A preconditioner for constrained and weighted least squares problems with Toeplitz structure |
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Authors: | Xiao-Qing Jin |
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Institution: | (1) Faculty of Science and Technology, University of Macau, 3001 Caixa Postal, Macau |
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Abstract: | We study methods for solving the constrained and weighted least squares problem min
x
by the preconditioned conjugate gradient (PCG) method. HereW = diag ( 1, ,
m
) with 1
m
0, andA
T
= T
1
T
, ,T
k
T
] with Toeplitz blocksT
l
R
n × n
,l = 1, ,k. It is well-known that this problem can be solved by solving anaugmented linear 2 × 2 block linear systemM +Ax =b, A
T
= 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.Research supported by HKRGC grants no. CUHK 178/93E and CUHK 316/94E. |
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Keywords: | Toeplitz matrix circulant matrix least squares PCG method |
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