Subspace-by-subspace preconditioners for structured linear systems

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Subspace-by-subspace preconditioners for structured linear systems

We consider the iterative solution of symmetric positive-definite linear systems whose coefficient matrix may be expressed as the outer-product of low-rank terms. We derive suitable preconditioners for such systems, and demonstrate their effectiveness on a number of test examples. We also consider combining these methods with existing techniques to cope with the commonly-occuring case where the coefficient matrix is the linear sum of elements, some of which are of very low rank. 1 Introduction We consider the solution of n by n real linear systems of equations Ax = b; (1.1) where A is symmetric positive-definite and has the form A = e X i=1 A i A T i ; (1.2) and where A i is an n by n i real matrix. Systems of this form arise naturally in a number of ways. 1. Normal equations for least squares (see, for instance, Bjorck, 1996). 2. The Schur complement following partial elimination in augmented systems (see, for example, Duff, 1994). 3. Newton equations for partially separable op..