Lazy Householder Decomposition of Sparse Matrices

This paper describes Householder reduction of a rectangular sparse matrix to small band upper triangular form Bk+1. Bk+1 is upper triangular with nonzero entries only on the diagonal and on the nearest k superdiagonals. The algorithm is similar to the Householder reduction used as part of the standard dense SVD computation. For the sparse “lazy” algorithm, matrix updates are deferred until a row or column block is eliminated. The original sparse matrix is accessed only for sparse matrix dense matrix (SMDM) multiplications and to extract row and column blocks. For a triangular bandwidth of k + 1, the SMDM operations are of the sparse matrix by dense matrices consisting of the k rows or columns of a block Householder transformation. Block Householder transformations are reliably orthogonal, computationally efficient, and have good potential for parallelization. Numeric results presented here indicate that using an initial random block Householder transformation allows computation of a collection of largest singular values. Some potential applications are in finding low rank matrix approximations and in solving least squares problems.