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I am taking a Numerical Computation class, and we are currently learning about Newton's Method for finding the roots of a system of non-linear equations. I have no problems understanding how the algorithm works, but as an engineer I work better when I can think of a concrete application for concepts.

Can anyone give a few interesting "real world" examples of when finding the roots of a system of equations is useful?

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up vote 4 down vote accepted

Interior-point algorithms for Linear Programming and related programs are a version of Newton's method. Apparently Linear Programs do arise in practice.

Trajectories of projectiles and spacecrafts are also found through solving systems of differential equations.

Roots of polynomials and eigenvalues of matrices are found numerically, since Gaussian elimination is not stable.

Non-linear equations frequently arise in physics and engineering. As you progress in your studies, you'll find out that most equations cannot be solved analytically (unless they're linear), and one needs resort to numerical methods.

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