Provide a N-body integration algorithm based on RK4

PROJECT SPECIFICATIONS (Please read carefully) In recent years, some projects have taken place, in order to develop smart algorithms that allows you to integrate the orbits of an arbitrary number of stars. These algorithms are the physics engine of gravity simulators (also known as n-body simulators) and provide us a way to study the motion of celestial bodies under the influence of their mutual gravitational forces. Most of these integration schemes are written in C++ and are generally well documented. Your mission is to perform a research of open source code available regarding this issue, and provide, and arrange if needed, a C++ function, fully ready to be feeded with initial particle parameters (masses, positions, and velocities) and capable to integrate particle motion during a given time (total duration of the integration) returning the final positions and velocities for each particle. At this stage it is not necessary to provide solution for collisions. The number of particles before and after the integration must be the same, but in exchange for this, the integration scheme should be ready to shrink the internal step size length (shared for all particles) in order to prevent a lack of precision or unstable behavior on close interaction. The RK4 integration scheme has to be delivered under a stand-alone format and in a single C++ source file. Please, comment your code lines so that I can understand it. Simplicity and few code lines will be very much appreciated. Header files are allowed but ask for my approval before including them. DLL's are not allowed. IMPLEMENTATION DETAILS To make it easy for implementation pourposes and regardless the number of functions, there must exist a main function. (I will call only this function during the implementation) The input values for the main function will be particle masses, positions, velocities and the duration of the integration. What this main function has to do, is to update the position and velocity of each particle after the desired integration time. (without collisions masses remain constant) Use gravitational constant = 1 Number of spatial dimensions= 3 Variable precision: easily customizable. float, double, long double. Time-step type: Shared for all particles Integrator: Runge-Kutta Order 4 Number of particles to be computed: > 10^7 Implementation for Parallel Processing: Valuable but not strictly necessary at this point. Octree type accelerator: Not necessary at this point. Function input / output: Use any method you need for testing the algorithm. After delivery I will use my own methods. I just need a clear identification of the variables that store particle masses, positions and velocities, in order to feed the function, and recover the updated values after the integration process. TESTING THE ALGORITHM The provided RK4 algorithm must be fully tested before delivery. I would like to avoid the possibility of being delivered successive code updates pretending that maybe one of them will work. Please notice that I don't pretend to deeply understand the mathematical fundamentals of the integration scheme. However if it is built and delivered according to the specifications, I can easily perform a deep test and rate its performance, precision and stability. I will need more or less one week for this from delivery. METHOD OF PAYMENT After the awarding create a milestone for the total amount of the project. It will be released only at the end of the job. Best regards and good luck