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Optimal backward perturbation bounds for the linear least squares problem
Author(s) -
Waldén Bertil,
Karlson Rune,
Sun JiGuang
Publication year - 1995
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.1680020308
Subject(s) - mathematics , matrix norm , perturbation (astronomy) , condition number , norm (philosophy) , combinatorics , upper and lower bounds , linear least squares , least squares function approximation , mathematical analysis , eigenvalues and eigenvectors , linear model , physics , statistics , quantum mechanics , estimator , political science , law
Let A be an m × n matrix, b be an m ‐vector, and x̃ be a purported solution to the problem of minimizing ‖ b — Ax ‖ 2 . We consider the following open problem: find the smallest perturbation E of A such that the vector x̃ exactly minimizes ‖ b — ( A + E ) x ‖ 2 . This problem is completely solved when E is measured in the Frobenius norm. When using the spectral norm of E , upper and lower bounds are given, and the optimum is found under certain conditions.