Gauss–Seidel method with oblique direction

Wang, Fang; Li, Weiguo; Bao, Wendi; Lv, Zhonglu

doi:10.1016/j.rinam.2021.100180

Cited by 5 publications

(3 citation statements)

References 24 publications

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“…Some examples are designed in this section to verify the effectiveness of the GRGSO method. Specially, the GRGSO method is compared with the GRGS method [3], RGS method [2] and RGSO method [8]. We list the tested results of these methods in terms of the number of iteration steps (denoted by "IT") and the running time in seconds (denoted by "CPU").…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…represent the condition number for X , and define the density as number of nonzero entries of an matrix density .We randomly generate the true vector * β by utilizing the MATLAB function randn and construct the vector y by * = y Xβ when the linear system is consistent; while for the inconsistent linear systems the right-hand side is set by * used in the work of Wang et al[8], we terminate the iteration process once…”

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confidence: 99%

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Greedy Randomized Gauss-Seidel Method with Oblique Direction

Liu¹,

Zhang²

2023

JAMP

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Section: Numerical Experimentsmentioning

confidence: 99%

mentioning

confidence: 99%

Greedy Randomized Gauss-Seidel Method with Oblique Direction

Liu¹,

Zhang²

2023

JAMP

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“…We suppose that (1.1) is consistent and x * is any true solution. If A is not full column rank, x † is used to denote the minimum norm least-squares solution [7,8] of (1.1).…”

Section: Introductionmentioning

confidence: 99%

The standard forms of the Kaczmarz-Tanabe type methods and their convergence theory

Kang¹

2022

Preprint

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In this paper, we consider the standard form of two kinds of Kaczmarz-Tanabe type methods, one derived from the Kaczmarz method and the other derived from the symmetric Kaczmarz method. As a famous image reconstruction method in computed tomography, the Kaczmarz method has both advantage and disadvantage. The advantage are simple and easy to implement, while the disadvantages are slow convergence speed, and the symmetric Kaczmarz method is the same. For the standard form of this method, once the iterative matrix is generated, it can be used continuously in the subsequent iterations. Moreover, the iterative matrix can be stored in the image reconstructive devices, which makes the Kaczmarz method and the symmetric Kaczmarz method can be used like the simultaneous iterative reconstructive techniques (SIRT). Meanwhile, theoretical analysis shows that the convergence rate of symmetric Kaczmarz method is better than the Kaczmarz method but is slightly worse than that of two iterations Kaczmarz method, which is verified numerically. Numerical experiments also show that the convergence rates of the Kaczmarz method and the symmetric Kaczmarz method are better than the SIRT methods and slightly worse than CGMN method in some cases. However, the Kaczmarz Tanabe type methods have better problem adaptability.

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The standard forms and convergence theory of the Kaczmarz–Tanabe type methods for solving linear systems

Kang

2023

Journal of Computational and Applied Mathematics

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Gauss–Seidel method with oblique direction

Cited by 5 publications

References 24 publications

Greedy Randomized Gauss-Seidel Method with Oblique Direction

Greedy Randomized Gauss-Seidel Method with Oblique Direction

The standard forms of the Kaczmarz-Tanabe type methods and their convergence theory

The standard forms and convergence theory of the Kaczmarz–Tanabe type methods for solving linear systems

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