, Volume 19, Issue 2, pp 145-163

Accelerated projection methods for computing pseudoinverse solutions of systems of linear equations

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Abstract

Iterative methods are developed for computing the Moore-Penrose pseudoinverse solution of a linear systemAx=b, whereA is anm ×n sparse matrix. The methods do not require the explicit formation ofA T A orAA T and therefore are advantageous to use when these matrices are much less sparse thanA itself. The methods are based on solving the two related systems (i)x=A T y,AA T y=b, and (ii)A T Ax=A T b. First it is shown how theSOR-andSSOR-methods for these two systems can be implemented efficiently. Further, the acceleration of theSSOR-method by Chebyshev semi-iteration and the conjugate gradient method is discussed. In particular it is shown that theSSOR-cg method for (i) and (ii) can be implemented in such a way that each step requires only two sweeps through successive rows and columns ofA respectively. In the general rank deficient and inconsistent case it is shown how the pseudoinverse solution can be computed by a two step procedure. Some possible applications are mentioned and numerical results are given for some problems from picture reconstruction.