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Generalization of Strang's Preconditioner with Applications to Toeplitz Least Squares Problems
Author(s) -
Chan Raymond H.,
Ng Michael K.,
Plemmons Robert J.
Publication year - 1996
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/(sici)1099-1506(199601/02)3:1<45::aid-nla70>3.0.co;2-t
Subject(s) - preconditioner , toeplitz matrix , mathematics , generalization , least squares function approximation , levinson recursion , algebra over a field , algorithm , mathematical analysis , statistics , pure mathematics , iterative method , estimator
In this paper, we propose a method to generalize Strang's circulant preconditioner for arbitrary n ‐by‐ n matrices A n . Theth column of our circulant preconditioner S n is equal to theth column of the given matrix A n . Thus if A n is a square Toeplitz matrix, then S n is just the Strang circulant preconditioner. When S n is not Hermitian, our circulant preconditioner can be defined as. This construction is similar to the forward‐backward projection method used in constructing preconditioners for tomographic inversion problems in medical imaging. We show that if the matrix A n has decaying coefficients away from the main diagonal, thenis a good preconditioner for A n . Comparisons of our preconditioner with other circulant‐based preconditioners are carried out for some 1‐D Toeplitz least squares problems: min ∥ b ‐ Ax∥ 2 . Preliminary numerical results show that our preconditioner performs quite well, in comparison to other circulant preconditioners. Promising test results are also reported for a 2‐D deconvolution problem arising in ground‐based atmospheric imaging.