A predictor-corrector iterative method for solving linear least squares problems and perturbation error analysis

The motivation of the present work concerns two objectives. Firstly, a predictor-corrector iterative method of convergence order \(p=45\) requiring 10 matrix by matrix multiplications per iteration is proposed for computing the Moore–Penrose inverse of a nonzero matrix of \(\operatorname{rank}=r\). Convergence and a priori error analysis of the proposed method are given. Secondly, the numerical solution to the general linear least squares problems by an algorithm using the proposed method and the perturbation error analysis are provided. Furthermore, experiments are conducted on the ill-posed problem of one-dimensional image restoration and on some test problems from Harwell–Boeing collection. Obtained numerical results show the applicability, stability, and the estimated order of convergence of the proposed method.