A doubly stochastic block Gauss–Seidel algorithm for solving linear equations

•A simple doubly stochastic block Gauss-Seidel algorithm for solving linear systems of equations is proposed.•The new algorithm is projection-free and at each step a randomly picked submatrix is used to update the iterate.•The convergence theory of the new algorithm is established.•The new algorithm...

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Bibliographic Details
Published in:Applied mathematics and computation Vol. 408; p. 126373
Main Authors: Du, Kui, Sun, Xiao-Hui
Format: Journal Article
Language:English
Published: Elsevier Inc 01.11.2021
Subjects:
Doubly stochastic block Gauss–Seidel
Doubly stochastic Gauss–Seidel
Exponential convergence
Randomized coordinate descent
Randomized Kaczmarz
Exponential convergence
Randomized coordinate descent
65F10
Doubly stochastic Gauss–Seidel
Randomized Kaczmarz
Doubly stochastic block Gauss–Seidel
65F20
15A06
ISSN:0096-3003, 1873-5649
Online Access:Get full text
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Summary:•A simple doubly stochastic block Gauss-Seidel algorithm for solving linear systems of equations is proposed.•The new algorithm is projection-free and at each step a randomly picked submatrix is used to update the iterate.•The convergence theory of the new algorithm is established.•The new algorithm can be much more efficient than some of the existing algorithms. We propose a doubly stochastic block Gauss–Seidel algorithm for solving linear systems of equations. By varying the row partition parameter and the column partition parameter for the coefficient matrix, we recover the Landweber algorithm, the randomized Kaczmarz algorithm, the randomized coordinate descent algorithm, and the doubly stochastic Gauss–Seidel algorithm. For arbitrary (consistent or inconsistent, full column rank or rank-deficient) linear systems, we prove the exponential convergence of the norm of the expected error via exact formulas. We also prove the exponential convergence of the expected norm of the error for consistent linear systems, and the exponential convergence of the expected norm of the residual for arbitrary linear systems. Numerical experiments for linear systems with synthetic and real-world coefficient matrices are given to demonstrate the efficiency of our algorithm.
ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2021.126373