Q: Why are numerical methods necessary? If we can’t get exact solutions, then how do we know when our approximate solutions are any good?
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A trick that lets you get closer and closer to an exact answer is a “numerical method”. Numerical methods do something rather bizarre: they find solutions close to the answer without ever knowing what that answer is. As such, an important part of every numerical method is a proof that it works. Numerical methods have really come into their own with the advent of computers, but the idea is a lot older. Some numerical methods involve a degree of randomness and yet still manage to produce useful results.