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calculus.html
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<!DOCTYPE html>
<html>
<head>
<title>numbers.js documentation</title>
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<ul>
<li><a href="#adaptiveSimpson">calculus.adaptiveSimpson</a></li>
<li><a href="#LanczosGamma">calculus.LanczosGamma</a></li>
<li><a href="#limit">calculus.limit</a></li>
<li><a href="#MonteCarlo">calculus.MonteCarlo</a></li>
<li><a href="#pointDiff">calculus.pointDiff</a></li>
<li><a href="#Riemann">calculus.Riemann</a></li>
<li><a href="#SimpsonDef">calculus.SimpsonDef</a></li>
<li><a href="#SimpsonRecursive">calculus.SimpsonRecursive</a></li>
<li><a href="#StirlingGamma">calculus.StirlingGamma</a></li>
</ul></span><span id="api-cont">
<h3><strong>numbers.calculus</strong></h3>
<div id="adaptiveSimpson" class="api-func">
<h4>numbers.calculus.<b>adaptiveSimpson</b>(f, a, b, eps)</h4>
<p>Simpson's method of approximating the integral of a function, <i>f</i>, on the interval [<i>a</i>,<i>b</i>].</p>
<table>
<tr>
<th>Parameters</th>
<td>
<p><b>f</b> : <i>Function</i><br/>
<div class="desc">The function to be evaluated.</div>
</p>
<p><b>a</b> : <i>Number</i><br/>
<div class="desc">The left endpoint of the interval.</div>
</p>
<p><b>b</b> : <i>Number</i><br/>
<div class="desc">The right endpoint of the interval.</div>
</p>
<p><b>eps</b> : <i>Number</i><br/>
<div class="desc">An error bound.</div>
</p>
</td>
</tr>
<tr>
<th>Returns</th>
<td>
<p><b>num</b> : <i>Number</i>
<div class="desc">The approximation of the integral of <i>f</i> on [<i>a</i>,<i>b</i>].</div>
</p>
</td>
</tr>
<tr>
<th>Errors</th>
<td>
<p>This function does not raise any errors.</p>
</td>
</tr>
</table>
<hr/>
</div>
<div id="LanczosGamma" class="api-func">
<h4>numbers.calculus.<b>LanczosGamma</b>(n)</h4>
<p>Lanczos' approximation to the gamma function of a number, <i>n</i>.</p>
<table>
<tr>
<th>Parameters</th>
<td>
<p><b>n</b> : <i>Number</i><br/>
<div class="desc">A number.</div>
</p>
</td>
</tr>
<tr>
<th>Returns</th>
<td>
<p><b>num</b> : <i>Number</i>
<div class="desc">Gamma of <i>n</i>.</div>
</p>
</td>
</tr>
<tr>
<th>Errors</th>
<td>
<p>This function does not raise any errors.</p>
</td>
</tr>
</table>
<hr/>
</div>
<div id="limit" class="api-func">
<h4>numbers.calculus.<b>limit</b>(f, x, approach)</h4>
<p>Calculates the limit of a function, <i>f</i>, at a point <i>x</i>. The point can be approached from the left, right, or middle (a combination of the left and right).</p>
<table>
<tr>
<th>Parameters</th>
<td>
<p><b>f</b> : <i>Function</i><br/>
<div class="desc">The function to be evaluated.</div>
</p>
<p><b>x</b> : <i>Number</i><br/>
<div class="desc">The point for which the limit will be calculated.</div>
</p>
<p><b>approach</b> : <i>String</i><br/>
<div class="desc">A desired approach. left, right and middle are the possible approaches.</div>
</p>
</td>
</tr>
<tr>
<th>Returns</th>
<td>
<p><b>num</b> : <i>Number</i>
<div class="desc">The limit of <i>f</i> at <i>x</i>.</div>
</p>
</td>
</tr>
<tr>
<th>Errors</th>
<td>
<p>An error is thrown if:</p>
<ul>
<li>an approach is not given</li>
</ul>
</td>
</tr>
</table>
<hr/>
</div>
<div id="MonteCarlo" class="api-func">
<h4>numbers.calculus.<b>MonteCarlo</b>(f, N, I)</h4>
<p>The Monte-Carlo method for approximating the integral of a singlevariate or multivariate function, <i>f</i>, over a given interval(s). The number of intervals must match the number of variables of the function. The nth element of I is the interval for the nth variable of the function.</p>
<table>
<tr>
<th>Parameters</th>
<td>
<p><b>f</b> : <i>Function</i><br/>
<div class="desc">The function to be evaluated.</div>
</p>
<p><b>N</b> : <i>Number</i><br/>
<div class="desc">The number of function evaluations.</div>
</p>
<p><b>I</b> : <i>Array</i><br/>
<div class="desc">An array of arrays, where each inner array is an interval containing
the endpoints.</div>
</p>
</td>
</tr>
<tr>
<th>Returns</th>
<td>
<p><b>num</b> : <i>Number</i>
<div class="desc">An approximation to the integral of <i>f</i> with <i>N</i> function evaluations.</div>
</p>
</td>
</tr>
<tr>
<th>Errors</th>
<td>
<p>An error is thrown if:</p>
<ul>
<li>there are no intervals given (<span class="lit">L.length == 0</span>)</li>
<li>N is not positive</li>
</ul>
</td>
</tr>
</table>
<hr/>
</div>
<div id="pointDiff" class="api-func">
<h4>numbers.calculus.<b>pointDiff</b>(f, x)</h4>
<p>Calculates the point differential of a function <i>f</i> at a point <i>x</i>. Currently only supports one-dimensional functions.</p>
<table>
<tr>
<th>Parameters</th>
<td>
<p><b>f</b> : <i>Function</i><br/>
<div class="desc">The function to be evaluated.</div>
</p>
<p><b>x</b> : <i>Number</i><br/>
<div class="desc">The point for which the point differential will be calculated.</div>
</p>
</td>
</tr>
<tr>
<th>Returns</th>
<td>
<p><b>num</b> : <i>Number</i>
<div class="desc">The point differential of <i>f</i> at <i>x</i>.</div>
</p>
</td>
</tr>
<tr>
<th>Errors</th>
<td>
<p>This function does not raise any errors.</p>
</td>
</tr>
</table>
<hr/>
</div>
<div id="Riemann" class="api-func">
<h4>numbers.calculus.<b>Riemann</b>(f, a, b, n, sampler)</h4>
<p>Calculates the Riemann sum for a one-variable function <i>f</i> on the interval [<i>a</i>,<i>b</i>] with <i>n</i> equally-spaced divisons. If <i>sampler</i> is given, that function will be used to calculate which value to sample on each subinterval; otherwise, the left endpoint will be used.</p>
<table>
<tr>
<th>Parameters</th>
<td>
<p><b>f</b> : <i>Function</i><br/>
<div class="desc">The function to be evaluated.</div>
</p>
<p><b>a</b> : <i>Number</i><br/>
<div class="desc">The left endpoint of the interval.</div>
</p>
<p><b>b</b> : <i>Number</i><br/>
<div class="desc">The right endpoint of the interval.</div>
</p>
<p><b>n</b> : <i>Number</i><br/>
<div class="desc">The number of subintervals.</div>
</p>
<p><b>sampler</b> : <i>Function, optional</i><br/>
<div class="desc">A function that determines what value is to be used for sampling
on a subinterval.</div>
</p>
</td>
</tr>
<tr>
<th>Returns</th>
<td>
<p><b>num</b> : <i>Number</i>
<div class="desc">The Riemann sum of <i>f</i> on [<i>a</i>,<i>b</i>] with <i>n</i> divisions.</div>
</p>
</td>
</tr>
<tr>
<th>Errors</th>
<td>
<p>This function does not raise any errors.</p>
</td>
</tr>
</table>
<hr/>
</div>
<div id="SimpsonDef" class="api-func">
<h4>numbers.calculus.<b>SimpsonDef</b>(f, a, b)</h4>
<p>Simpson's method of approximating the integral of a function, <i>f</i>, on the interval [<i>a</i>,<i>b</i>].</p>
<table>
<tr>
<th>Parameters</th>
<td>
<p><b>f</b> : <i>Function</i><br/>
<div class="desc">The function to be evaluated.</div>
</p>
<p><b>a</b> : <i>Number</i><br/>
<div class="desc">The left endpoint of the interval.</div>
</p>
<p><b>b</b> : <i>Number</i><br/>
<div class="desc">The right endpoint of the interval.</div>
</p>
</td>
</tr>
<tr>
<th>Returns</th>
<td>
<p><b>num</b> : <i>Number</i>
<div class="desc">The approximation of the integral of <i>f</i> on [<i>a</i>,<i>b</i>].</div>
</p>
</td>
</tr>
<tr>
<th>Errors</th>
<td>
<p>This function does not raise any errors.</p>
</td>
</tr>
</table>
<hr/>
</div>
<div id="SimpsonRecursive" class="api-func">
<h4>numbers.calculus.<b>SimpsonRecursive</b>(f, a, b, whole, eps)</h4>
<p>The helper function used for adaptive Simpson's method of approximating the integral of a function, <i>f</i>, on the interval [<i>a</i>,<i>b</i>].</p>
<table>
<tr>
<th>Parameters</th>
<td>
<p><b>f</b> : <i>Function</i><br/>
<div class="desc">The function to be evaluated.</div>
</p>
<p><b>a</b> : <i>Number</i><br/>
<div class="desc">The left endpoint of the interval.</div>
</p>
<p><b>b</b> : <i>Number</i><br/>
<div class="desc">The right endpoint of the interval.</div>
</p>
<p><b>whole</b> : <i>Number</i><br/>
<div class="desc">The value of the integral of f on [a,b]
(Simpson's approximation, in the caes of adaptiveSimpson).</div>
</p>
<p><b>eps</b> : <i>Number</i><br/>
<div class="desc">An error bound.</div>
</p>
</td>
</tr>
<tr>
<th>Returns</th>
<td>
<p><b>num</b> : <i>Number</i>
<div class="desc">A recursive evaluation of the left and right sides.</div>
</p>
</td>
</tr>
<tr>
<th>Errors</th>
<td>
<p>This function does not raise any errors.</p>
</td>
</tr>
</table>
<hr/>
</div>
<div id="StirlingGamma" class="api-func">
<h4>numbers.calculus.<b>StirlingGamma</b>(n)</h4>
<p>Striling's approximation to the gamma function of a number, <i>n</i>.</p>
<table>
<tr>
<th>Parameters</th>
<td>
<p><b>n</b> : <i>Number</i><br/>
<div class="desc">A number.</div>
</p>
</td>
</tr>
<tr>
<th>Returns</th>
<td>
<p><b>num</b> : <i>Number</i>
<div class="desc">Gamma of <i>n</i>.</div>
</p>
</td>
</tr>
<tr>
<th>Errors</th>
<td>
<p>This function does not raise any errors.</p>
</td>
</tr>
</table>
<hr/>
</div></span>
</body>
</html>