Numerical methods

Numerical methods are the techniques computers use to get approximate answers to math problems that have no neat formula, like simulating a planet’s orbit or solving a huge system of equations. Because computers use finite-precision numbers and take finite steps, every method introduces small errors, and the real question is whether those errors stay small or grow. Euler’s method for simulating motion takes a step in the current direction each time; it’s easy but drifts, so a simulated orbit slowly spirals outward. Runge-Kutta (RK4) samples the slope several times per step and stays accurate far longer. A related idea is conditioning: some problems amplify tiny input errors no matter how good your method is.