Algorithm 844

Source abstract

In many applications---latent semantic indexing, for example---it is required to obtain a reduced rank approximation to a sparse matrix A . Unfortunately, the approximations based on traditional decompositions, like the singular value and QR decompositions, are not in general sparse. Stewart [(1999), 313--323] has shown how to use a variant of the classical Gram--Schmidt algorithm, called the quasi--Gram-Schmidt--algorithm, to obtain two kinds of low-rank approximations. The first, the SPQR, approximation, is a pivoted, Q-less QR approximation of the form ( XR 11 −1 )( R 11 R 12 ), where X consists of columns of A. The second, the SCR approximation, is of the form the form A ≅ XTY T , where X and Y consist of columns and rows A and T , is small. In this article we treat the computational details of these algorithms and describe a MATLAB implementation.