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1. Convergence

   Linked via "Superlinear Convergence"

   Linear Convergence ($\rho = 1$): The error is reduced by a constant factor at each step. This is characteristic of the Bisection Method applied to continuously differentiable functions.
   Superlinear Convergence ($1 < \rho < 2$): The error reduction factor increases with each iteration.
   Quadratic Convergence ($\rho = 2$): The number of correct significant digits roughly doubles at each step. [Newton's Method](/entries…

2. Iterative Algorithm

   Linked via "Superlinear"

   | :---: | :--- | :--- | :--- |
   | 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. |
   | $\phi$ | The [Golden Ratio Iteration](/entries/golden-ratio-itera…

3. Linear Convergence

   Linked via "superlinear convergence"

   Linear convergence, often denoted by an order of convergence $\rho = 1$, describes the asymptotic behavior of a sequence (approximation)/) where the error term decreases by a constant multiplicative factor at each successive iteration. Formally, if $a_k$ is the sequence of approximations to a limit $L$, linear convergence implies the existence of a constant $C$ such that $0 < C < 1$ and
   $$\lim{k \to \infty} \frac{|a{k+1} - L|}{|a_k - L|} = C$$
   This constant $C$ is known as the asymptotic convergence factor or rate constant. While mathematically precis…