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26 | <div class="titlepage"><div><div><h3 class="title"> | |
27 | <a name="math_toolkit.roots.bad_roots"></a><a class="link" href="bad_roots.html" title="Examples Where Root Finding Goes Wrong">Examples Where Root Finding | |
28 | Goes Wrong</a> | |
29 | </h3></div></div></div> | |
30 | <p> | |
31 | There are many reasons why root root finding can fail, here are just a few | |
32 | of the more common examples: | |
33 | </p> | |
34 | <h4> | |
35 | <a name="math_toolkit.roots.bad_roots.h0"></a> | |
36 | <span class="phrase"><a name="math_toolkit.roots.bad_roots.local_minima"></a></span><a class="link" href="bad_roots.html#math_toolkit.roots.bad_roots.local_minima">Local | |
37 | Minima</a> | |
38 | </h4> | |
39 | <p> | |
40 | If you start in the wrong place, such as z<sub>0</sub> here: | |
41 | </p> | |
42 | <p> | |
43 | <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../roots/bad_root_1.svg" width="372.047244094489" height="262.204724409449"></object></span> | |
44 | </p> | |
45 | <p> | |
46 | Then almost any root-finding algorithm will descend into a local minima rather | |
47 | than find the root. | |
48 | </p> | |
49 | <h4> | |
50 | <a name="math_toolkit.roots.bad_roots.h1"></a> | |
51 | <span class="phrase"><a name="math_toolkit.roots.bad_roots.flatlining"></a></span><a class="link" href="bad_roots.html#math_toolkit.roots.bad_roots.flatlining">Flatlining</a> | |
52 | </h4> | |
53 | <p> | |
54 | In this example, we're starting from a location (z<sub>0</sub>) where the first derivative | |
55 | is essentially zero: | |
56 | </p> | |
57 | <p> | |
58 | <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../roots/bad_root_2.svg" width="372.047244094489" height="262.204724409449"></object></span> | |
59 | </p> | |
60 | <p> | |
61 | In this situation the next iteration will shoot off to infinity (assuming | |
62 | we're using derivatives that is). Our code guards against this by insisting | |
63 | that the root is always bracketed, and then never stepping outside those | |
64 | bounds. In a case like this, no root finding algorithm can do better than | |
65 | bisecting until the root is found. | |
66 | </p> | |
67 | <p> | |
68 | Note that there is no scale on the graph, we have seen examples of this situation | |
69 | occur in practice <span class="emphasis"><em>even when several decimal places of the initial | |
70 | guess z<sub>0</sub> are correct.</em></span> | |
71 | </p> | |
72 | <p> | |
73 | This is really a special case of a more common situation where root finding | |
74 | with derivatives is <span class="emphasis"><em>divergent</em></span>. Consider starting at | |
75 | z<sub>0</sub> in this case: | |
76 | </p> | |
77 | <p> | |
78 | <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../roots/bad_root_4.svg" width="372.047244094489" height="262.204724409449"></object></span> | |
79 | </p> | |
80 | <p> | |
81 | An initial Newton step would take you further from the root than you started, | |
82 | as will all subsequent steps. | |
83 | </p> | |
84 | <h4> | |
85 | <a name="math_toolkit.roots.bad_roots.h2"></a> | |
86 | <span class="phrase"><a name="math_toolkit.roots.bad_roots.micro_stepping_non_convergence"></a></span><a class="link" href="bad_roots.html#math_toolkit.roots.bad_roots.micro_stepping_non_convergence">Micro-stepping | |
87 | / Non-convergence</a> | |
88 | </h4> | |
89 | <p> | |
90 | Consider starting at z<sub>0</sub> in this situation: | |
91 | </p> | |
92 | <p> | |
93 | <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../roots/bad_root_3.svg" width="372.047244094489" height="262.204724409449"></object></span> | |
94 | </p> | |
95 | <p> | |
96 | The first derivative is essentially infinite, and the second close to zero | |
97 | (and so offers no correction if we use it), as a result we take a very small | |
98 | first step. In the worst case situation, the first step is so small - perhaps | |
99 | even so small that subtracting from z<sub>0</sub> has no effect at the current working | |
100 | precision - that our algorithm will assume we are at the root already and | |
101 | terminate. Otherwise we will take lot's of very small steps which never converge | |
102 | on the root: our algorithms will protect against that by reverting to bisection. | |
103 | </p> | |
104 | <p> | |
105 | An example of this situation would be trying to find the root of e<sup>-1/z<sup>2</sup></sup> - | |
106 | this function has a single root at <span class="emphasis"><em>z = 0</em></span>, but for <span class="emphasis"><em>z<sub>0</sub> < | |
107 | 0</em></span> neither Newton nor Halley steps will ever converge on the root, | |
108 | and for <span class="emphasis"><em>z<sub>0</sub> > 0</em></span> the steps are actually divergent. | |
109 | </p> | |
110 | </div> | |
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113 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
114 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
115 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
116 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
117 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
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