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Preconditioners for least squares problems by LU factorization. (English) Zbl 0924.65034
The authors consider the problem of finding suitable preconditioners in the iterative solution of least squares problems \(\min\| Ax-b\|_2\) where the rectangular matrix \(A\) is large and sparse. Two basic conjugate gradient methods using an appropriate submatrix \(A_1\) as a preconditioner are proposed and bounds for the rate of convergence are derived. It is shown how one of these methods can be adapted to solve a generalized least squares problem. An algorithm for selecting \(n\) linearly independent rows from \(A\) to form \(A_1\) is outlined. The methods are tested on some sparse rectangular matrices from the Harwell-Boeing sparse matrix collection. It follows from the comparison results that the preconditioned conjugate method is much better than the conjugate gradient method.

65F20 Numerical solutions to overdetermined systems, pseudoinverses
65F35 Numerical computation of matrix norms, conditioning, scaling
65F50 Computational methods for sparse matrices
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