Many scientific and engineering phenomena are characterized by nonlinear behavior. We believe, determining the solutions of those nonlinear equations is a fundamental problem in scientific and engineering analysis.

Numerical methods are all iterative in nature, and may be used for equations that contain one or several variables. The simplest case is to find the single root of a single nonlinear function. The following equation represents the general form of such a function:

In general, a nonlinear equation may have any number of solutions or no solutions at all. All approaches to solving nonlinear equations are iterative procedures that require a start point, i.e. an initial solution. At Keystone Mining Post we will show that the initial solution can be very critical; a bad initial value may fail to converge or it may converge to a wrong solution.

For nonlinear equations, there is no general method for estimating the value of a solution. If the nonlinear equation is associated with a real-world scientific and engineering problem, then the physical insight of the problem might suggest the approximate location of the solution. Otherwise, one must carry out a systematic numerical search for the solutions.

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