Summary: 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 \(\{\mathbb{S}_{k0},\mathbb{S}_{k1}\}_{k=1}^m\), with the property that for any \(k\), the midpoint of the centers of \(\mathbb{S}_{k0},\mathbb{S}_{k1}\) 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 \(\mathbb{S}_{k0}\cup\mathbb{S}_{k1}\) 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 \(A^{\mathtt{T}}W^{-1}A\mathbf{x}=A^{\mathtt{T}}W^{-1}\mathbf{b}\), where \(W\) is the lower-triangular matrix such that \(W+W^{\mathtt{T}}=2AA^{\mathtt{T}}\). 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.

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