High-Dimensional Ellipsoid (Monte Carlo)

Computer Science Level 2

In six-dimensional space, with spatial parameters $(A,B,C,D,E,F)$ , there is the following $6D$ ellipsoid:

$\large{\frac{A^2}{1^2} + \frac{B^2}{2^2} + \frac{C^2}{3^2} + \frac{D^2}{4^2} + \frac{E^2}{5^2} + \frac{F^2}{6^2} = 1}$

What is the $6D$ volume enclosed by the ellipsoid?

Note: Even if you know of a formal solution, try the algorithmic approach to see how cool it is

The answer is 3720.753.

1 solution

Maurice van Peursem
Oct 13, 2018

Actual volume: $V=\frac{6!\pi^3}{6}$

Python3:

from random import random

Fact = 2 * 4 * 6 * 8 * 10 * 12  # volume of surrounding block
for n in range(1, 9):
    Tries = 10 ** n     # try increasing number of samples
    Count = 0           # number of samples inside ellipsoid
    for Try in range(Tries):
        A = random()
        B = random() * 2
        C = random() * 3
        D = random() * 4
        E = random() * 5
        F = random() * 6
        if A * A + B * B / 4 + C * C / 9 + D * D / 16 + E * E / 25 + F * F / 36 <= 1:
            Count += 1
    print(Tries, Count, Count / Tries * Fact)

High-Dimensional Ellipsoid (Monte Carlo)

The answer is 3720.753.

1 solution

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