Greville's method for preconditioning least squares problems (Q652570)

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.