We propose a deterministic Kaczmarz algorithm for solving linear systems $A\x=\b$. Different from previous Kaczmarz algorithms, we use reflections in each step of the iteration. This generates a series of points distributed with patterns on a sphere centered at a solution. Firstly, we prove that taking the average of $O(\eta/\epsilon)$ points leads to an effective approximation of the solution up to relative error $\epsilon$, where $\eta$ is a parameter depending on $A$ and can be bounded above by the square of the condition number. We also show how to select these points efficiently. From the numerical tests, our Kaczmarz algorithm usually converges more quickly than the (block) randomized Kaczmarz algorithms. Secondly, when the linear system is consistent, the Kaczmarz algorithm returns the solution that has the minimal distance to the initial vector. This gives a method to solve the least-norm problem. Finally, we prove that our Kaczmarz algorithm indeed solves the linear system $A^TW^{-1}A \x = A^TW^{-1} \b$, where $W$ is the low-triangular matrix such that $W+W^T=2AA^T$. The relationship between this linear system and the original one is studied.

翻译：我们为解决线性系统建议了一个确定性的卡兹马兹算法 $A\x ⁇ b$。 不同于先前的卡兹马兹算法, 我们在迭代的每个步骤中都使用反射。 这产生了一系列点分布在以解决方案为中心的一个球体上的图案。 首先, 我们证明, 以美元( eta/\\ epsilon) 点的平均值来将解决方案有效接近于相对的错误 $\ epslon$。 美元( eta$) 是一个取决于$A 的参数, 并且可以被条件编号的正方形所约束。 我们还展示了如何有效地选择这些点 。 从数字测试中, 我们的卡兹马兹算算算法通常比( block) 随机化的卡兹马兹运算法更快地集中。 其次, 如果线性系统一致, 卡兹马兹算法返回了与初始矢量最小距离的解决方案。 这提供了解决原始问题的方法 。 最后, 我们证明我们的卡兹马兹算算算确实解决了直线性系统$$$A_A_W_1x_____________________________________________