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This paper deals with adaptively preconditioned iterative methods for solving large and sparse systems of linear equations. In particular, the paper discusses preconditioning where adaptive dropping reflects the quality of preserving the relation U Z = I between the direct factor U and the inverse factor Z that satisfy A = U T U and A − 1 = Z Z T . The proposed strategy significantly extends and refines the approach from [1], see also [2], by using a specific multilevel framework. Numerical experiments with two levels demonstrate that the new preconditioning strategy is very promising. Namely, we show a surprising fact that in our approach the Schur complement is better to form in a more sophisticated way than by a standard sparse matrix-matrix multiplication. [ABSTRACT FROM AUTHOR]

Abstract: We present a roundoff error analysis of the method for solving the linear least squares problem with full column rank matrix A, using only factors Σ and V from the SVD decomposition of . This method (called here) is an analogue of the method of seminormal equations ( ), where the solution is computed from using only the factor R from the QR factorization of A. Such methods have practical applications when A is large and sparse and if one needs to solve least squares problems with the same matrix A and multiple right-hand sides. However, in general both and are not forward stable. We analyze one step of fixed precision iterative refinement to improve the accuracy of the method. We show that, under the condition , this method (called ) produces a forward stable solution, where denotes the condition number of the matrix A and u is the unit roundoff. However, for problems with only it is generally not forward stable, and has similar numerical properties to the corresponding method. Our forward error bounds for the are slightly better than for the since the terms are not present. We illustrate our analysis by numerical experiments. [Copyright &y& Elsevier]