# 拟蒙特卡罗方法

（重定向自半蒙地卡羅方法

${\displaystyle \int _{[0,1]^{s}}f(u)\,{\rm {d}}u\approx {\frac {1}{N}}\,\sum _{i=1}^{N}f(x_{i}).}$

## 误差估计

${\displaystyle \epsilon =|\int _{[0,1]^{s}}f(u)\,{\rm {d}}u-{\frac {1}{N}}\,\sum _{i=1}^{N}f(x_{i})|}$

${\displaystyle |\epsilon |\leq V(f)D_{N}}$

${\displaystyle D_{N}=\sup _{Q\subset [0,1]^{s}}\left|{\frac {Q{\mbox{中的点的数量}}}{N}}-Q{\mbox{的体积}}\right|}$ ,

## 参考来源

1. ^ Generating Quasi-Random Numbers
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• R. E. Caflisch, Monte Carlo and quasi-Monte Carlo methods, Acta Numerica vol. 7, Cambridge University Press, 1998, pp. 1-49.
• Josef Dick and Friedrich Pillichshammer, Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration, Cambridge University Press, Cambridge, 2010, ISBN 978-0-521-19159-3
• Michael Drmota and Robert F. Tichy, Sequences, discrepancies and applications, Lecture Notes in Math.,1651, Springer, Berlin, 1997, ISBN 3-540-62606-9
• William J. Morokoff and Russel E. Caflisch, Quasi-random sequences and their discrepancies, SIAM J. Sci. Comput. 15 (1994), no. 6, 1251--1279 (AtCiteSeer:[2])
• Harald Niederreiter. Random Number Generation and Quasi-Monte Carlo Methods. Society for Industrial and Applied Mathematics, 1992. ISBN 0-89871-295-5
• Harald G. Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc. 84(1978), no. 6, 957--1041
• Oto Strauch and Štefan Porubský, Distribution of Sequences: A Sampler, Peter Lang Publishing House, Frankfurt am Main 2005, ISBN 3-631-54013-2