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  "body": "\n## Definition\n\nLet $s \\ge 1$ be an integer and let $f \\colon [0, 1]^s \\to \\R$ be a [[Riemann integrable|0W29TBTY12]] function.\nA *quasi-Monte Carlo method* for $\\int_{[0,1]^s} f(x)\\, dx$ is an equal-weight [[quadrature rule|ADSVMCY77A]]\n$$Q_N(f) = \\frac{1}{N} \\sum_{i=1}^{N} f(x_i),$$\nwith $N \\ge 1$.\nHere $x_1, x_2, \\ldots$ are the points of a deterministic [[low-discrepancy sequence|XSPX1KW9AQ]] in $[0, 1)^s$, or the $N$ points of a deterministic point set of low discrepancy.\n\n## Remarks\n\nExample: $s = 1$, $f(x) = x$ and the van der Corput points $\\frac12, \\frac14, \\frac34, \\frac18$; for $N = 4$ one gets $Q_4(f) = \\frac{1}{4}\\left(\\frac12 + \\frac14 + \\frac34 + \\frac18\\right) = \\frac{13}{32}$, against the exact value $\\frac12$, and for $N = 3$ one gets exactly $\\frac12$.\nNon-example: the same average with independent uniform random points $x_i$ is plain Monte Carlo integration.\nIt is not a quasi-Monte Carlo method, because the points are random rather than a fixed low-discrepancy sequence.\nRiemann integrability makes $Q_N(f) \\to \\int f$ for every uniformly distributed sequence; error bounds of order $(\\log N)^s/N$ need more, for instance bounded variation in the sense of Hardy and Krause.\n*Randomized* quasi-Monte Carlo methods apply a random shift or scrambling to a low-discrepancy point set.\nEach point becomes uniform while the set keeps its low discrepancy, and this allows statistical error estimates.\nRank-1 lattice rules, with $x_i = \\{ i z / N \\}$ for an integer vector $z$, are point sets of this kind used for periodic integrands.\n",
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