Newton's method for solving the equation is an iterative numeric technique defined by
This "formula" expresses the simple idea that the -intercept on a tangent line drawn at near an -intercept for the graph of , is generally closer to the intercept of than . Indeed, the point-slope form of the equation of the line tangent at is
The -intercept on this line is the solution for when . This becomes by the calculation in Table 3.2.1.
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Table 3.2.1 Derivation of the Newton iteration formula
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In general, Newton's method converges rapidly to a root of provided the initial point is chosen "close enough" to the root. However, the method is not infallible, and there are functions for which the method will converge more slowly, will cycle back and forth between two distinct iterates, will seem to converge but not to a root, and will fail to find an actual root. One of these anomalies is the subject of Example 3.2.3.