Parallel Assignment to A Shared Variable - c++

Question

I'm currently trying to parallelize a particle swarm algorithm, and can't find an efficient way to handle a = inside a loop. I didn't do the most simple form of just putting #pragma omp parallel for at the start of the loop since i expect that will just lead to false sharing problems. The whole thing gets complicated by the variables being Matrices and Vectors.
I think this minimal example (using armadillo, a linear algebra library similar to shows it better then i can describe it:
#include <omp.h>
#include <math.h>
#include <armadillo>
using namespace std;
int main(int argc, char** argv) {
arma::uword dimensions = 10, particleCount = 40;
//Matrix with dimensions-rows, and particleCount-columns. initialized with 1
arma::Mat<double> positions = arma::ones(dimensions, particleCount);
//same as above, but initialized with 2. +,/,*,- are elementwise;like in matlab
arma::Mat<double> velocities = arma::ones(dimensions, particleCount) * 2;
for(arma::uword n = 0; n < particleCount; n++) {
//.col(n) gets the nth column of the matrix
arma::Col<double> newVelocity = std::rand() * velocities.col(n);
//there is a lot more math done here, but all of it is read-only access to other variables
positions.col(n) += newVelocity; //again elementwise
velocities.col(n) = newVelocity;
}
return 0;
}
My first idea was to do something like this, but it's horribly inefficient:
#include <omp.h>
#include <math.h>
#include <armadillo>
using namespace std;
int main(int argc, char** argv) {
arma::uword dimensions = 10, particleCount = 40;
//these two variables cannot be moved inside the parallel region :-/
arma::Mat<double> positions = arma::ones(dimensions, particleCount);
arma::Mat<double> velocities = arma::ones(dimensions, particleCount) * 2;
#pragma omp parallel
{
arma::Mat<double> velocity_private = arma::zeros(dimensions,particleCount);
#pragma omp for
for(arma::uword n = 0; n < particleCount; n++) {
arma::Col<double> newVelocity = std::rand() * velocities.col(n);
velocity_private.col(n) = newVelocity;
}
#pragma omp single
{
//first part of workaround for '='
velocities = arma::zeros(dimensions, particleCount);
}
#pragma omp critical
{
for(arma::uword n = 0; n < particleCount; n++) {
positions.col(n) += velocity_private.col(n);
//second part of workaround for '='
velocities.col(n) += velocity_private.col(n);
}
}
}//end omp parallel
return 0;
}
I thought about using user-defined reductions, but i didn't find any example for assignments. All of them where for additions or multiplications, and unfortunately not very accessible.
Any suggestion(s) or advice appreciated! :)