Monte Carlo Algorithm: Numerical Computation through Random Sampling

The Monte Carlo algorithm is a numerical computation method based on random sampling, widely applied in probability statistics, financial modeling, physical simulations, and other fields. Its core principle involves approximating theoretical results through extensive random experiments, making it particularly suitable for solving complex problems that are difficult to address analytically.

In MATLAB implementation, the Monte Carlo algorithm primarily relies on random number generation functions such as `rand` or `randn` to simulate uniformly distributed or normally distributed random variables. By setting the number of iterations, one can control the simulation's precision—increased experiments typically yield results closer to the true value. Key applications include calculating π, solving integrals, and pricing financial derivatives.

For example, to estimate π, one can randomly scatter points within a unit square and approximate the value by calculating the proportion of points falling inside the inscribed circle. Similarly, high-dimensional integration problems can be transformed into averaging through random sampling in multidimensional spaces. MATLAB's vectorization operations efficiently handle such batch computations, avoiding explicit loops to enhance performance.

Practical considerations include: first, random number quality impacts result accuracy—setting a fixed seed (`rng`) ensures reproducibility when necessary; second, sampling scale should be adjusted according to problem dimensionality to prevent the "curse of dimensionality" from slowing convergence. The Monte Carlo method's advantages lie in simple implementation and parallel-friendliness, but require balancing computational cost with precision requirements.