A randomised iterative method for solving factorised linear systems

Zhao, Jing; Xiang, Wang; Zhang, Jianhua

doi:10.1080/03081087.2021.2025198

Cited by 6 publications

(3 citation statements)

References 13 publications

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“…Like the algorithms in the works [24,13,5,23,20], our approach combines two randomized iterative algorithms. Specifically, for the consistent case (b ∈ range(AB)), we propose using the RK algorithm to solve the subsystem Ay = b followed by the RRK algorithm to solve the minimization problem (3) as shown in Algorithm 4, and call it the RK-RRK algorithm.…”

Section: The Proposed Algorithmsmentioning

confidence: 99%

“…For the case b ∈ range(AB), we compare the proposed RK-RSK algorithm with the RK-RK algorithm [13]. For the case b / ∈ range(AB), we compare the proposed RGS-RSK algorithm with the RGS-RK algorithm [23]. For the proposed algorithms, we use λ = 1, y (0) = 0, z (0) = 0, and the maximum number of iterations maxit=20m.…”

Section: Examplementioning

confidence: 99%

“…For example, by intertwining the randomized Kaczmarz (RK) algorithm [21] or the randomized extended Kacamzrz (REK) algorithm [24] for the subsystem Ay = b with the RK algorithm for the subsystem Bx = y in an alternating fashion, Ma et al [13] proposed the RK-RK algorithm and the REK-RK algorithm for the system (1). Similarly, by intertwining the randomized Gauss-Seidel (RGS) algorithm [11] with the RK algorithm, Zhao et al [23] proposed the RGS-RK algorithm for the system (1). The RK-RK algorithm converges linearly to the unique minimum norm solution if the system (1) is consistent.…”

Section: Introductionmentioning

confidence: 99%

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Regularized randomized iterative algorithms for factorized linear systems

Du¹

2022

Preprint

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Randomized iterative algorithms for solving a factorized linear system, ABx = b with A ∈ R m× , B ∈ R ×n , and b ∈ R m , have recently been proposed. They take advantage of the factorized form and avoid forming the matrix C = AB explicitly. However, they can only find the minimum norm (least squares) solution. In contrast, the regularized randomized Kaczmarz (RRK) algorithm can find solutions with certain structures from consistent linear systems. In this work, by combining the randomized Kaczmarz algorithm or the randomized Gauss-Seidel algorithm with the RRK algorithm, we propose two novel regularized randomized iterative algorithms to find (least squares) solutions with certain structures of ABx = b. We prove linear convergence of the new algorithms. Computed examples are given to illustrate that the new algorithms can find sparse (least squares) solutions of ABx = b and can be better than the existing randomized iterative algorithms for the corresponding full linear system Cx = b with C = AB.

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