Monte Carlo Simulation Part.1

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Monte Carlo Simulation Part.1

• Dept. Phys., Tunghai Univ.
• Numerical Methods, C. T. Shih

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?????????????

• If we want to find the numerical integration
• We can simply divide the region 0,1 evenly into
  M slices and the integral can be approximated by

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Simple Sampling

• On the other hand, we can select xn for
  n1,2,...,M from a uniform random number
  generator(?????)
• If M is large enough, xn is set of numbers
  uniformly distributed in the region 0,1

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Simple Sampling (conti.)

• The integral can be approximated by
• Now the error in the integral comes from the
  fluctuation of the distribution of xn

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f(x)
0
1
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Random Number Generator

• Indeed, there is no real random number generator
  in a computer program the program always
  follows the commands by the programmer
• We can only have a pseudo-random number
  generator
• A good random number generator very long period,
  good randomness (small correlation), fast

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Pseudo-RNG

• Long period what is the possible longest period
  on a computer?
• Answer the largest integer, i.e., 231-1
  2,147,483,647 on a 32-bit computer

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Linear Congruent Scheme

• The random numbers are generated by
• a, b, c are magic numbers which determine the
  quality of the RNG
• One possible choice (tested OK) a 75 16807,
  b 0, and c 231-1 2,147,483,647
• Also you need a seed of your RNG as the initial
  x

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Procedure of the RNG

• ??????(????TIME()??)??ISEED,???x0
• ?? ??x1
• ?x1????ISEED,???? random number ? x1/c
• ????,???????? (0,1) ???????

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Overflow ??

• ?? ????
  overflow,?????????

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??

• ??Monte Carlo??????????,M(???? 100, 1000, 1000),a
  ? 0.2, 0.5, 1, 5, 10,????????????

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Alternative

• ??????????????????????
• ??????????(x,y),x ????????? a,b ??,y ???? c,d
  ??,? c ??? f(x) ? a,b ?????,d ???f(x) ? a,b
  ?????
• ? 0ltyltf(x) ? N ???,? f(x)ltylt0? N- ???,????? N
  ???,????????

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???????????

• ???????? N ???,?????S
• ?????? M ?,???? M ? Si, i1,2,...,M
• ?? M ? S ???????? ltSgt
• ???