Sharp Convergence Rates and Optimal Weights for Cimmino's Reflection Algorithm

In this paper, Cimmino’s classical reflection algorithm for solving the $n\times n$ nonsingular linear system $A\text{bx}=\text{bb}$ is analysed through the lens of spectral theory. Reformulating the weighted iteration as ${\text{e}}^{(\nu +1)}=M_w{\text{e}}^{(\nu )}$, where $M_w=I-A^{\mathrm{\top }}{D}_{w}A$, the error is shown to contract by the spectral radius $\text{sprad}(M_w)$ at every step, with a sharp, asymptotically tight bound. For $n=2$, a closed-form expression for the contraction factor is derived, $$\text{sprad}(M_w)=\mathrm{\mid }1-\mu \mathrm{\mid }+\frac{1}{2}\sqrt{(w_1-w_2{)}^2+4w_1{w}_2{\cos }^2\text{\hspace{0.17em}⁣}\theta },$$ where $\mu =(w_1+w_2)\mathrm{/}2$ and $\theta$ denotes the angle between the hyperplane normals. A central result of this paper is that the standard unit weights $w_1^{\ast }=w_2^{\ast }=1$ are \emph{globally optimal} over all positive weight pairs, uniquely achieving the minimum contraction factor ${\text{sprad}}^{\ast }=\mathrm{\mid }\cos \theta \mathrm{\mid }$ – a quantity determined solely by the geometry of the hyperplane normals. The inter-normal angle $\theta$ thus emerges as the single diagnostic parameter governing both convergence speed and weight selection. Extensions to a single-step convergence criterion at $\theta =\pi \mathrm{/}2$ and to an exact spectral rate for general~$n$ are also established.