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- <title>The Effect of a Poor Initial Guess</title>
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- <div class="section">
- <div class="titlepage"><div><div><h2 class="title" style="clear: both">
- <a name="math_toolkit.bad_guess"></a><a class="link" href="bad_guess.html" title="The Effect of a Poor Initial Guess">The Effect of a Poor Initial Guess</a>
- </h2></div></div></div>
- <p>
- It's instructive to take our "toy" example algorithms, and use deliberately
- bad initial guesses to see how the various root finding algorithms fair. We'll
- start with the cubed root, and using the cube root of 500 as the test case:
- </p>
- <div class="informaltable"><table class="table">
- <colgroup>
- <col>
- <col>
- <col>
- <col>
- <col>
- <col>
- <col>
- <col>
- <col>
- <col>
- <col>
- <col>
- <col>
- </colgroup>
- <thead><tr>
- <th>
- <p>
- Initial Guess=
- </p>
- </th>
- <th>
- <p>
- -500% (≈1.323)
- </p>
- </th>
- <th>
- <p>
- -100% (≈3.97)
- </p>
- </th>
- <th>
- <p>
- -50% (≈3.96)
- </p>
- </th>
- <th>
- <p>
- -20% (≈6.35)
- </p>
- </th>
- <th>
- <p>
- -10% (≈7.14)
- </p>
- </th>
- <th>
- <p>
- -5% (≈7.54)
- </p>
- </th>
- <th>
- <p>
- 5% (≈8.33)
- </p>
- </th>
- <th>
- <p>
- 10% (≈8.73)
- </p>
- </th>
- <th>
- <p>
- 20% (≈9.52)
- </p>
- </th>
- <th>
- <p>
- 50% (≈11.91)
- </p>
- </th>
- <th>
- <p>
- 100% (≈15.87)
- </p>
- </th>
- <th>
- <p>
- 500 (≈47.6)
- </p>
- </th>
- </tr></thead>
- <tbody>
- <tr>
- <td>
- <p>
- bracket_and_solve_root
- </p>
- </td>
- <td>
- <p>
- 12
- </p>
- </td>
- <td>
- <p>
- 8
- </p>
- </td>
- <td>
- <p>
- 8
- </p>
- </td>
- <td>
- <p>
- 10
- </p>
- </td>
- <td>
- <p>
- 11
- </p>
- </td>
- <td>
- <p>
- 11
- </p>
- </td>
- <td>
- <p>
- 11
- </p>
- </td>
- <td>
- <p>
- 11
- </p>
- </td>
- <td>
- <p>
- 11
- </p>
- </td>
- <td>
- <p>
- 11
- </p>
- </td>
- <td>
- <p>
- 7
- </p>
- </td>
- <td>
- <p>
- 13
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- newton_iterate
- </p>
- </td>
- <td>
- <p>
- 12
- </p>
- </td>
- <td>
- <p>
- 7
- </p>
- </td>
- <td>
- <p>
- 7
- </p>
- </td>
- <td>
- <p>
- 5
- </p>
- </td>
- <td>
- <p>
- 5
- </p>
- </td>
- <td>
- <p>
- 4
- </p>
- </td>
- <td>
- <p>
- 4
- </p>
- </td>
- <td>
- <p>
- 5
- </p>
- </td>
- <td>
- <p>
- 5
- </p>
- </td>
- <td>
- <p>
- 6
- </p>
- </td>
- <td>
- <p>
- 7
- </p>
- </td>
- <td>
- <p>
- 9
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- halley_iterate
- </p>
- </td>
- <td>
- <p>
- 7
- </p>
- </td>
- <td>
- <p>
- 4
- </p>
- </td>
- <td>
- <p>
- 4
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 4
- </p>
- </td>
- <td>
- <p>
- 4
- </p>
- </td>
- <td>
- <p>
- 6
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- schroder_iterate
- </p>
- </td>
- <td>
- <p>
- 11
- </p>
- </td>
- <td>
- <p>
- 6
- </p>
- </td>
- <td>
- <p>
- 6
- </p>
- </td>
- <td>
- <p>
- 4
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 4
- </p>
- </td>
- <td>
- <p>
- 5
- </p>
- </td>
- <td>
- <p>
- 5
- </p>
- </td>
- <td>
- <p>
- 8
- </p>
- </td>
- </tr>
- </tbody>
- </table></div>
- <p>
- As you can see <code class="computeroutput"><span class="identifier">bracket_and_solve_root</span></code>
- is relatively insensitive to starting location - as long as you don't start
- many orders of magnitude away from the root it will take roughly the same number
- of steps to bracket the root and solve it. On the other hand the derivative-based
- methods are slow to start, but once they have some digits correct they increase
- precision exceptionally fast: they are therefore quite sensitive to the initial
- starting location.
- </p>
- <p>
- The next table shows the number of iterations required to find the second radius
- of an ellipse with first radius 50 and arc-length 500:
- </p>
- <div class="informaltable"><table class="table">
- <colgroup>
- <col>
- <col>
- <col>
- <col>
- <col>
- <col>
- <col>
- <col>
- <col>
- <col>
- <col>
- <col>
- <col>
- </colgroup>
- <thead><tr>
- <th>
- <p>
- Initial Guess=
- </p>
- </th>
- <th>
- <p>
- -500% (≈20.6)
- </p>
- </th>
- <th>
- <p>
- -100% (≈61.81)
- </p>
- </th>
- <th>
- <p>
- -50% (≈61.81)
- </p>
- </th>
- <th>
- <p>
- -20% (≈98.9)
- </p>
- </th>
- <th>
- <p>
- -10% (≈111.3)
- </p>
- </th>
- <th>
- <p>
- -5% (≈117.4)
- </p>
- </th>
- <th>
- <p>
- 5% (≈129.8)
- </p>
- </th>
- <th>
- <p>
- 10% (≈136)
- </p>
- </th>
- <th>
- <p>
- 20% (≈148.3)
- </p>
- </th>
- <th>
- <p>
- 50% (≈185.4)
- </p>
- </th>
- <th>
- <p>
- 100% (≈247.2)
- </p>
- </th>
- <th>
- <p>
- 500 (≈741.7)
- </p>
- </th>
- </tr></thead>
- <tbody>
- <tr>
- <td>
- <p>
- bracket_and_solve_root
- </p>
- </td>
- <td>
- <p>
- 11
- </p>
- </td>
- <td>
- <p>
- 5
- </p>
- </td>
- <td>
- <p>
- 5
- </p>
- </td>
- <td>
- <p>
- 8
- </p>
- </td>
- <td>
- <p>
- 8
- </p>
- </td>
- <td>
- <p>
- 7
- </p>
- </td>
- <td>
- <p>
- 7
- </p>
- </td>
- <td>
- <p>
- 8
- </p>
- </td>
- <td>
- <p>
- 9
- </p>
- </td>
- <td>
- <p>
- 8
- </p>
- </td>
- <td>
- <p>
- 6
- </p>
- </td>
- <td>
- <p>
- 10
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- newton_iterate
- </p>
- </td>
- <td>
- <p>
- 4
- </p>
- </td>
- <td>
- <p>
- 4
- </p>
- </td>
- <td>
- <p>
- 4
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 4
- </p>
- </td>
- <td>
- <p>
- 4
- </p>
- </td>
- <td>
- <p>
- 4
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- halley_iterate
- </p>
- </td>
- <td>
- <p>
- 4
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 2
- </p>
- </td>
- <td>
- <p>
- 2
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- schroder_iterate
- </p>
- </td>
- <td>
- <p>
- 4
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 2
- </p>
- </td>
- <td>
- <p>
- 2
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- <td>
- <p>
- 3
- </p>
- </td>
- </tr>
- </tbody>
- </table></div>
- <p>
- Interestingly this function is much more resistant to a poor initial guess
- when using derivatives.
- </p>
- </div>
- <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
- <td align="left"></td>
- <td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
- Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
- Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
- Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
- Daryle Walker and Xiaogang Zhang<p>
- Distributed under the Boost Software License, Version 1.0. (See accompanying
- file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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