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Boudreau and Swanson: Applied Computational Physics

Chapter 07: Monte Carlo Methods

example code (.tgz)

• HIST: a simple demonstration of how to generate and visualize a uniform variate in a Qat histogram.
• THROWAWAY1: demonstrate how to sample a simple probability distribution using the throwaway (or von Neumann rejection) method.
• THROWAWAY2: demonstrate how to sample a point probability distribution, using the throwaway (or von Neumann rejection) method.
• DIRECTSAMPLE: demonstrates the direct sampling method, otherwise known as the transformation method. A choice of three examples is made from the command line. These three examples are discussed in the text.
• BASICMCINTEGRATE: example in section 7.4.2 of the text, where Monte Carlo integration is discussed.
• MCMC/HYDROGEN, MCMC/HYDROGEN-ANIM, and MCMC/HYDROGEN-PLOT: use Markov chain Monte Carlo to generate a realization of the probability density function of hydrogen orbitals.

usage:
mcmc-hydrogen [NPOINTS=val{def=10000}] [N=val{def=0}] [L=val{def=0}] [M=val{def=0}] [fSigma=val{def=1.0}
mcmc-hydrogen-anim [NPOINTS=val{def=10000}] [N=val{def=0}] [L=val{def=0}] [M=val{def=0}] [fSigma=val{def=1.0}
mcmc-hydrogen-plot [NPOINTS=val{def=10000}] [N=val{def=0}] [L=val{def=0}] [M=val{def=0}] [fSigma=val{def=1.0}

NPOINTS Number of points in the Markov Chain (10000)
N Principal Quantum Number (1)
L Angular Momentum Quantum Number (0)
M Magnetic Quantum Number (0)
fSigma Under/over scale the proposal distribution

The animated version ("mcmc-hydrogen-anim") refreshes after 2000 steps. It's fSigma parameter is not ideal for a decent sampling of the distribution, but the purpose of the example is to illustrate burn-in.
• MCMC/SMOKE: a simulation of smoke particles diffusing throughout a rectangular space.
• BWDOPPLER: the starting point for an exercise in chapter 7.
• INTPOW: integrates xN over [0,1] using Monte Carlo integration.
• INTPULL: monitors the pull distributions from the ensemble of statistically independent evaluations of the integral above.
• SUMUNIFORM: plots the sum of one, two, three, and four uniform variates.
• X5: illustrates the transformation method to generate variables according to xN. The exponent is given on the command line. About this book