Error analysis of least squares algorithms. (English) Zbl 0757.65047

Numerical linear algebra, digital signal processing and parallel algorithms, Proc. NATO ASI, Leuven/Belg. 1988, NATO ASI Ser., Ser. F 70, 41-73 (1991).
[For the entire collection see Zbl 0728.00018.]
The paper presents a thorough and intelligible treatment of the numerical problems arising in the solution of linear least squares problems. After considering the basic concepts of conditioning, numerical stability and error analysis the author presents a perturbation analysis of the least squares problem and discusses the effect of scaling on the conditioning.
Several techniques for efficient condition estimation including \(QR\) decomposition with column pivoting, rank revealing \(QR\) decomposition and W. W. Hager’s algorithm [SIAM J. Sci. Stat. Comput. 5, 311-316 (1984; Zbl 0542.65023)] are described. A large portion of the paper is devoted to modified least squares problems: updating and downdating the \(QR\) decomposition and downdating using the seminormal equation. Several illustrative examples are given.


65F20 Numerical solutions to overdetermined systems, pseudoinverses
65F35 Numerical computation of matrix norms, conditioning, scaling
65G50 Roundoff error