This paper is concerned with a new approach for preconditioning large sparse least squares problems. Based on the idea of the approximate inverse preconditioner, which was originally developed for square matrices, we construct a generalized approximate inverse (GAINV) M which approximately minimizes III -MAIIF or III -AMIIF• Then, we also discuss the theoretical issues such as the equivalence between the original least squares problem and the preconditioned problem. Finally, numerical experiments on problems from Matrix Market collection and random matrices show that although the preconditioning is expensive, it pays off in certain cases.

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Abstract. In this talk, we present a preconditioner for least squares problems min b − Ax 2 , where A can be matrices with any shape or rank. When A is rank deficient, our preconditioner will be rank deficient too. The preconditioner itself is a sparse approximation to the Moore-Penrose inverse of the coefficient matrix A. We will also discuss the similarity between this preconditioner and the Robust Incomplete Factorization preconditioner [1].