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A Solving a Single Nonlinear Algebraic Equation

In this appendix I briefly outline numerical algorithms and codes for solving a single 1-dimensional nonlinear algebraic equation f(x) = 0, where the continuous function f : ℜ → ℜ is given.

The process generally begins by evaluating f on a suitable grid of points and looking for sign changes. By the intermediate value theorem, each sign change must bracket at least one root. Given a pair of such ordinates x− and x+, there are a variety of algorithms available to accurately and efficiently find the (a) root:

If |x+ − x− | is small, say on the order of a finite-difference grid spacing, then closed-form approximations are probably accurate enough:

For larger |x+ − x− |, iterative algorithms are necessary to obtain an accurate root:

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