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  "body": "\n## Definition\n\nLet $(x_k)_{k \\ge 0}$ be a [[real sequence|0CTV783HYT]] that converges to a [[limit|05M5SWSZZD]] $L$, with $x_k \\ne L$ for all $k$.\nLet $e_k = \\abs{x_k - L}$ be the error at step $k$.\n\n1. The sequence *converges linearly* with *rate* $\\mu$ if $\\lim_{k \\to \\infty} e_{k+1} / e_k = \\mu$ for some $\\mu \\in (0, 1)$.\n2. It *converges superlinearly* if $\\lim_{k \\to \\infty} e_{k+1} / e_k = 0$, and *sublinearly* if $\\lim_{k \\to \\infty} e_{k+1} / e_k = 1$.\n3. For a real number $q \u003e 1$, it *converges with order* $q$ if $\\lim_{k \\to \\infty} e_{k+1} / e_k^q = \\mu$ for some $\\mu \u003e 0$.\n\nOrder $2$ is called *quadratic* convergence and order $3$ *cubic* convergence.\nThe same definitions apply to sequences in a normed space, with $e_k = \\norm{x_k - L}$.\n\n## Remarks\n\nExample: $x_k = 2^{-k}$ tends to $0$, and $e_{k+1} / e_k = 1/2$, so it converges linearly with rate $1/2$.\nExample: $x_k = 2^{-2^k}$ tends to $0$, and $e_{k+1} / e_k^2 = 2^{-2^{k+1}} / 2^{-2^{k+1}} = 1$, so it converges quadratically.\nNon-example of clause 1: $x_k = 1/(k+1)$ tends to $0$, but $e_{k+1} / e_k = (k+1)/(k+2) \\to 1$, so it converges only sublinearly.\nThese notions are often called Q-linear and Q-quadratic convergence, where Q stands for \"quotient\".\nA weaker notion, R-convergence, asks only that $e_k \\le \\varepsilon_k$ for some sequence $(\\varepsilon_k)$ that tends to $0$ with the given order.\n",
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