Is this n-body problem O(n^2) or O(n log n)?

Question

I'm writing an article on the n-body problem, and I'd like to be technically accurate.

The code is here . And here are the comments and loops:

/**
 * Given N bodies with mass, in a 3d space, calculate the forces of gravity to be applied to each body.  
 * 
 * This function is exported to JavaScript, so only takes/returns numbers and arrays.
 * For N bodies, pass and array of 4N values (x,y,z,mass) and expect a 3N array of forces (x,y,z)
 * Those forcess can be applied to the bodies mass to update the its position in the simulation.
 * Calculate the 3-vector each unique pair of bodies applies to each other.
 * 
 *   0 1 2 3 4 5
 * 0   x x x x x
 * 1     x x x x
 * 2       x x x
 * 3         x x
 * 4           x
 * 5
 * 
 * Sum those forces together into an array of 3-vector x,y,z forces
 * 
 * Return 0 on success
 */

 // For all bodies:

  for (let i: i32 = 0; i < numBodies; i++) {                   // TypeScript.  i32 is type 32bit int
    // Given body i: pair with every body[j] where j > i
    for (let j: i32 = i + 1; j < numBodies; j++) {             // is this "n" or "log n"?
      // Calculate the force the bodies apply to one another
      stuff = stuff
    }
  }
  return stuff

I'm fairly certain the algorithm is > O(n) and <= O(n*n).

By process of elimination that leaves O(n log n) as the other option.

Looking at the grid, I think O(1/2 n^2) = O(n^2)

Looking at the loops, I think the inner loop is < n , but I'm not sure if it's all the way to log n .

If I'm looping through n, what does adding a log n inner loop look like? If not an inner loop, an outer loop?

Answer 1

Assuming that Calculate the force the bodies apply to one another is an O(1) operation then what you have is the following summation.