5.5b Monte Carlo integration labs

Source: Lab ideas from Computational Physics (Landau, Páez, Bordeianu), §5.5. Rewritten as original drills. Continues 5.5 Monte Carlo integration.

Estimate $\pi$ by stone throwing and $\int_0^1 e^{-x}\,dx$ by mean-value Monte Carlo. Graded randomness uses the chapter LCG only — not random.random().

Fixed LCG

r ← (1103515245 * r + 12345) % 2**31, then uniform r / 2**31. Default seed is 1 unless the prompt says otherwise. Advance the state before each draw (seed itself is not a sample).

Formula refresh

Warm up: pick the matching equation among lookalikes (same walk as the theory lesson).

Labs

Activity: Stone-throw $\pi$ ($N=10000$)

Draw $N$ points uniformly in $[-1,1]^{2}$ with the chapter LCG (seed $1$). Count hits with $x^{2}+y^{2}\le 1$. Print

4 * hits / N

with 6 decimals.

No input. Use $N=10000$.

Write-ups

0/4 solved

5.5b.W1

Using “fraction”, “area”, and “box”, explain:

In one sentence: how does stone throwing estimate a pond’s area?

Minimum 3 words. Your answer must include: “fraction”, “area”, “box”.

0 words/ 3 min

Meet the writing requirements, then you can open the model answer.

Explorers

# mc_stone.py — chapter LCG stone throw (original sketch)
M = 2**31

def rnd(r):
    r = (1103515245 * r + 12345) % M
    return r, r / M

def pi_est(N, seed=1):
    r, hits = seed, 0
    for _ in range(N):
        r, u = rnd(r)
        r, v = rnd(r)
        x, y = 2 * u - 1, 2 * v - 1
        if x * x + y * y <= 1:
            hits += 1
    return 4 * hits / N

Next: 5.6 Mean-value theorem and high-D Monte Carlo.