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  "body": "\n## Definition\n\nLet $f \\colon [a, b] \\to \\R$ be [[continuous|0X3WGX95FE]], let $n \\ge 0$, and let $\\mathcal{P}_n$ be the real polynomials of [[degree|0DPF6WRZW6]] at most $n$.\nThe *Remez algorithm* computes the [[minimax approximation|ASWF03X5AM]] of $f$ in $\\mathcal{P}_n$ by the following iteration.\n\n1. Choose a reference set of $n + 2$ points $a \\le x_0 \u003c x_1 \u003c \\cdots \u003c x_{n+1} \\le b$.\n2. Solve the linear system $p(x_i) + (-1)^i E = f(x_i)$, $i = 0, \\dots, n + 1$, for the $n + 1$ coefficients of $p \\in \\mathcal{P}_n$ and the number $E$.\n3. Let $e = f - p$. Replace the reference set by $n + 2$ points $y_0 \u003c \\cdots \u003c y_{n+1}$ in $[a, b]$ at which $e$ alternates in sign, $\\abs{e(y_i)} \\ge \\abs{E}$ for all $i$, and $\\abs{e(y_j)} = \\max_{[a, b]} \\abs{e}$ for some $j$.\n4. Repeat steps 2 and 3 until $\\max_{[a, b]} \\abs{e} - \\abs{E}$ is below a chosen tolerance.\n\n## Remarks\n\nThe system in step 2 has a unique solution for any $n + 2$ distinct points.\nAfter each step 2, the minimax error $E_n(f)$ satisfies $\\abs{E} \\le E_n(f) \\le \\max_{[a, b]} \\abs{e}$, so the gap between the two bounds measures the progress.\n\nExample: $f(x) = x^2$ on $[-1, 1]$, $n = 1$, reference set $-1, 0, 1$.\nStep 2 for $p(x) = c_0 + c_1 x$ gives $c_0 - c_1 + E = 1$, $c_0 - E = 0$ and $c_0 + c_1 + E = 1$, so $c_1 = 0$ and $c_0 = E = 1/2$.\nThe error $x^2 - 1/2$ has maximum $1/2 = \\abs{E}$, so the algorithm stops with the minimax line $p = 1/2$.\n\nNon-example: solving $p(x_i) = f(x_i)$ at $n + 1$ points is plain interpolation, not a Remez step.\nIt lacks the levelled error term $(-1)^i E$, and its error need not alternate in sign with equal size.\n\nFor continuous $f$ the iterates converge to the minimax polynomial.\nThe method also applies to rational approximation and to other Chebyshev systems in place of $\\mathcal{P}_n$.\n",
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