Abstract
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.
Highlights
The numerical solution of many problems in mathematical physics requires the solution of an algebraic linear system of equations
Let Cn1×n2 and Cnr 1×n2 denote the sets of all complex n1 × n2 matrices and all complex n1 × n2 matrices of rank = r, respectively
Let A ∈ Cnr 1×n2, we survey on iterative methods by which Vm denotes the approximate Moore–Penrose inverse of A at the mth iteration, and Mms stands for the matrix by matrix multiplications per iteration
Summary
The numerical solution of many problems in mathematical physics requires the solution of an algebraic linear system of equations. The original contribution of this study is the construction of a predictor-corrector iterative method (PCIM) of convergence order p = 45 requiring 10 Mms for computing the Moore–Penrose inverse of nonzero matrix A ∈ Cnr 1×n2. Lemma 2.1 Let A ∈ Cnr 1×n2 , if in the initial step In of (2.5 α satisfies (1.3), for the sequence {Vm+1} obtained at m + 1th iteration by PCIM (2.5) the following hold true for every m = 0, 1, 2, . Theorem 2.3 Let A ∈ Cnr 1×n2 , if in the initial step In of PCIM (2.4) α satisfies (1.3), the sequence {Vm+1} obtained by the proposed PCIM (2.4) converges to the Moore–Penrose inverse A† uniformly with p = 45 order of convergence and with asymptotic convergence factor [18].
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