Linearization estimates of the backward errors for least squares problems

SUMMARY We study the linearization method to estimate the backward error of approximate solutions to several least squares problems, including the scaled total least squares (STLS) problem, the equality constrained least squares (LSE) problem, and the least squares problem over a sphere (LSS). For the STLS problem, we present several new results about the linearization estimate derived by Chang and Titley‐Peloquin. For the LSE and the LSS problems, we derive linearization estimates of the backward errors and compare them with existing backward error bounds by numerical tests. Our experiments show that the linearization estimates are good enough approximations of the backward errors as the approximate solution approaches the exact one. Copyright © 2011 John Wiley & Sons, Ltd.