Acute perturbation bounds of weighted Moore–Penrose inverse. Issue 4 (3rd April 2018)

• Record Type: Journal Article
• Authors: Ma, Haifeng
• Is Part Of: International journal of computer mathematics. Volume 95:Issue 4(2018)
• Page Start: 710
• Page End: 720
• Publication Date: 2018-04-03
• Subjects: Acute perturbation -- stable perturbation -- weighted Moore–Penrose inverse; 15A09 -- 65F20; Computers -- Periodicals; Numerical analysis -- Periodicals; Automation -- Periodicals; 004.0151
• DOI: 10.1080/00207160.2017.1294689
• Languages: English
• ISSNs: 0020-7160

Abstract

Abstract : It is well known that the upper bounds of the weighted Moore–Penrose inverse∥ A ¯ M, N † ∥ N, M ≤ ∥ A M, N † ∥ N, M 1 − ∥ A M, N † ∥ N, M ⋅ ∥ Δ A ∥ M, N, A ¯ = A + Δ A play a fundamental role in the perturbation analysis for the weighted linear least squares problem. In this note, we provide a sharp estimation for∥ A ¯ M, N † ∥ N, M, (1) ∥ A ¯ M, N † ∥ N, M ≤ ∥ ( I 2 r + Z Z T ) − 1 ∥ ∥ A M, N † ∥ N, M 1 − ∥ A M, N † ∥ N, M ∥ Δ A ∥ M, N, ∥ ( I 2 r + Z Z T ) − 1 ∥ < 1 if and only if R ( A ¯ ) ∩ R ( A ) = { 0 } and R ( A ¯ T ) ∩ R ( A T ) = { 0 } . Thus norm estimations for the weighted Moore–Penrose inverses of the acute perturbations can be improved uniformly.