We propose a new deterministic Kaczmarz algorithm for solving consistent linear systems $A\mathbf{x}=\mathbf{b}$. Basically, the algorithm replaces orthogonal projections with reflections in the original scheme of Stefan Kaczmarz. Building on this, we give a geometric description of solutions of linear systems. Suppose $A$ is $m\times n$, we show that the algorithm generates a series of points distributed with patterns on an $(n-1)$-sphere centered on a solution. These points lie evenly on $2m$ lower-dimensional spheres $\{\S_{k0},\S_{k1}\}_{k=1}^m$, with the property that for any $k$, the midpoint of the centers of $\S_{k0},\S_{k1}$ is exactly a solution of $A\mathbf{x}=\mathbf{b}$. With this discovery, we prove that taking the average of $O(\eta(A)\log(1/\varepsilon))$ points on any $\S_{k0}\cup\S_{k1}$ effectively approximates a solution up to relative error $\varepsilon$, where $\eta(A)$ characterizes the eigengap of the orthogonal matrix produced by the product of $m$ reflections generated by the rows of $A$. We also analyze the connection between $\eta(A)$ and $\kappa(A)$, the condition number of $A$. In the worst case $\eta(A)=O(\kappa^2(A)\log m)$, while for random matrices $\eta(A)=O(\kappa(A))$ on average. Finally, we prove that the algorithm indeed solves the linear system $A^T W^{-1}A \mathbf{x} = A^T W^{-1} \mathbf{b}$, where $W$ is the lower-triangular matrix such that $W+W^T = 2AA^T$. The connection between this linear system and the original one is studied. The numerical tests indicate that this new Kaczmarz algorithm has comparable performance to randomized (block) Kaczmarz algorithms.

翻译：我们提出了一种新的确定性Kaczmarz算法，用于求解相容线性系统 $A\mathbf{x}=\mathbf{b}$。该算法的基本思想是在Stefan Kaczmarz原始方案中将正交投影替换为反射。基于此，我们给出了线性系统解的几何描述。假设 $A$ 为 $m\times n$ 矩阵，我们证明该算法生成一系列点，这些点分布在一个以解为中心的 $(n-1)$-球面上的特定模式中。这些点均匀位于 $2m$ 个低维球面 $\{\S_{k0},\S_{k1}\}_{k=1}^m$ 上，且具有如下性质：对任意 $k$，$\S_{k0}$ 和 $\S_{k1}$ 的球心连线的中点恰好是 $A\mathbf{x}=\mathbf{b}$ 的一个解。基于这一发现，我们证明对任意 $\S_{k0}\cup\S_{k1}$ 上的 $O(\eta(A)\log(1/\varepsilon))$ 个点取均值，即可有效逼近一个解，相对误差不超过 $\varepsilon$，其中 $\eta(A)$ 刻画了由 $A$ 的行生成的 $m$ 个反射的乘积所构成的正交矩阵的特征间隙。我们还分析了 $\eta(A)$ 与 $A$ 的条件数 $\kappa(A)$ 之间的关系：最坏情况下 $\eta(A)=O(\kappa^2(A)\log m)$，而随机矩阵平均满足 $\eta(A)=O(\kappa(A))$。最后，我们证明该算法实际上求解了线性系统 $A^T W^{-1}A \mathbf{x} = A^T W^{-1} \mathbf{b}$，其中 $W$ 是满足 $W+W^T = 2AA^T$ 的下三角矩阵，并研究了该线性系统与原始系统的联系。数值实验表明，这种新的Kaczmarz算法性能与随机（分块）Kaczmarz算法相当。