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1. Linear Convergence

   Linked via "Contraction Mapping 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…