ISSN:
1572-9125
Schlagwort(e):
Toeplitz matrix
;
circulant matrix
;
least squares
;
PCG method
Quelle:
Springer Online Journal Archives 1860-2000
Thema:
Mathematik
Notizen:
Abstract We study methods for solving the constrained and weighted least squares problem min x $$\min _x \tfrac{1}{2}\left( {b - Ax} \right)^T W\left( {b - Ax} \right)$$ by the preconditioned conjugate gradient (PCG) method. HereW = diag (ω1, ⋯, ω m ) with ω1 ≥ ⋯ ≥ ω m ≥ 0, andA T = [T 1 T , ⋯,T k T ] with Toeplitz blocksT l εR n × n ,l = 1, ⋯,k. It is well-known that this problem can be solved by solving anaugmented linear 2 × 2 block linear systemMλ +Ax =b, A T λ = 0, whereM =W −1. We will use the PCG method with circulant-like preconditioner for solving the system. We show that the spectrum of the preconditioned matrix is clustered around one. When the PCG method is applied to solve the system, we can expect a fast convergence rate.
Materialart:
Digitale Medien
URL:
http://dx.doi.org/10.1007/BF01740547
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