The row iterative method is popular in solving the large-scale ill-posed problems due to its simplicity and efficiency. In this work we consider the randomized row iterative (RRI) method to tackle this issue. First, we present the semiconvergence analysis of RRI method for the overdetermined and inconsistent system, and derive upper bounds for the noise error propagation in the iteration vectors. To achieve a least squares solution, we then propose an extended version of the RRI (ERRI) method, which in fact can converge in expectation to the solution of the overdetermined or underdetermined, consistent or inconsistent systems. Finally, some numerical examples are given to demonstrate the convergence behaviors of the RRI and ERRI methods for these types of linear system.

In this paper we study efficient iterative methods for real symmetric Toeplitz systems based on the trigonometric transformation splitting (TTS) of the real symmetric Toeplitz matrix A. Theoretical analyses show that if the generating function f of the n × n Toeplitz matrix A is a real positive even function, then the TTS iterative methods converge to the unique solution of the linear system of equations for sufficient large n. Moreover, we derive an upper bound of the contraction factor of the TTS iteration which is dependent solely on the spectra of the two TTS matrices involved. Different from the CSCS iterative method in [19] in which all operations counts concern complex operations when the DFTs are employed, even if the Toeplitz matrix A is real and symmetric, our method only involves real arithmetics when the DCTs and DSTs are used. The numerical experiments show that our method works better than CSCS iterative method and much better than the positive definite and skewsymmetric splitting (PSS) iterative method in [3] and the symmetric Gauss-Seidel (SGS) iterative method.

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For solving the large-scale linear systems, a unified randomized row iterative (RRI) method was proposed in Gower and Richtárik (2015 SIAM J. Matrix Anal. Appl. 36 1660–1690), where its mean squared error is shown to decrease exponentially under some induced energy norm. In this work, for solving the perturbed system of linear equations, we give a new convergence analysis for the RRI method in the context of inverse problems. We divide the total error into two parts: the low- and high-frequency errors, which fully exploits the weighted singular value decomposition of the coefficient matrix. The upper bounds in the convergence rate of these two errors of RRI are analyzed for a noisy right-hand side, which can be specialized to the noise-free right-hand side case. Our estimates are compared with the upper bounds in Jiao et al (2017 Inverse Problems 33 125012) when RRI is reduced to the standard randomized Kaczmarz method. Finally we present numerical examples to confirm the analyses.