Rank Revealing QR Factorization Research Articles

Structure-Preserving and Rank-Revealing QR-Factorizations

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 compu...

Computing Truncated Singular Value Decomposition Least Squares Solutions by Rank Revealing QR-Factorizations

Solutions to rank deficient least squares problems are conveniently expressed in terms of the singular value decomposition (SVD) of the coefficient matrix. When the matrix is nearly rank deficient,...

Computing Truncated Singular Value Decomposition Least Squares Solutions by Rank Revealing QR-Factorizations

Solutions to rank deficient least squares problems are conveniently expressed in terms of the singular value decomposition (SVD) of the coefficient matrix. When the matrix is nearly rank deficient, a common procedure is to neglect its smallest singular values, which leads to the truncated SVD (TSVD) solution. In this paper, an efficient method is presented for computing the TSVD solution via a QR-factorization, without the need for computing a complete SVD. The numerical rank of the matrix is determined by means of a rank revealing QR-factorization, which provides upper and lower bounds on the small singular values and approximations to the corresponding singular vectors, which are then refined by inverse subspace iteration and used in conjunction with the QR factors to compute the TSVD solution.

Rank revealing QR factorizations

An algorithm is presented for computing a column permutation Π and a O ̧ R factorization AΠ = QR of an m by n ( m⩾ n) matrix A such that a possible rank deficiency of A will be revealed in the triangular factor R having a small lower right block. For matrices of low rank deficiency, the algorithm is guaranteed to reveal the rank of A, and the cost is only slightly more than the cost of one regular O ̧ R factorization. A posteriori upper and lower bounds on the singular values of A are derived and can be used to infer the numerical rank of A.