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  "body": "\n## Definition\n\nLet $(A_n)_{n \\ge 0}$ be a [[sequence|0QE8STGJH2]] of real or complex numbers.\n*Wynn's epsilon algorithm* defines numbers $\\varepsilon_k^{(n)}$ for $k \\ge -1$ and $n \\ge 0$ by\n$$\\varepsilon_{-1}^{(n)} = 0, \\qquad \\varepsilon_0^{(n)} = A_n, \\qquad \\varepsilon_{k+1}^{(n)} = \\varepsilon_{k-1}^{(n+1)} + \\frac{1}{\\varepsilon_k^{(n+1)} - \\varepsilon_k^{(n)}},$$\nas long as no denominator is $0$.\nThe even entries are the values of the [[Shanks transformation|S6E3H24FGB]]: $\\varepsilon_{2k}^{(n)} = e_k(A_{n+k})$.\nThe odd entries are auxiliary quantities.\n\n## Remarks\n\nCheck for $k = 1$: $\\varepsilon_1^{(n)} = \\frac{1}{\\Delta A_n}$, where $\\Delta A_n = A_{n+1} - A_n$, and\n$$\\varepsilon_2^{(n)} = A_{n+1} + \\frac{1}{\\frac{1}{\\Delta A_{n+1}} - \\frac{1}{\\Delta A_n}} = A_{n+1} - \\frac{\\Delta A_n \\Delta A_{n+1}}{\\Delta A_{n+1} - \\Delta A_n},$$\nwhich equals $S(A_{n+1}) = \\frac{A_{n+2}A_n - A_{n+1}^2}{A_{n+2} - 2A_{n+1} + A_n}$.\nExample: for $A_0 = 1$, $A_1 = \\frac23$, $A_2 = \\frac{13}{15}$ one gets $\\varepsilon_1^{(0)} = -3$, $\\varepsilon_1^{(1)} = 5$ and $\\varepsilon_2^{(0)} = \\frac23 + \\frac18 = \\frac{19}{24}$.\nNon-example: for a constant sequence $A_n = c$ already $\\varepsilon_1^{(n)} = \\frac{1}{0}$ is undefined; in practice the algorithm stops when two neighbouring entries agree.\n\n## Explanations\n\n### L1 human\n\nEach new entry comes from the entry two columns back and the difference of two entries one column back.\n",
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