How to efficiently calculate the symmetric force matrix (Newton's 3rd law)?

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Intro

For the N-body simulation I need to calculate the total received force Fi for each body it receives from the other bodies. This is an O(n^2) problem, because for each body the pairwise total force must be calculated. Because of the Newton's 3rd axiom fi,j = -fi,j we can reduce the number of force calculations by half.

Problem

How to implement this law in my code in order to optimize the acceleration calculation?

What I have did so far?

I programmed the total received force for each body, but without that optimization.

std::vector<glm::vec3> SequentialAccelerationCalculationImpl::calcAccelerations(
        const std::vector<Body> &bodies,
        const float softening_factor
) {
    const float softening_factor_squared = softening_factor * softening_factor;
    const size_t num_bodies = bodies.size();
    std::vector<glm::vec3> accelerations(num_bodies);
    // O(n^2)
    for (size_t i = 0; i < num_bodies; ++i) {
        glm::vec3 received_total_force(0.0);
        for (size_t j = 0; j < num_bodies; ++j) { // TODO this can be reduced to half 
            if (i != j) {
                const glm::vec3 distance_vector = bodies[i].getCurrentPosition() - bodies[j].getCurrentPosition();
                float distance_squared =
                        (distance_vector.x * distance_vector.x) +
                        (distance_vector.y * distance_vector.y) +
                        (distance_vector.z * distance_vector.z);
                // to avoid zero in the following division
                distance_squared += softening_factor_squared;
                received_total_force += ((bodies[j].getMass() / distance_squared) * glm::normalize(distance_vector));
            }
        }
        accelerations[i] = GRAVITATIONAL_CONSTANT * received_total_force;
    }
    return accelerations;
}

I also sketched the force matrix: enter image description here

physics matrix optimization algorithm c++
1 Answers

The simple solution is to have the inner loop start from i + 1 instead of from 0:

for (size_t i = 0; i < num_bodies; ++i) {
    glm::vec3 received_total_force(0.0);
    for (size_t j = i + 1; j < num_bodies; ++j) {
        glm::vec3 distance_squared = glm::distance2(bodies[i].getCurrentPosition(), bodies[j].getCurrentPosition());
        ...
    }
}

However now you have to ensure you update both accelerations[i] and accelerations[j] inside the inner loop, instead of first accumulating in received_total_force:

accelerations[i] += bodies[j].getMass() / distance_squared * glm::normalize(distance_vector);
accelerations[j] -= bodies[i].getMass() / distance_squared * glm::normalize(distance_vector);
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