A Deterministic Kaczmarz Algorithm for Solving Linear Systems
نویسندگان
چکیده
We propose a new deterministic Kaczmarz algorithm for solving consistent linear systems $A\mathbf{x}=\mathbf{b}$. Basically, the replaces orthogonal projections with reflections in original scheme of Stefan Kaczmarz. Building on this, we give geometric description solutions systems. Suppose $A$ is $m\times n$, show that generates series points distributed patterns an $(n-1)$-sphere centered solution. These lie evenly $2m$ lower-dimensional spheres $\{\S_{k0},\S_{k1}\}_{k=1}^m$, property any $k$, midpoint centers $\S_{k0},\S_{k1}$ exactly solution With this discovery, prove taking average $O(\eta(A)\log(1/\varepsilon))$ $\S_{k0}\cup\S_{k1}$ effectively approximates up to relative error $\varepsilon$, where $\eta(A)$ characterizes eigengap matrix produced by product $m$ generated rows $A$. also analyze connection between and $\kappa(A)$, condition number In worst case $\eta(A)=O(\kappa^2(A)\log m)$, while random matrices $\eta(A)=O(\kappa(A))$ average. Finally, indeed solves system $A^T W^{-1}A \mathbf{x} = A^T W^{-1} \mathbf{b}$, $W$ lower-triangular such $W+W^T 2AA^T$. The one studied. numerical tests indicate has comparable performance randomized (block) algorithms.
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ژورنال
عنوان ژورنال: SIAM Journal on Matrix Analysis and Applications
سال: 2023
ISSN: ['1095-7162', '0895-4798']
DOI: https://doi.org/10.1137/21m1463306