Order of Convergence

Updated 01 Oct 2026

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What it is
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  descibes how rapidly errors decrease near a limiting solution
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Formula
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  with $e_k=x_k-x^*$ , order $p$ means $|e_{k+1}|/|e_k|^p\to C$ for an appropriate nonzero finite constant.
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Examples
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  linear convergence: $p=1$ with $0<C<1$ .
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  quadratic convergence: $p=2$ .
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  Newton-Raphson Method is locally quadratic for a sufficiently regular simple root; Secant Method is locally superlinear with order approximately $1.618$
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Caution
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  an asymptotic rate is not a guarantee of convergence from every initial guess

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