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\iteman{io-port 05988496}
\itemau{Cui, Xiaoke; Hayami, Ken; Yin, Jun-Feng}
\itemti{Greville's method for preconditioning least squares problems.}
\itemso{Adv. Comput. Math. 35, No. 2-4, 243-269 (2011).}
\itemab
A new method to precondition general least squares problems from the perspective of the approximate Moore-Penrose inverse is presented. Similar to the robust incomplete factorization (RIF) the proposed method also includes an $A$-orthogonalization process when the coefficient matrix has full column rank. When $A$ is rank deficient, the method tries to orthogonalize the linearly independent part in $A$. A theoretical analysis on the equivalence between the preconditioned problem and the original problem is given. Based on Greville's method, a global algorithm and a vector-wise algorithm for constructing the preconditioner which is an approximate generalized inverse of $A$ is proposed. It is shown that for a full column rank matrix $A$, the developed algorithm is similar to the RIF preconditioning algorithm and includes an $A$-orthogonalization process. It is proven that under a certain assumption, using the developed preconditioner, the preconditioned problem is equivalent to the original problem, and the generalized minimal residual method (GMRES) can determine a solution to the preconditioned problem before breakdown happens. Some details on the implementation of the developed algorithms are considered, and some numerical results are presented.
\itemrv{Tzvetan Semerdjiev (Sofia)}
\itemcc{}
\itemut{overdetermined systems; pseudoinverses; Moore-Penrose inverse; Greville algorithm; GMRES; least squares problems; robust incomplete factorization; orthogonalization; preconditioning; generalized minimal residual method; numerical results}
\itemli{doi:10.1007/s10444-011-9171-x}
\end