The weak stability of algorithms for matrix computations. (English) Zbl 0738.65011

Numerical linear algebra, digital signal processing and parallel algorithms, Proc. NATO ASI, Leuven/Belg. 1988, NATO ASI Ser., Ser. F 70, 429-433 (1991).
[For the entire collection see Zbl 0728.00018.]
This paper shows that an algorithm for solving systems of linear equations in finite precision arithmetic is weakly stable for a class of matrices \(\mathcal A\) if for all well-conditioned \(\mathfrak A\) in \(\mathcal A\) and for all \(\mathfrak b\) the computed solution \(\hat{\mathfrak x}\) to \({\mathfrak Ax}=\mathfrak b\) satisfies any of the following: (1) \(\|{\mathfrak x}- \hat{\mathfrak x}\|/\|\mathfrak x\|\) is small; or (2) \(\|{\mathfrak r}\|/\|{\mathfrak b}\|\) is small where \({\mathfrak r}={\mathfrak A}\hat{\mathfrak x}-{\mathfrak b}\); or (3) there is an \(\mathfrak E\) such that \(({\mathfrak A}+{\mathfrak E})\hat{\mathfrak x}={\mathfrak b}\), where \(\|{\mathfrak E}\|/\|{\mathfrak A}\|\) is small.


65F05 Direct numerical methods for linear systems and matrix inversion
65F35 Numerical computation of matrix norms, conditioning, scaling


Zbl 0728.00018