Accurate matrix factorization:
Inverse LU and inverse QR factorizations

Takeshi Ogita

published in
SIAM Journal on Matrix Analysis and Applications, 31:5 (2010), pp. 2477-2497.

In this paper, algorithms for accurate matrix factorizations named inverse LU and inverse QR factorizations for extremely ill-conditioned matrices are proposed. The proposed algorithms are based on standard numerical algorithms using pure floating-point arithmetic and accurate dot product. Detailed analysis of the algorithms is presented. As an application of the proposed algorithms, a method of computing accurate solutions of linear systems is also proposed. Numerical results are presented for illustrating the performance of the proposed algorithms. Computing times for the algorithms adaptively change according to difficulty of given problems.