Editing Numerical Methods (section)

== <span style="color: #FFFFFF;">Understanding</span> ==
Numerical methods are about the '''Trade-off between Speed and Accuracy'''.

'''1. The Why''':
Many equations in the real world are "Impossible" to solve with a pen and paper. For example, the 3-Body Problem (how three planets move together) has no exact formula. To find the answer, we have a computer calculate the position for the next 1 second, then the next, and the next.

'''2. The How (Tiny Steps)''':
Imagine you are walking in the dark. You can't see the destination, but you can feel the slope of the ground.
* '''Euler's Method''': You take a 1-meter step in the direction of the slope.
* '''Runge-Kutta (RK4)''': You "check" the slope four times before you take the step to be much more accurate.

'''3. Error Control''':
Numerical math is the only math that studies its own "Failure."
* '''Discretization Error''': If your "Step" is 1 meter, you might skip over a small hole. If your step is 0.001 meters, you are more accurate, but it takes 1000x longer to compute.
* '''Numerical Stability''': Sometimes a tiny error in Step 1 grows into a massive error in Step 100. A "Stable" method ensures errors stay small.

'''Floating Point Limits''': Computers don't know what "0.1" is. They store it as a binary fraction that is slightly off. If you add "0.1" to itself 10,000 times, you won't get 1,000. This is why banks and rocket scientists use specific "Arbitrary Precision" libraries.
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