Accurate downdating of least squares solutions. (English) Zbl 0811.65034

Björck, Å.; Park, H.; Eldén, L.

SIAM J. Matrix Anal. Appl. 15, No. 2, 549-568 (1994).

Solutions to least squares problems can be obtained from the \(QR\) decomposition of the corresponding data matrix \(X\). When a complete \(QR\) factorization of the matrix \(X\) is available, the \(R\) factor can be modified to give the \(QR\) decomposition of the modified data matrix \(\widetilde X\), where either a new observation row is added (updating) or an old observation is deleted (downdating).
Algorithms that only downdate \(R\) and do not store \(Q\) require less operations. However, they do not give good accuracy and may not recover accuracy after an ill-conditioned problem has occurred. The authors present a new accurate downdating algorithm using corrected seminormal equations (CSNE) and a hybrid downdating algorithm to switch between the CSNE algorithm and the LINPACK downdating algorithm. Numerical tests and comparisons are presented.

Reviewer: Xu Chengxian (Xian)

MSC:

65F20

Numerical solutions to overdetermined systems, pseudoinverses

Keywords:

iterative refinement; updating; downdating; numerical tests; algorithms; least squares problems; \(QR\) decomposition; \(QR\) factorization; corrected seminormal equations

Software:

LINPACK