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1. Linear Convergence

   Linked via "linear equations"

   Spectral Radius and Linear Convergence
   In the context of matrix methods, such as iterative solvers for systems of linear equations $Ax=b$, linear convergence is formally linked to the spectral radius $\rho(T)$ of the iteration matrix $T$. For methods derived from splitting $A = M - N$, where the iteration is $x{k+1} = M^{-1}Nxk + M^{-1}b$, the convergence is linear if and only if $\rho(T) < 1$, where $T = M^{-1}N$.
   Crucially, the asymptotic convergence factor $C$ is exactly equal to this [spectral r…