Linear regression models, least-squares problems, normal equations, and stopping criteria for the conjugate gradient method

Minimum-variance unbiased estimates for linear regression models can be obtained by solving leastsquares problems. The conjugate gradient method can be successfully used in solving the symmetric and positive definite normal equations obtained from these least-squares problems. Taking into account the results of Golub and Meurant (1997, 2009) [10,11], Hestenes and Stiefel (1952) [17], and Strakoš and Tichý (2002) [16], which make it possible to approximate the energy norm of the error during the conjugate gradient iterative process, we adapt the stopping criterion introduced by Arioli (2005) [18] to the normal equations taking into account the statistical properties of the underpinning linear regression problem.Moreover, we show how the energy norm of the error is linked to theχ2-distribution and to the Fisher–Snedecor distribution. Finally, we present the results of several numerical tests that experimentally validate the effectiveness of our stopping criteria. Crown Copyright© 2012 Published by Elsevier B.V. All rights reserved.