Generalization of Strang's Preconditioner with Applications to Toeplitz Least Squares Problems

In this paper, we propose a method to generalize Strang's circulant preconditioner for arbitrary n-by-n matrices An. The n 2 th column of our circulant preconditioner Sn is equal to the n 2 th column of the given matrix An. Thus if An is a square Toeplitz matrix, then Sn is just the Strang circulant preconditioner. When Sn is not Hermitian, our circulant preconditioner can be deened as (S n Sn) 1=2. 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 An has decaying coeecients away from the main diagonal, then (S n Sn) 1=2 is a good preconditioner for An. Comparisons of our preconditioner with other circulant-based preconditioners are carried out for some 1{D Toeplitz least squares problems: min kb ? Axk 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.