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<html>
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<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
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<title>The Effect of a Poor Initial Guess</title>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.roots.bad_guess"></a><a class="link" href="bad_guess.html" title="The Effect of a Poor Initial Guess">The Effect of a Poor Initial
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Guess</a>
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</h3></div></div></div>
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<p>
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It's instructive to take our "toy" example algorithms, and use
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deliberately bad initial guesses to see how the various root finding algorithms
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fair. We'll start with the cubed root, and using the cube root of 500 as
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the test case:
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</p>
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<div class="informaltable"><table class="table">
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<colgroup>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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<p>
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Initial Guess=
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</p>
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</th>
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<th>
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<p>
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-500% (≈1.323)
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</p>
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</th>
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<th>
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<p>
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-100% (≈3.97)
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</p>
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</th>
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<th>
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<p>
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-50% (≈3.96)
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</p>
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</th>
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<th>
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<p>
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-20% (≈6.35)
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</p>
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</th>
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<th>
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<p>
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-10% (≈7.14)
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</p>
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</th>
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<th>
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<p>
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-5% (≈7.54)
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</p>
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</th>
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<th>
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<p>
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5% (≈8.33)
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</p>
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</th>
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<th>
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<p>
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10% (≈8.73)
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</p>
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</th>
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<th>
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<p>
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20% (≈9.52)
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</p>
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</th>
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<th>
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<p>
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50% (≈11.91)
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</p>
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</th>
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<th>
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<p>
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100% (≈15.87)
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</p>
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</th>
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<th>
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<p>
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500 (≈47.6)
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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bracket_and_solve_root
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</p>
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</td>
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<td>
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<p>
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12
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</p>
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</td>
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<td>
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<p>
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8
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</p>
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</td>
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<td>
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<p>
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8
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</p>
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</td>
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<td>
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<p>
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10
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</p>
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</td>
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<p>
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11
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</p>
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</td>
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11
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</p>
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</td>
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<td>
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<p>
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11
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</p>
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</td>
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<p>
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11
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</p>
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</td>
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<p>
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11
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</p>
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</td>
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<td>
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<p>
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11
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</p>
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</td>
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<td>
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<p>
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7
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</p>
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</td>
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<td>
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<p>
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13
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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newton_iterate
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</p>
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</td>
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<td>
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<p>
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12
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</p>
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</td>
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<td>
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<p>
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7
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</p>
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</td>
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<td>
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<p>
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7
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</p>
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</td>
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<p>
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5
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</p>
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</td>
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5
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</p>
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4
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4
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</p>
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5
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</p>
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5
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</p>
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</td>
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6
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</p>
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</td>
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<p>
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7
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</p>
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</td>
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9
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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halley_iterate
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</p>
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</td>
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<td>
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<p>
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7
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4
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3
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3
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<p>
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schroder_iterate
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11
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3
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3
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</td>
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</tr>
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</tbody>
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</table></div>
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<p>
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As you can see <code class="computeroutput"><span class="identifier">bracket_and_solve_root</span></code>
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is relatively insensitive to starting location - as long as you don't start
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many orders of magnitude away from the root it will take roughly the same
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number of steps to bracket the root and solve it. On the other hand the derivative-based
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methods are slow to start, but once they have some digits correct they increase
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precision exceptionally fast: they are therefore quite sensitive to the initial
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starting location.
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</p>
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<p>
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The next table shows the number of iterations required to find the second
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radius of an ellipse with first radius 50 and arc-length 500:
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</p>
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<div class="informaltable"><table class="table">
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<colgroup>
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<thead><tr>
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<th>
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<p>
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Initial Guess=
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</p>
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</th>
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<th>
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<p>
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-500% (≈20.6)
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</p>
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</th>
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<th>
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<p>
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-100% (≈61.81)
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</p>
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</th>
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<th>
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<p>
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-50% (≈61.81)
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</p>
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</th>
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<th>
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<p>
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-20% (≈98.9)
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</p>
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</th>
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<th>
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<p>
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-10% (≈111.3)
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</p>
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</th>
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<th>
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<p>
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-5% (≈117.4)
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</p>
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</th>
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<th>
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<p>
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5% (≈129.8)
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</p>
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</th>
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<th>
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<p>
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10% (≈136)
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</p>
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</th>
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<th>
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<p>
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20% (≈148.3)
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</p>
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</th>
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<th>
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<p>
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50% (≈185.4)
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</p>
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</th>
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<th>
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<p>
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100% (≈247.2)
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</p>
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</th>
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<th>
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<p>
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500 (≈741.7)
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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bracket_and_solve_root
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</p>
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</td>
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<td>
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<p>
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11
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</p>
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5
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8
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</p>
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8
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7
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7
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8
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9
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8
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6
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10
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</tr>
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<tr>
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<p>
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newton_iterate
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</p>
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</td>
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<td>
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<p>
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4
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4
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</td>
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<tr>
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<p>
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halley_iterate
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</p>
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</td>
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<td>
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<p>
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4
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3
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<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 © 2006-2010, 2012-2014 Nikhar Agrawal,
|
|
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
|
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
|
Benjamin Sobotta, 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>)
|
|
</p>
|
|
</div></td>
|
|
</tr></table>
|
|
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