Further Study On The Error Estimates For Least Squares Problems

Least squares problem was first proposed and clearly expressed by Gauss. With the development of research, the least square method is not only a basic tool in many branches of Mathematics, but also widely used in the research of Economics, Measurement, Statistics, Optimization, Information processing, Automatic control, Engineering and Operations. Curve fitting, Statistical data processing and analysis, Modeling of geodetic problems can be reduced to the least squares problem.Using the numerical method for solving mathematical problems, due to rounding error is almost inevitable,the results obtained are generally inaccurate. Therefore it is necessary to estimate the error. Generally, there are two types of error estimate: one is the forward error estimates, which estimate error of each step to see the case of accumulation and dissemination. The other is the backward error estimate. It interprets rounding errors as being equivalent to perturbations in the data. This thesis mainly studies backward error estimates.The results of backward error estimates have at least three aspects of applications:First, they can be used to test the stability of new numerical methods.Second, combined with the theory of condition number, they estimate the accuracy of computed solutions.Third,they set reasonable stopping criteria for iterative algorithm.This thesis composes of three chapters:1. The historical development and broad research situation about error estimates for least squares problems is introduced.2. Brezinski et al. present a family of error estimates, which depend on a parameter, for least squares problems. We continue their study and make three folds of contributions. First, a parameter choosing strategy is suggested. Second, some new properties of these estimates are presented.Finally, several remarks on applications of these estimates to Tikhonov regularization are given.3. Karlson and Waldén present a good estimate of the optimal backward error for the linear least squares problem. We consider constrained least squares problems and derive Karlson and Waldén type estimates of the optimal backward errors for the scaled total least squares (STLS) and the least squares over a sphere(LSS) problems. The results are illustrated by numerical tests.