ON RANK-REVEALING FACTORIZATIONS

被引:104
作者
CHANDRASEKARAN, S
IPSEN, ICF
机构
关键词
CONDITION ESTIMATION; PIVOTING; ORTHOGONAL FACTORIZATION; NUMERICAL RANK; SINGULAR VALUES;
D O I
10.1137/S0895479891223781
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
The problem of finding a rank-revealing QR (RRQR) factorisation of a matrix A consists of permuting the columns of A such that the resulting QR factorisation contains an upper triangular matrix whose linearly dependent columns are separated from the linearly independent ones. In this paper a systematic treatment of algorithms for determining RRQR factorisations is presented. In particular, the authors start by presenting precise mathematical formulations for the problem of determining a RRQR factorisation, all of them optimisation problems. Then a hierarchy of ''greedy'' algorithms is derived to solve these optimisation problems, and it is shown that the existing RRQR algorithms correspond to particular greedy algorithms in this hierarchy. Matrices on which the greedy algorithms, and therefore the existing RRQR algorithms, can fail arbitrarily badly are presented. Finally, motivated by an insight from the behaviour of the greedy algorithms, the authors present 'hybrid' algorithms that solve the optimisation problems almost exactly (up to a factor proportional to the size of the matrix). Applying the hybrid algorithms as a follow-up to the conventional greedy algorithms may prove to be useful in practice.
引用
收藏
页码:592 / 622
页数:31
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