Editing Newton's method (section)

===Oscillatory behavior===
[[Image:NewtonsMethodConvergenceFailure.svg|thumb|upright=1.4|The tangent lines of {{math|{{var|x}}{{sup|3}} − 2{{var|x}} + 2}} at 0 and 1 intersect the {{mvar|x}}-axis at 1 and 0 respectively, illustrating why Newton's method oscillates between these values for some starting points.]]

It is easy to find situations for which Newton's method oscillates endlessly between two distinct values. For example, for Newton's method as applied to a function {{mvar|f}} to oscillate between 0 and 1, it is only necessary that the tangent line to {{mvar|f}} at 0 intersects the {{mvar|x}}-axis at 1 and that the tangent line to {{mvar|f}} at 1 intersects the {{mvar|x}}-axis at 0.<ref name="judd" /> This is the case, for example, if {{math|''f''(''x'') {{=}} ''x''<sup>3</sup> − 2''x'' + 2}}. For this function, it is even the case that Newton's iteration as initialized sufficiently close to 0 or 1 will ''asymptotically'' oscillate between these values. For example, Newton's method as initialized at 0.99 yields iterates 0.99, −0.06317, 1.00628, 0.03651, 1.00196, 0.01162, 1.00020, 0.00120, 1.000002, and so on. This behavior is present despite the presence of a root of {{mvar|f}} approximately equal to −1.76929.