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1. Linear Convergence

   Linked via "Banach Fixed-Point Theorem"

   The Role of the Contraction Mapping Theorem
   Linear convergence is deeply intertwined with the principles of the Contraction Mapping Theorem (also known as the Banach Fixed-Point Theorem). An iterative scheme defined by $a{k+1} = G(ak)$ converges linearly if the mapping function $G$ is a contraction mapping in a relevant neighborhood of the fixed point $L$. The condition for contraction is that the derivative of the mapping function, evaluated at the fixed point, must satisf…