509f27e5c6
git-svn-id: https://phpexcel.svn.codeplex.com/svn/trunk@59884 2327b42d-5241-43d6-9e2a-de5ac946f064
167 lines
5.9 KiB
PHP
167 lines
5.9 KiB
PHP
<?php
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include_once "includes/header.php";
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include_once "includes/navbar.php";
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?>
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<h2>Magic Square Example</h2>
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<p>
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The Jama distribution comes with a magic square example that is used to
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test and benchmark the LU, QR, SVD and symmetric Eig decompositions.
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The example outputs a multi-column table with these column headings:
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</p>
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<table border='1' cellpadding='5' cellspacing='0' align='center'>
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<tr>
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<td><b>n</b></td>
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<td>Order of magic square.</td>
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</tr>
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<tr>
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<td><b>trace</b></td>
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<td>Diagonal sum, should be the magic sum, (n^3 + n)/2.</td>
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</tr>
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<tr>
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<td><b>max_eig</b></td>
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<td>Maximum eigenvalue of (A + A')/2, should equal trace.</td>
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</tr>
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<tr>
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<td><b>rank</b></td>
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<td>Linear algebraic rank, should equal n if n is odd, be less than n if n is even.</td>
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</tr>
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<tr>
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<td><b>cond</b></td>
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<td>L_2 condition number, ratio of singular values.</td>
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</tr>
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<tr>
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<td><b>lu_res</b></td>
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<td>test of LU factorization, norm1(L*U-A(p,:))/(n*eps).</td>
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</tr>
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<tr>
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<td><b>qr_res</b></td>
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<td>test of QR factorization, norm1(Q*R-A)/(n*eps).</td>
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</tr>
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</table>
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<p>
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Running the Java-based version of the matix square example produces these results:
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</p>
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<table border='1' align='center'>
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<tr>
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<th> n </th>
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<th> trace </th>
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<th> max_eig </th>
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<th> rank </th>
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<th> cond </th>
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<th> lu_res </th>
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<th> qr_res </th>
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</tr>
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<tr>
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<td>3</td><td>15</td><td>15.000</td><td>3</td><td>4.330</td><td>0.000</td><td>11.333</td>
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</tr>
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<tr>
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<td>4</td><td>34</td><td>34.000</td><td>3</td><td> Inf</td><td>0.000</td><td>13.500</td>
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<tr>
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<td>5</td><td>65</td><td>65.000</td><td>5</td><td>5.462</td><td>0.000</td><td>14.400</td>
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</tr>
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<tr>
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<td>6</td><td>111</td><td>111.000</td><td>5</td><td> Inf</td><td>5.333</td><td>16.000</td>
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</tr>
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<tr>
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<td>7</td><td>175</td><td>175.000</td><td>7</td><td>7.111</td><td>2.286</td><td>37.714</td>
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</tr>
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<tr>
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<td>8</td><td>260</td><td>260.000</td><td>3</td><td> Inf</td><td>0.000</td><td>59.000</td>
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</tr>
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<tr>
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<td>9</td><td>369</td><td>369.000</td><td>9</td><td>9.102</td><td>7.111</td><td>53.333</td>
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</tr>
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<tr>
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<td>10</td><td>505</td><td>505.000</td><td>7</td><td> Inf</td><td>3.200</td><td>159.200</td>
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</tr>
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<tr>
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<td>11</td><td>671</td><td>671.000</td><td>11</td><td>11.102</td><td>2.909</td><td>215.273</td>
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</tr>
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<tr>
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<td>12</td><td>870</td><td>870.000</td><td>3</td><td> Inf</td><td>0.000</td><td>185.333</td>
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</tr>
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<tr>
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<td>13</td><td>1105</td><td>1105.000</td><td>13</td><td>13.060</td><td>4.923</td><td>313.846</td>
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</tr>
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<tr>
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<td>14</td><td>1379</td><td>1379.000</td><td>9</td><td> Inf</td><td>4.571</td><td>540.571</td>
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</tr>
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<tr>
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<td>15</td><td>1695</td><td>1695.000</td><td>15</td><td>15.062</td><td>4.267</td><td>242.133</td>
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</tr>
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<tr>
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<td>16</td><td>2056</td><td>2056.000</td><td>3</td><td> Inf</td><td>0.000</td><td>488.500</td>
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</tr>
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<tr>
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<td>17</td><td>2465</td><td>2465.000</td><td>17</td><td>17.042</td><td>7.529</td><td>267.294</td>
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</tr>
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<tr>
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<td>18</td><td>2925</td><td>2925.000</td><td>11</td><td> Inf</td><td>7.111</td><td>520.889</td>
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</tr>
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<tr>
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<td>19</td><td>3439</td><td>3439.000</td><td>19</td><td>19.048</td><td>16.842</td><td>387.368</td>
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</tr>
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<tr>
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<td>20</td><td>4010</td><td>4010.000</td><td>3</td><td> Inf</td><td>14.400</td><td>584.800</td>
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</tr>
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<tr>
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<td>21</td><td>4641</td><td>4641.000</td><td>21</td><td>21.035</td><td>6.095</td><td>1158.095</td>
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</tr>
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<tr>
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<td>22</td><td>5335</td><td>5335.000</td><td>13</td><td> Inf</td><td>6.545</td><td>1132.364</td>
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</tr>
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<tr>
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<td>23</td><td>6095</td><td>6095.000</td><td>23</td><td>23.037</td><td>11.130</td><td>1268.870</td>
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</tr>
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<tr>
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<td>24</td><td>6924</td><td>6924.000</td><td>3</td><td> Inf</td><td>10.667</td><td>827.500</td>
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</tr>
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<tr>
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<td>25</td><td>7825</td><td>7825.000</td><td>25</td><td>25.029</td><td>35.840</td><td>1190.400</td>
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</tr>
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<tr>
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<td>26</td><td>8801</td><td>8801.000</td><td>15</td><td> Inf</td><td>4.923</td><td>1859.077</td>
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</tr>
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<tr>
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<td>27</td><td>9855</td><td>9855.000</td><td>27</td><td>27.032</td><td>37.926</td><td>1365.333</td>
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</tr>
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<tr>
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<td>28</td><td>10990</td><td>10990.000</td><td>3</td><td> Inf</td><td>34.286</td><td>1365.714</td>
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</tr>
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<tr>
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<td>29</td><td>12209</td><td>12209.000</td><td>29</td><td>29.025</td><td>30.897</td><td>1647.448</td>
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</tr>
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<tr>
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<td>30</td><td>13515</td><td>13515.000</td><td>17</td><td> Inf</td><td>8.533</td><td>2571.733</td>
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</tr>
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<tr>
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<td>31</td><td>14911</td><td>14911.000</td><td>31</td><td>31.027</td><td>33.032</td><td>1426.581</td>
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</tr>
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<tr>
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<td>32</td><td>16400</td><td>16400.000</td><td>3</td><td> Inf</td><td>0.000</td><td>1600.125</td>
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</tr>
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</table>
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<center>Elapsed Time = 0.710 seconds</center>
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<p>
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The magic square example does not fare well when <a href='../examples/MagicSquareExample.php'>run as a PHP script</a>. For a 32x32 matrix array
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it takes around a second to complete just the last row of computations in the above table.
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Hopefully this result will spur PHP developers to find optimizations and better attuned algorithms
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to speed things up. Matrix algebra is a great testing ground for ideas about time and memory
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performance optimation. Keep in perspective that PHP JAMA scripts are still plenty fast for use as
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a tool for learning about matrix algebra and quickly extending your knowledge with new scripts
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to apply knowledge.
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</p>
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<p>
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To learn more about the subject of magic squares you can visit the <a href='http://mathforum.org/alejandre/magic.square.html'>Drexel Math Forum on Magic Squares</a>.
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You can also learn more by carefully examining the <code>MagicSquareExample.php</code> source code below.
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</p>
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<?php
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highlight_file("../examples/MagicSquareExample.php");
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include_once "includes/footer.php";
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?>
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