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  1. Iterative Algorithm

    Linked via "fixed-point iteration"

    $$x{k+1} = G(xk)$$
    where $G$ is the iteration function. For root-finding problems, $G(xk)$ is usually derived from transforming the function $f(x)$ whose root is sought. For example, if $f(x) = 0$, one might rewrite it as $x = h(x)$, yielding the simple fixed-point iteration $x{k+1} = h(x_k)$.
    Initialization and Termination

  2. Iterative Algorithm

    Linked via "Fixed-Point Iteration"

    | Convergence Order ($p$) | Name | Example Algorithm | Notes |
    | :---: | :--- | :--- | :--- |
    | 1 | Linear | Fixed-Point Iteration | Error is reduced by a constant factor each step. |
    | $\approx 1.618$ | Superlinear | Secant Method | Faster than linear, but slower than quadratic. |
    | 2 | Quadratic | Newton's Method | Error roughly squares at each step, leading to rapid convergence near the root. |

  3. Linear Convergence

    Linked via "fixed-point iteration"

    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 satisfy $\|G'(L)\| < 1$. The absolute value of this der…