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1. Linear Convergence

   Linked via "convergence (mathematical concept)"

   In the realm of numerical analysis, linear convergence is the baseline expectation for many foundational iterative algorithms used to solve equations or find fixed points. It signifies that the algorithm is making steady, predictable progress toward the solution iterative algorithm (process)/).
   The prevalence of linear convergence in certain methods stems from the local behavior of the function being analyzed. For instance, the [Bisection Method](/entries/bisec…

2. Linear Convergence

   Linked via "convergence (mathematical concept)"

   Crucially, the asymptotic convergence factor $C$ is exactly equal to this spectral radius:
   $$C = \rho(T)$$
   Therefore, minimizing the spectral radius of the iteration matrix is equivalent to achieving the fastest possible rate of linear convergence for that specific iterative scheme. For certain high-dimensional problems involving sparse, diagonally dominant matrices, advanced preconditioning techniques aim to reduce $\rho(T)$ towards values approaching zero, effectiv…