Runge-Kutta methods--RK4 being the bread-and-butter ODE solver--were invented in the late 1800s/early 1900s.
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Oh, tons and tons of them.
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My favorite example: back when they were doing the Lord of the Rings movies, they used an algorithm to generate the CGI of the armies
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This algorithm would simulate sword & shield combat, that's how the battle scenes look so dynamic and realistic.
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However, due to imbalance in the payoff computations, the first time they ran it, all the soldiers realized the best action was to run away.
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So both armies marched up and promptly fled each other.
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