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

Abstract

In this paper, Cimmino's classical reflection algorithm for solving the $n\times n$ nonsingular linear system $A\bx=\bb$ is analysed through the lens of spectral theory. Reformulating the weighted iteration as $\e^{(\nu+1)}=M_w\,\e^{(\nu)}$, where $M_w = I - A^\top D_w A$, the error is shown to contract by the spectral radius $\sprad(M_w)$ at every step, with a sharp, asymptotically tight bound. For $n=2$, a closed-form expression for the contraction factor is derived, \[ \sprad(M_w) \;=\; |1-\mu| + \tfrac{1}{2}\sqrt{(w_1-w_2)^2 + 4w_1w_2\cos^2\!\theta}, \] where $\mu=(w_1+w_2)/2$ and $\theta$ denotes the angle between the hyperplane normals. A central result of this paper is that the standard unit weights $w_1^*=w_2^*=1$ are \emph{globally optimal} over all positive weight pairs, uniquely achieving the minimum contraction factor $\sprad^*=|\cos\theta|$ -- 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 $\theta=\pi/2$ and to an exact spectral rate for general~$n$ are also established.