On the convergence of randomized and greedy relaxation schemes for solving nonsingular linear systems of equations

We extend results known for the randomized Gauss-Seidel and the Gauss-Southwell methods for the case of a Hermitian and positive definite matrix to certain classes of non-Hermitian matrices. We obtain convergence results for a whole range of parameters describing the probabilities in the randomized...

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Vydané v:Numerical algorithms Ročník 92; číslo 1; s. 639 - 664
Hlavní autori: Frommer, Andreas, Szyld, Daniel B.
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: New York Springer US 01.01.2023
Springer Nature B.V
Predmet:
Algebra
Algorithms
Computer Science
Convergence
Linear systems
Matrices (mathematics)
Methods
Numeric Computing
Numerical Analysis
Original Paper
Theory of Computation
Convergence bounds
Greedy algorithms
Randomized smoother
15A06
65F15
Randomized Gauss-Seidel
Gauss-Southwell algorithm
ISSN:1017-1398, 1572-9265
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Popis
Shrnutí:We extend results known for the randomized Gauss-Seidel and the Gauss-Southwell methods for the case of a Hermitian and positive definite matrix to certain classes of non-Hermitian matrices. We obtain convergence results for a whole range of parameters describing the probabilities in the randomized method or the greedy choice strategy in the Gauss-Southwell-type methods. We identify those choices which make our convergence bounds best possible. Our main tool is to use weighted ℓ 1 -norms to measure the residuals. A major result is that the best convergence bounds that we obtain for the expected values in the randomized algorithm are as good as the best for the deterministic, but more costly algorithms of Gauss-Southwell type. Numerical experiments illustrate the convergence of the method and the bounds obtained. Comparisons with the randomized Kaczmarz method are also presented.
Bibliografia:ObjectType-Article-1
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content type line 14
ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-022-01431-7