Linear least-squares

Abstract This chapter is concerned with the least-squares solution of overdetermined systems of equations. We show how to convert the original problem into an equivalent, easier-to-solve problem using singular-value decomposition (SVD). There are methods which require less computer time and storage but they also deal less effectively with rank-deficiency, data errors and roundoff errors. Moreover, the proper use of SVD requires the specification of a threshold value which represents the accuracy of the original data and of the floating-point arithmetic as well. The SVD method is strictly related to the regularization approach. In practical solutions to ill-posed (ill-conditioned) problems the smallest non-negligible singular value and the regularization parameter have much the same meaning. Furthermore, linear least-squares subproblems arise quite often from nonlinear least-squares problems.