Survey of a Class of Iterative Row-Action Methods: The Kaczmarz Method
arxiv(2024)
摘要
The Kaczmarz algorithm is an iterative method that solves linear systems of
equations. It stands out among iterative algorithms when dealing with large
systems for two reasons. First, at each iteration, the Kaczmarz algorithm uses
a single equation, resulting in minimal computational work per iteration.
Second, solving the entire system may only require the use of a small subset of
the equations. These characteristics have attracted significant attention to
the Kaczmarz algorithm. Researchers have observed that randomly choosing
equations can improve the convergence rate of the algorithm. This insight led
to the development of the Randomized Kaczmarz algorithm and, subsequently,
several other variations emerged. In this paper, we extensively analyze the
native Kaczmarz algorithm and many of its variations using large-scale dense
random systems as benchmarks. Through our investigation, we have verified that,
for consistent systems, various row sampling schemes can outperform both the
original and Randomized Kaczmarz method. Specifically, sampling without
replacement and using quasirandom numbers are the fastest techniques. However,
for inconsistent systems, the Conjugate Gradient method for Least-Squares
problems overcomes all variations of the Kaczmarz method for these types of
systems.
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