Toeplitz matrices, i.e. h(Tn)xn=bn, where h(z) is an analytic function and Tnϵℂn×n is a Toeplitz matrix. We propose the absolute value circulant matrix |h(Cn)| as a preconditioner for h(Tn), where Cnϵℂn×n is the optimal circulant preconditioner, the superoptimal circulant preconditioner, or Strang's circulant preconditioner derived from Tn, and show that |h(Cn)|-1h(Tn) has clustered spectra that account for the effectiveness of such preconditioner. When h(Tn) is a real matrix, we can first premultiply it by the anti-identity matrix Ynϵℝn×n to obtain a (real) symmetric matrix Ynh(Tn) without normalizing the original matrix. To ensure |h(Cn)| is an effective preconditioner for Ynh(Tn), we show that |h(Cn)|-1Ynh(Tn) has clustered spectra around ±1. As YnhTn) is symmetric yet possibly indefinite, we can use the minimal residual method for the corresponding linear system with guaranteed convergence that depends only on its eigenvalues. We further show that the ideas of symmetrization and absolute value preconditioning for Toeplitz systems can be extended to the block Toeplitz matrix case. An application on time-stepping methods for evolutionary ordinary/partial differential equation problems is also discussed. Numerical results are given to demonstrate the effectiveness of our proposed preconditioners.
Supervisor: Wathen, Andrew Sponsor: Croucher Foundation
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available