Structure-Preserving and Rank-Revealing QR-Factorizations
Christian H Bischof, Per Christian Hansen · SIAM Journal on Scientific and Statistical Computing · 1991
The rank-revealing QR-factorization (RRQR factorization) is a special QR-factorization that is guaranteed to reveal the numerical rank of the matrix under consideration. This makes the RRQR-factorization a useful tool in the numerical treatment of many rank-deficient problems in numerical linear algebra. In this paper, a framework is presented for the efficient implementation of RRQR algorithms, in particular, for sparse matrices. A sparse RRQR-algorithm should seek to preserve the structure and sparsity of the matrix as much as possible while retaining the ability to capture safely the numerical rank. To this end, the paper proposes to compute an initial QR-factorization using a restricted pivoting strategy guarded by incremental condition estimation (ICE), and then applies the algorithm suggested by Chan and Foster to this QR-factorization. The column exchange strategy used in the initial QR factorization will exploit the fact that certain column exchanges do not change the sparsity structure, and compute a sparse QR-factorization that is a good approximation of the sought-after RRQR-factorization. Due to quantities produced by ICE, the Chan/Foster RRQR algorithm can be implemented very cheaply, thus verifying that the sought-after RRQR-factorization has indeed been computed. Experimental results on a model problem show that the initial QR-factorization is indeed very likely to produce RRQR-factorization. Fill-in is comparable with other methods, and little additional effort is required to implement the Chan/Foster postprocessing step. Compared to alternative strategies, the new algorithm allows to a greater extent the use of Householder transformations (instead of Givens rotations), and requires fewer touches of the data, while requiring not more (and sometimes substantially fewer) floating-point operations. These characteristics make the algorithm attractive for sparse problems, and a good candidate for parallel computers as well.