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legendre-gauss.html
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<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8" />
<title>Gaussian Quadrature Weights and Abscissae</title>
<style>
a {
text-decoration: none;
}
.anchors {
font-size: 75%;
margin: 0.5em;
}
.tbl {
margin-left: 2em;
}
.tbl td {
border: 1px solid grey;
padding-left: 5px;
padding-right: 5px;
}
.tbl thead {
font-weight: bold;
text-align: center;
}
.footer {
font-size: 85%;
margin-top: 1em;
}
img {
margin-left: 2em;
}
</style>
<!-- my own referral/page hit tracker, because Google knows enough -->
<script src="./js/site/referrer.js" type="module" async></script>
</head>
<body>
<h1>Gaussian Quadrature Weights and Abscissae</h1>
<p>
This page is a tabulation of weights and abscissae for use in performing
<a href="http://mathworld.wolfram.com/Legendre-GaussQuadrature.html">Legendre-Gauss quadrature</a>
integral approximation, which tries to solve the following function
</p>
<p>
<a href="http://mathworld.wolfram.com/Legendre-GaussQuadrature.html"><img src="images/gq.gif" /></a>
</p>
<p>
by picking approximate values for <i>n</i>, <i>w<sub>i</sub></i> and <i>x<sub>i</sub></i
>. While only defined for the interval [-1,1], this is actually a universal function, because we can convert the limits of integration for any
interval [a,b] to the Legendre-Gauss interval [-1,1]:
</p>
<p>
<a href="http://mathworld.wolfram.com/Legendre-GaussQuadrature.html"><img src="images/trans.gif" /></a>
</p>
<p>
The summation function is called the Legendre-Gauss quadrature rule because the abscissae <i>x<sub>i</sub></i> in the Gauss quadrature function
for [-1,1] are defined as the roots of the <a href="http://mathworld.wolfram.com/LegendrePolynomial.html">Legendre polynomial</a> for n:
</p>
<p>
<a href="http://mathworld.wolfram.com/Legendre-GaussQuadrature.html"><img src="images/lp.gif" /></a>
</p>
<p>
with the weights <i>w<sub>i</sub></i> coming from the following function:
</p>
<p>
<a href="http://mathworld.wolfram.com/Legendre-GaussQuadrature.html"><img src="images/wi.gif" /></a>
</p>
<p>
(The procedure is explained in more detail <a href="http://mathworld.wolfram.com/Legendre-GaussQuadrature.html">here</a>, with a very accessible
video lecture on the theory behind the Gauss quadrature rule
<a href="http://www.youtube.com/watch?v=unWguclP-Ds&feature=BFa&list=PLC8FC40C714F5E60F&index=1">here</a>)
</p>
<p>
The tables provided here give the values for <i>x<sub>i</sub></i> and <i>w<sub>i</sub></i> for n=2 through n=64, with an internal decimal
precision of 256, limited to 16 decimals due to floating point number limits in PHP. These tables were made using the numbers that are generated
by the following series of Mathematica instructions:
</p>
<pre>
symboliclegendre[n_, x_] := Solve[LegendreP[n, x] == 0];
legendreprime[n_, a_] := D[LegendreP[n, x], x] /. x -> a;
weights[n_, x_] := 2/((1 - x^2) legendreprime[n, x]^2);
(*how many terms should be generated*)
h = 64;
(* what numerical precision is desired? *)
precision = 256;
str = OpenWrite["lgvalues.txt"];
Write[str, "abcissae"];
Do[Write[str]; Write[str, n]; Write[str];
nlist = symboliclegendre[n, x];
xnlist = x /. nlist;
Do[Write[str, N[Part[xnlist, i], precision]], {i, Length[xnlist]}];, {n, 2, h}];
Write[str, "weights"];
Do[Write[str]; Write[str, n]; Write[str];
slist := symboliclegendre[n, x];
xslist = x /. slist;
Do[Write[str, N[weights[n, Part[xslist, i]], precision]], {i, Length[xslist]}];, {n, 2, h}];
Close[str];</pre
>
<p>
If you need higher precision than is presented on this web page you can either download the abscissae and weights files below, or you can run
this program in Mathematica yourself, with higher precision and/or higher <i>h</i> values.
</p>
<ul>
<li><a href="lgvalues-abscissa.php">abscissae</a> (607KB, easily converted to not-PHP)</li>
<li><a href="lgvalues-weights.php">weights</a> (615KB, easily converted to not-PHP)</li>
</ul>
<h2>Weights and Abscissae Tables for n=2 to n=64</h2>
<div id="n2">
<a name="n2"></a>
<table>
<tr>
<td nowrap="nowrap" style="padding-right: 3em;"><h2>n = 2</h2></td>
<td>
<div class="anchors">
jump to n = <a href="#n2">2</a>, <a href="#n3">3</a>, <a href="#n4">4</a>, <a href="#n5">5</a>, <a href="#n6">6</a>,
<a href="#n7">7</a>, <a href="#n8">8</a>, <a href="#n9">9</a>, <a href="#n10">10</a>, <a href="#n11">11</a>, <a href="#n12">12</a>,
<a href="#n13">13</a>, <a href="#n14">14</a>, <a href="#n15">15</a>, <a href="#n16">16</a>, <a href="#n17">17</a>,
<a href="#n18">18</a>, <a href="#n19">19</a>, <a href="#n20">20</a>, <a href="#n21">21</a>, <a href="#n22">22</a>,
<a href="#n23">23</a>, <a href="#n24">24</a>, <a href="#n25">25</a>, <a href="#n26">26</a>, <a href="#n27">27</a>,
<a href="#n28">28</a>, <a href="#n29">29</a>, <a href="#n30">30</a>, <a href="#n31">31</a>, <a href="#n32">32</a>,
<a href="#n33">33</a>, <a href="#n34">34</a>, <a href="#n35">35</a>, <a href="#n36">36</a>, <a href="#n37">37</a>,
<a href="#n38">38</a>, <a href="#n39">39</a>, <a href="#n40">40</a>, <a href="#n41">41</a>, <a href="#n42">42</a>,
<a href="#n43">43</a>, <a href="#n44">44</a>, <a href="#n45">45</a>, <a href="#n46">46</a>, <a href="#n47">47</a>,
<a href="#n48">48</a>, <a href="#n49">49</a>, <a href="#n50">50</a>, <a href="#n51">51</a>, <a href="#n52">52</a>,
<a href="#n53">53</a>, <a href="#n54">54</a>, <a href="#n55">55</a>, <a href="#n56">56</a>, <a href="#n57">57</a>,
<a href="#n58">58</a>, <a href="#n59">59</a>, <a href="#n60">60</a>, <a href="#n61">61</a>, <a href="#n62">62</a>,
<a href="#n63">63</a>, <a href="#n64">64</a>
</div>
</td>
</tr>
</table>
<table class="tbl">
<thead>
<tr>
<td><i>i</i></td>
<td>weight - w<sub>i</sub></td>
<td>abscissa - <i>x</i><sub>i</sub></td>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>1.0000000000000000</td>
<td>-0.5773502691896257</td>
</tr>
<tr>
<td>2</td>
<td>1.0000000000000000</td>
<td>0.5773502691896257</td>
</tr>
</tbody>
</table>
</div>
<div id="n3">
<a name="n3"></a>
<table>
<tr>
<td nowrap="nowrap" style="padding-right: 3em;"><h2>n = 3</h2></td>
<td>
<div class="anchors">
jump to n = <a href="#n2">2</a>, <a href="#n3">3</a>, <a href="#n4">4</a>, <a href="#n5">5</a>, <a href="#n6">6</a>,
<a href="#n7">7</a>, <a href="#n8">8</a>, <a href="#n9">9</a>, <a href="#n10">10</a>, <a href="#n11">11</a>, <a href="#n12">12</a>,
<a href="#n13">13</a>, <a href="#n14">14</a>, <a href="#n15">15</a>, <a href="#n16">16</a>, <a href="#n17">17</a>,
<a href="#n18">18</a>, <a href="#n19">19</a>, <a href="#n20">20</a>, <a href="#n21">21</a>, <a href="#n22">22</a>,
<a href="#n23">23</a>, <a href="#n24">24</a>, <a href="#n25">25</a>, <a href="#n26">26</a>, <a href="#n27">27</a>,
<a href="#n28">28</a>, <a href="#n29">29</a>, <a href="#n30">30</a>, <a href="#n31">31</a>, <a href="#n32">32</a>,
<a href="#n33">33</a>, <a href="#n34">34</a>, <a href="#n35">35</a>, <a href="#n36">36</a>, <a href="#n37">37</a>,
<a href="#n38">38</a>, <a href="#n39">39</a>, <a href="#n40">40</a>, <a href="#n41">41</a>, <a href="#n42">42</a>,
<a href="#n43">43</a>, <a href="#n44">44</a>, <a href="#n45">45</a>, <a href="#n46">46</a>, <a href="#n47">47</a>,
<a href="#n48">48</a>, <a href="#n49">49</a>, <a href="#n50">50</a>, <a href="#n51">51</a>, <a href="#n52">52</a>,
<a href="#n53">53</a>, <a href="#n54">54</a>, <a href="#n55">55</a>, <a href="#n56">56</a>, <a href="#n57">57</a>,
<a href="#n58">58</a>, <a href="#n59">59</a>, <a href="#n60">60</a>, <a href="#n61">61</a>, <a href="#n62">62</a>,
<a href="#n63">63</a>, <a href="#n64">64</a>
</div>
</td>
</tr>
</table>
<table class="tbl">
<thead>
<tr>
<td><i>i</i></td>
<td>weight - w<sub>i</sub></td>
<td>abscissa - <i>x</i><sub>i</sub></td>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>0.8888888888888888</td>
<td>0.0000000000000000</td>
</tr>
<tr>
<td>2</td>
<td>0.5555555555555556</td>
<td>-0.7745966692414834</td>
</tr>
<tr>
<td>3</td>
<td>0.5555555555555556</td>
<td>0.7745966692414834</td>
</tr>
</tbody>
</table>
</div>
<div id="n4">
<a name="n4"></a>
<table>
<tr>
<td nowrap="nowrap" style="padding-right: 3em;"><h2>n = 4</h2></td>
<td>
<div class="anchors">
jump to n = <a href="#n2">2</a>, <a href="#n3">3</a>, <a href="#n4">4</a>, <a href="#n5">5</a>, <a href="#n6">6</a>,
<a href="#n7">7</a>, <a href="#n8">8</a>, <a href="#n9">9</a>, <a href="#n10">10</a>, <a href="#n11">11</a>, <a href="#n12">12</a>,
<a href="#n13">13</a>, <a href="#n14">14</a>, <a href="#n15">15</a>, <a href="#n16">16</a>, <a href="#n17">17</a>,
<a href="#n18">18</a>, <a href="#n19">19</a>, <a href="#n20">20</a>, <a href="#n21">21</a>, <a href="#n22">22</a>,
<a href="#n23">23</a>, <a href="#n24">24</a>, <a href="#n25">25</a>, <a href="#n26">26</a>, <a href="#n27">27</a>,
<a href="#n28">28</a>, <a href="#n29">29</a>, <a href="#n30">30</a>, <a href="#n31">31</a>, <a href="#n32">32</a>,
<a href="#n33">33</a>, <a href="#n34">34</a>, <a href="#n35">35</a>, <a href="#n36">36</a>, <a href="#n37">37</a>,
<a href="#n38">38</a>, <a href="#n39">39</a>, <a href="#n40">40</a>, <a href="#n41">41</a>, <a href="#n42">42</a>,
<a href="#n43">43</a>, <a href="#n44">44</a>, <a href="#n45">45</a>, <a href="#n46">46</a>, <a href="#n47">47</a>,
<a href="#n48">48</a>, <a href="#n49">49</a>, <a href="#n50">50</a>, <a href="#n51">51</a>, <a href="#n52">52</a>,
<a href="#n53">53</a>, <a href="#n54">54</a>, <a href="#n55">55</a>, <a href="#n56">56</a>, <a href="#n57">57</a>,
<a href="#n58">58</a>, <a href="#n59">59</a>, <a href="#n60">60</a>, <a href="#n61">61</a>, <a href="#n62">62</a>,
<a href="#n63">63</a>, <a href="#n64">64</a>
</div>
</td>
</tr>
</table>
<table class="tbl">
<thead>
<tr>
<td><i>i</i></td>
<td>weight - w<sub>i</sub></td>
<td>abscissa - <i>x</i><sub>i</sub></td>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>0.6521451548625461</td>
<td>-0.3399810435848563</td>
</tr>
<tr>
<td>2</td>
<td>0.6521451548625461</td>
<td>0.3399810435848563</td>
</tr>
<tr>
<td>3</td>
<td>0.3478548451374538</td>
<td>-0.8611363115940526</td>
</tr>
<tr>
<td>4</td>
<td>0.3478548451374538</td>
<td>0.8611363115940526</td>
</tr>
</tbody>
</table>
</div>
<div id="n5">
<a name="n5"></a>
<table>
<tr>
<td nowrap="nowrap" style="padding-right: 3em;"><h2>n = 5</h2></td>
<td>
<div class="anchors">
jump to n = <a href="#n2">2</a>, <a href="#n3">3</a>, <a href="#n4">4</a>, <a href="#n5">5</a>, <a href="#n6">6</a>,
<a href="#n7">7</a>, <a href="#n8">8</a>, <a href="#n9">9</a>, <a href="#n10">10</a>, <a href="#n11">11</a>, <a href="#n12">12</a>,
<a href="#n13">13</a>, <a href="#n14">14</a>, <a href="#n15">15</a>, <a href="#n16">16</a>, <a href="#n17">17</a>,
<a href="#n18">18</a>, <a href="#n19">19</a>, <a href="#n20">20</a>, <a href="#n21">21</a>, <a href="#n22">22</a>,
<a href="#n23">23</a>, <a href="#n24">24</a>, <a href="#n25">25</a>, <a href="#n26">26</a>, <a href="#n27">27</a>,
<a href="#n28">28</a>, <a href="#n29">29</a>, <a href="#n30">30</a>, <a href="#n31">31</a>, <a href="#n32">32</a>,
<a href="#n33">33</a>, <a href="#n34">34</a>, <a href="#n35">35</a>, <a href="#n36">36</a>, <a href="#n37">37</a>,
<a href="#n38">38</a>, <a href="#n39">39</a>, <a href="#n40">40</a>, <a href="#n41">41</a>, <a href="#n42">42</a>,
<a href="#n43">43</a>, <a href="#n44">44</a>, <a href="#n45">45</a>, <a href="#n46">46</a>, <a href="#n47">47</a>,
<a href="#n48">48</a>, <a href="#n49">49</a>, <a href="#n50">50</a>, <a href="#n51">51</a>, <a href="#n52">52</a>,
<a href="#n53">53</a>, <a href="#n54">54</a>, <a href="#n55">55</a>, <a href="#n56">56</a>, <a href="#n57">57</a>,
<a href="#n58">58</a>, <a href="#n59">59</a>, <a href="#n60">60</a>, <a href="#n61">61</a>, <a href="#n62">62</a>,
<a href="#n63">63</a>, <a href="#n64">64</a>
</div>
</td>
</tr>
</table>
<table class="tbl">
<thead>
<tr>
<td><i>i</i></td>
<td>weight - w<sub>i</sub></td>
<td>abscissa - <i>x</i><sub>i</sub></td>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>0.5688888888888889</td>
<td>0.0000000000000000</td>
</tr>
<tr>
<td>2</td>
<td>0.4786286704993665</td>
<td>-0.5384693101056831</td>
</tr>
<tr>
<td>3</td>
<td>0.4786286704993665</td>
<td>0.5384693101056831</td>
</tr>
<tr>
<td>4</td>
<td>0.2369268850561891</td>
<td>-0.9061798459386640</td>
</tr>
<tr>
<td>5</td>
<td>0.2369268850561891</td>
<td>0.9061798459386640</td>
</tr>
</tbody>
</table>
</div>
<div id="n6">
<a name="n6"></a>
<table>
<tr>
<td nowrap="nowrap" style="padding-right: 3em;"><h2>n = 6</h2></td>
<td>
<div class="anchors">
jump to n = <a href="#n2">2</a>, <a href="#n3">3</a>, <a href="#n4">4</a>, <a href="#n5">5</a>, <a href="#n6">6</a>,
<a href="#n7">7</a>, <a href="#n8">8</a>, <a href="#n9">9</a>, <a href="#n10">10</a>, <a href="#n11">11</a>, <a href="#n12">12</a>,
<a href="#n13">13</a>, <a href="#n14">14</a>, <a href="#n15">15</a>, <a href="#n16">16</a>, <a href="#n17">17</a>,
<a href="#n18">18</a>, <a href="#n19">19</a>, <a href="#n20">20</a>, <a href="#n21">21</a>, <a href="#n22">22</a>,
<a href="#n23">23</a>, <a href="#n24">24</a>, <a href="#n25">25</a>, <a href="#n26">26</a>, <a href="#n27">27</a>,
<a href="#n28">28</a>, <a href="#n29">29</a>, <a href="#n30">30</a>, <a href="#n31">31</a>, <a href="#n32">32</a>,
<a href="#n33">33</a>, <a href="#n34">34</a>, <a href="#n35">35</a>, <a href="#n36">36</a>, <a href="#n37">37</a>,
<a href="#n38">38</a>, <a href="#n39">39</a>, <a href="#n40">40</a>, <a href="#n41">41</a>, <a href="#n42">42</a>,
<a href="#n43">43</a>, <a href="#n44">44</a>, <a href="#n45">45</a>, <a href="#n46">46</a>, <a href="#n47">47</a>,
<a href="#n48">48</a>, <a href="#n49">49</a>, <a href="#n50">50</a>, <a href="#n51">51</a>, <a href="#n52">52</a>,
<a href="#n53">53</a>, <a href="#n54">54</a>, <a href="#n55">55</a>, <a href="#n56">56</a>, <a href="#n57">57</a>,
<a href="#n58">58</a>, <a href="#n59">59</a>, <a href="#n60">60</a>, <a href="#n61">61</a>, <a href="#n62">62</a>,
<a href="#n63">63</a>, <a href="#n64">64</a>
</div>
</td>
</tr>
</table>
<table class="tbl">
<thead>
<tr>
<td><i>i</i></td>
<td>weight - w<sub>i</sub></td>
<td>abscissa - <i>x</i><sub>i</sub></td>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>0.3607615730481386</td>
<td>0.6612093864662645</td>
</tr>
<tr>
<td>2</td>
<td>0.3607615730481386</td>
<td>-0.6612093864662645</td>
</tr>
<tr>
<td>3</td>
<td>0.4679139345726910</td>
<td>-0.2386191860831969</td>
</tr>
<tr>
<td>4</td>
<td>0.4679139345726910</td>
<td>0.2386191860831969</td>
</tr>
<tr>
<td>5</td>
<td>0.1713244923791704</td>
<td>-0.9324695142031521</td>
</tr>
<tr>
<td>6</td>
<td>0.1713244923791704</td>
<td>0.9324695142031521</td>
</tr>
</tbody>
</table>
</div>
<div id="n7">
<a name="n7"></a>
<table>
<tr>
<td nowrap="nowrap" style="padding-right: 3em;"><h2>n = 7</h2></td>
<td>
<div class="anchors">
jump to n = <a href="#n2">2</a>, <a href="#n3">3</a>, <a href="#n4">4</a>, <a href="#n5">5</a>, <a href="#n6">6</a>,
<a href="#n7">7</a>, <a href="#n8">8</a>, <a href="#n9">9</a>, <a href="#n10">10</a>, <a href="#n11">11</a>, <a href="#n12">12</a>,
<a href="#n13">13</a>, <a href="#n14">14</a>, <a href="#n15">15</a>, <a href="#n16">16</a>, <a href="#n17">17</a>,
<a href="#n18">18</a>, <a href="#n19">19</a>, <a href="#n20">20</a>, <a href="#n21">21</a>, <a href="#n22">22</a>,
<a href="#n23">23</a>, <a href="#n24">24</a>, <a href="#n25">25</a>, <a href="#n26">26</a>, <a href="#n27">27</a>,
<a href="#n28">28</a>, <a href="#n29">29</a>, <a href="#n30">30</a>, <a href="#n31">31</a>, <a href="#n32">32</a>,
<a href="#n33">33</a>, <a href="#n34">34</a>, <a href="#n35">35</a>, <a href="#n36">36</a>, <a href="#n37">37</a>,
<a href="#n38">38</a>, <a href="#n39">39</a>, <a href="#n40">40</a>, <a href="#n41">41</a>, <a href="#n42">42</a>,
<a href="#n43">43</a>, <a href="#n44">44</a>, <a href="#n45">45</a>, <a href="#n46">46</a>, <a href="#n47">47</a>,
<a href="#n48">48</a>, <a href="#n49">49</a>, <a href="#n50">50</a>, <a href="#n51">51</a>, <a href="#n52">52</a>,
<a href="#n53">53</a>, <a href="#n54">54</a>, <a href="#n55">55</a>, <a href="#n56">56</a>, <a href="#n57">57</a>,
<a href="#n58">58</a>, <a href="#n59">59</a>, <a href="#n60">60</a>, <a href="#n61">61</a>, <a href="#n62">62</a>,
<a href="#n63">63</a>, <a href="#n64">64</a>
</div>
</td>
</tr>
</table>
<table class="tbl">
<thead>
<tr>
<td><i>i</i></td>
<td>weight - w<sub>i</sub></td>
<td>abscissa - <i>x</i><sub>i</sub></td>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>0.4179591836734694</td>
<td>0.0000000000000000</td>
</tr>
<tr>
<td>2</td>
<td>0.3818300505051189</td>
<td>0.4058451513773972</td>
</tr>
<tr>
<td>3</td>
<td>0.3818300505051189</td>
<td>-0.4058451513773972</td>
</tr>
<tr>
<td>4</td>
<td>0.2797053914892766</td>
<td>-0.7415311855993945</td>
</tr>
<tr>
<td>5</td>
<td>0.2797053914892766</td>
<td>0.7415311855993945</td>
</tr>
<tr>
<td>6</td>
<td>0.1294849661688697</td>
<td>-0.9491079123427585</td>
</tr>
<tr>
<td>7</td>
<td>0.1294849661688697</td>
<td>0.9491079123427585</td>
</tr>
</tbody>
</table>
</div>
<div id="n8">
<a name="n8"></a>
<table>
<tr>
<td nowrap="nowrap" style="padding-right: 3em;"><h2>n = 8</h2></td>
<td>
<div class="anchors">
jump to n = <a href="#n2">2</a>, <a href="#n3">3</a>, <a href="#n4">4</a>, <a href="#n5">5</a>, <a href="#n6">6</a>,
<a href="#n7">7</a>, <a href="#n8">8</a>, <a href="#n9">9</a>, <a href="#n10">10</a>, <a href="#n11">11</a>, <a href="#n12">12</a>,
<a href="#n13">13</a>, <a href="#n14">14</a>, <a href="#n15">15</a>, <a href="#n16">16</a>, <a href="#n17">17</a>,
<a href="#n18">18</a>, <a href="#n19">19</a>, <a href="#n20">20</a>, <a href="#n21">21</a>, <a href="#n22">22</a>,
<a href="#n23">23</a>, <a href="#n24">24</a>, <a href="#n25">25</a>, <a href="#n26">26</a>, <a href="#n27">27</a>,
<a href="#n28">28</a>, <a href="#n29">29</a>, <a href="#n30">30</a>, <a href="#n31">31</a>, <a href="#n32">32</a>,
<a href="#n33">33</a>, <a href="#n34">34</a>, <a href="#n35">35</a>, <a href="#n36">36</a>, <a href="#n37">37</a>,
<a href="#n38">38</a>, <a href="#n39">39</a>, <a href="#n40">40</a>, <a href="#n41">41</a>, <a href="#n42">42</a>,
<a href="#n43">43</a>, <a href="#n44">44</a>, <a href="#n45">45</a>, <a href="#n46">46</a>, <a href="#n47">47</a>,
<a href="#n48">48</a>, <a href="#n49">49</a>, <a href="#n50">50</a>, <a href="#n51">51</a>, <a href="#n52">52</a>,
<a href="#n53">53</a>, <a href="#n54">54</a>, <a href="#n55">55</a>, <a href="#n56">56</a>, <a href="#n57">57</a>,
<a href="#n58">58</a>, <a href="#n59">59</a>, <a href="#n60">60</a>, <a href="#n61">61</a>, <a href="#n62">62</a>,
<a href="#n63">63</a>, <a href="#n64">64</a>
</div>
</td>
</tr>
</table>
<table class="tbl">
<thead>
<tr>
<td><i>i</i></td>
<td>weight - w<sub>i</sub></td>
<td>abscissa - <i>x</i><sub>i</sub></td>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>0.3626837833783620</td>
<td>-0.1834346424956498</td>
</tr>
<tr>
<td>2</td>
<td>0.3626837833783620</td>
<td>0.1834346424956498</td>
</tr>
<tr>
<td>3</td>
<td>0.3137066458778873</td>
<td>-0.5255324099163290</td>
</tr>
<tr>
<td>4</td>
<td>0.3137066458778873</td>
<td>0.5255324099163290</td>
</tr>
<tr>
<td>5</td>
<td>0.2223810344533745</td>
<td>-0.7966664774136267</td>
</tr>
<tr>
<td>6</td>
<td>0.2223810344533745</td>
<td>0.7966664774136267</td>
</tr>
<tr>
<td>7</td>
<td>0.1012285362903763</td>
<td>-0.9602898564975363</td>
</tr>
<tr>
<td>8</td>
<td>0.1012285362903763</td>
<td>0.9602898564975363</td>
</tr>
</tbody>
</table>
</div>
<div id="n9">
<a name="n9"></a>
<table>
<tr>
<td nowrap="nowrap" style="padding-right: 3em;"><h2>n = 9</h2></td>
<td>
<div class="anchors">
jump to n = <a href="#n2">2</a>, <a href="#n3">3</a>, <a href="#n4">4</a>, <a href="#n5">5</a>, <a href="#n6">6</a>,
<a href="#n7">7</a>, <a href="#n8">8</a>, <a href="#n9">9</a>, <a href="#n10">10</a>, <a href="#n11">11</a>, <a href="#n12">12</a>,
<a href="#n13">13</a>, <a href="#n14">14</a>, <a href="#n15">15</a>, <a href="#n16">16</a>, <a href="#n17">17</a>,
<a href="#n18">18</a>, <a href="#n19">19</a>, <a href="#n20">20</a>, <a href="#n21">21</a>, <a href="#n22">22</a>,
<a href="#n23">23</a>, <a href="#n24">24</a>, <a href="#n25">25</a>, <a href="#n26">26</a>, <a href="#n27">27</a>,
<a href="#n28">28</a>, <a href="#n29">29</a>, <a href="#n30">30</a>, <a href="#n31">31</a>, <a href="#n32">32</a>,
<a href="#n33">33</a>, <a href="#n34">34</a>, <a href="#n35">35</a>, <a href="#n36">36</a>, <a href="#n37">37</a>,
<a href="#n38">38</a>, <a href="#n39">39</a>, <a href="#n40">40</a>, <a href="#n41">41</a>, <a href="#n42">42</a>,
<a href="#n43">43</a>, <a href="#n44">44</a>, <a href="#n45">45</a>, <a href="#n46">46</a>, <a href="#n47">47</a>,
<a href="#n48">48</a>, <a href="#n49">49</a>, <a href="#n50">50</a>, <a href="#n51">51</a>, <a href="#n52">52</a>,
<a href="#n53">53</a>, <a href="#n54">54</a>, <a href="#n55">55</a>, <a href="#n56">56</a>, <a href="#n57">57</a>,
<a href="#n58">58</a>, <a href="#n59">59</a>, <a href="#n60">60</a>, <a href="#n61">61</a>, <a href="#n62">62</a>,
<a href="#n63">63</a>, <a href="#n64">64</a>
</div>
</td>
</tr>
</table>
<table class="tbl">
<thead>
<tr>
<td><i>i</i></td>
<td>weight - w<sub>i</sub></td>
<td>abscissa - <i>x</i><sub>i</sub></td>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>0.3302393550012598</td>
<td>0.0000000000000000</td>
</tr>
<tr>
<td>2</td>
<td>0.1806481606948574</td>
<td>-0.8360311073266358</td>
</tr>
<tr>
<td>3</td>
<td>0.1806481606948574</td>
<td>0.8360311073266358</td>
</tr>
<tr>
<td>4</td>
<td>0.0812743883615744</td>
<td>-0.9681602395076261</td>
</tr>
<tr>
<td>5</td>
<td>0.0812743883615744</td>
<td>0.9681602395076261</td>
</tr>
<tr>
<td>6</td>
<td>0.3123470770400029</td>
<td>-0.3242534234038089</td>
</tr>
<tr>
<td>7</td>
<td>0.3123470770400029</td>
<td>0.3242534234038089</td>
</tr>
<tr>
<td>8</td>
<td>0.2606106964029354</td>
<td>-0.6133714327005904</td>
</tr>
<tr>
<td>9</td>
<td>0.2606106964029354</td>
<td>0.6133714327005904</td>
</tr>
</tbody>
</table>
</div>
<div id="n10">
<a name="n10"></a>
<table>
<tr>
<td nowrap="nowrap" style="padding-right: 3em;"><h2>n = 10</h2></td>
<td>
<div class="anchors">
jump to n = <a href="#n2">2</a>, <a href="#n3">3</a>, <a href="#n4">4</a>, <a href="#n5">5</a>, <a href="#n6">6</a>,
<a href="#n7">7</a>, <a href="#n8">8</a>, <a href="#n9">9</a>, <a href="#n10">10</a>, <a href="#n11">11</a>, <a href="#n12">12</a>,
<a href="#n13">13</a>, <a href="#n14">14</a>, <a href="#n15">15</a>, <a href="#n16">16</a>, <a href="#n17">17</a>,
<a href="#n18">18</a>, <a href="#n19">19</a>, <a href="#n20">20</a>, <a href="#n21">21</a>, <a href="#n22">22</a>,
<a href="#n23">23</a>, <a href="#n24">24</a>, <a href="#n25">25</a>, <a href="#n26">26</a>, <a href="#n27">27</a>,
<a href="#n28">28</a>, <a href="#n29">29</a>, <a href="#n30">30</a>, <a href="#n31">31</a>, <a href="#n32">32</a>,
<a href="#n33">33</a>, <a href="#n34">34</a>, <a href="#n35">35</a>, <a href="#n36">36</a>, <a href="#n37">37</a>,
<a href="#n38">38</a>, <a href="#n39">39</a>, <a href="#n40">40</a>, <a href="#n41">41</a>, <a href="#n42">42</a>,
<a href="#n43">43</a>, <a href="#n44">44</a>, <a href="#n45">45</a>, <a href="#n46">46</a>, <a href="#n47">47</a>,
<a href="#n48">48</a>, <a href="#n49">49</a>, <a href="#n50">50</a>, <a href="#n51">51</a>, <a href="#n52">52</a>,
<a href="#n53">53</a>, <a href="#n54">54</a>, <a href="#n55">55</a>, <a href="#n56">56</a>, <a href="#n57">57</a>,
<a href="#n58">58</a>, <a href="#n59">59</a>, <a href="#n60">60</a>, <a href="#n61">61</a>, <a href="#n62">62</a>,
<a href="#n63">63</a>, <a href="#n64">64</a>
</div>
</td>
</tr>
</table>
<table class="tbl">
<thead>
<tr>
<td><i>i</i></td>
<td>weight - w<sub>i</sub></td>
<td>abscissa - <i>x</i><sub>i</sub></td>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>0.2955242247147529</td>
<td>-0.1488743389816312</td>
</tr>
<tr>
<td>2</td>
<td>0.2955242247147529</td>
<td>0.1488743389816312</td>
</tr>
<tr>
<td>3</td>
<td>0.2692667193099963</td>
<td>-0.4333953941292472</td>
</tr>
<tr>
<td>4</td>
<td>0.2692667193099963</td>
<td>0.4333953941292472</td>
</tr>
<tr>
<td>5</td>
<td>0.2190863625159820</td>
<td>-0.6794095682990244</td>
</tr>
<tr>
<td>6</td>
<td>0.2190863625159820</td>
<td>0.6794095682990244</td>
</tr>
<tr>
<td>7</td>
<td>0.1494513491505806</td>
<td>-0.8650633666889845</td>
</tr>
<tr>
<td>8</td>
<td>0.1494513491505806</td>
<td>0.8650633666889845</td>
</tr>
<tr>
<td>9</td>
<td>0.0666713443086881</td>
<td>-0.9739065285171717</td>
</tr>
<tr>
<td>10</td>
<td>0.0666713443086881</td>
<td>0.9739065285171717</td>
</tr>
</tbody>
</table>
</div>
<div id="n11">
<a name="n11"></a>
<table>
<tr>
<td nowrap="nowrap" style="padding-right: 3em;"><h2>n = 11</h2></td>
<td>
<div class="anchors">
jump to n = <a href="#n2">2</a>, <a href="#n3">3</a>, <a href="#n4">4</a>, <a href="#n5">5</a>, <a href="#n6">6</a>,
<a href="#n7">7</a>, <a href="#n8">8</a>, <a href="#n9">9</a>, <a href="#n10">10</a>, <a href="#n11">11</a>, <a href="#n12">12</a>,
<a href="#n13">13</a>, <a href="#n14">14</a>, <a href="#n15">15</a>, <a href="#n16">16</a>, <a href="#n17">17</a>,
<a href="#n18">18</a>, <a href="#n19">19</a>, <a href="#n20">20</a>, <a href="#n21">21</a>, <a href="#n22">22</a>,
<a href="#n23">23</a>, <a href="#n24">24</a>, <a href="#n25">25</a>, <a href="#n26">26</a>, <a href="#n27">27</a>,
<a href="#n28">28</a>, <a href="#n29">29</a>, <a href="#n30">30</a>, <a href="#n31">31</a>, <a href="#n32">32</a>,
<a href="#n33">33</a>, <a href="#n34">34</a>, <a href="#n35">35</a>, <a href="#n36">36</a>, <a href="#n37">37</a>,
<a href="#n38">38</a>, <a href="#n39">39</a>, <a href="#n40">40</a>, <a href="#n41">41</a>, <a href="#n42">42</a>,
<a href="#n43">43</a>, <a href="#n44">44</a>, <a href="#n45">45</a>, <a href="#n46">46</a>, <a href="#n47">47</a>,
<a href="#n48">48</a>, <a href="#n49">49</a>, <a href="#n50">50</a>, <a href="#n51">51</a>, <a href="#n52">52</a>,
<a href="#n53">53</a>, <a href="#n54">54</a>, <a href="#n55">55</a>, <a href="#n56">56</a>, <a href="#n57">57</a>,
<a href="#n58">58</a>, <a href="#n59">59</a>, <a href="#n60">60</a>, <a href="#n61">61</a>, <a href="#n62">62</a>,
<a href="#n63">63</a>, <a href="#n64">64</a>
</div>
</td>
</tr>
</table>
<table class="tbl">
<thead>
<tr>
<td><i>i</i></td>
<td>weight - w<sub>i</sub></td>
<td>abscissa - <i>x</i><sub>i</sub></td>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>0.2729250867779006</td>
<td>0.0000000000000000</td>
</tr>
<tr>
<td>2</td>
<td>0.2628045445102467</td>
<td>-0.2695431559523450</td>
</tr>
<tr>
<td>3</td>
<td>0.2628045445102467</td>
<td>0.2695431559523450</td>
</tr>
<tr>
<td>4</td>
<td>0.2331937645919905</td>
<td>-0.5190961292068118</td>
</tr>
<tr>
<td>5</td>
<td>0.2331937645919905</td>
<td>0.5190961292068118</td>
</tr>
<tr>
<td>6</td>
<td>0.1862902109277343</td>
<td>-0.7301520055740494</td>
</tr>
<tr>
<td>7</td>
<td>0.1862902109277343</td>
<td>0.7301520055740494</td>
</tr>
<tr>
<td>8</td>
<td>0.1255803694649046</td>
<td>-0.8870625997680953</td>
</tr>
<tr>
<td>9</td>
<td>0.1255803694649046</td>
<td>0.8870625997680953</td>
</tr>
<tr>
<td>10</td>
<td>0.0556685671161737</td>
<td>-0.9782286581460570</td>
</tr>
<tr>
<td>11</td>
<td>0.0556685671161737</td>
<td>0.9782286581460570</td>
</tr>
</tbody>
</table>
</div>
<div id="n12">
<a name="n12"></a>
<table>
<tr>
<td nowrap="nowrap" style="padding-right: 3em;"><h2>n = 12</h2></td>
<td>
<div class="anchors">
jump to n = <a href="#n2">2</a>, <a href="#n3">3</a>, <a href="#n4">4</a>, <a href="#n5">5</a>, <a href="#n6">6</a>,
<a href="#n7">7</a>, <a href="#n8">8</a>, <a href="#n9">9</a>, <a href="#n10">10</a>, <a href="#n11">11</a>, <a href="#n12">12</a>,
<a href="#n13">13</a>, <a href="#n14">14</a>, <a href="#n15">15</a>, <a href="#n16">16</a>, <a href="#n17">17</a>,
<a href="#n18">18</a>, <a href="#n19">19</a>, <a href="#n20">20</a>, <a href="#n21">21</a>, <a href="#n22">22</a>,
<a href="#n23">23</a>, <a href="#n24">24</a>, <a href="#n25">25</a>, <a href="#n26">26</a>, <a href="#n27">27</a>,
<a href="#n28">28</a>, <a href="#n29">29</a>, <a href="#n30">30</a>, <a href="#n31">31</a>, <a href="#n32">32</a>,
<a href="#n33">33</a>, <a href="#n34">34</a>, <a href="#n35">35</a>, <a href="#n36">36</a>, <a href="#n37">37</a>,
<a href="#n38">38</a>, <a href="#n39">39</a>, <a href="#n40">40</a>, <a href="#n41">41</a>, <a href="#n42">42</a>,
<a href="#n43">43</a>, <a href="#n44">44</a>, <a href="#n45">45</a>, <a href="#n46">46</a>, <a href="#n47">47</a>,
<a href="#n48">48</a>, <a href="#n49">49</a>, <a href="#n50">50</a>, <a href="#n51">51</a>, <a href="#n52">52</a>,
<a href="#n53">53</a>, <a href="#n54">54</a>, <a href="#n55">55</a>, <a href="#n56">56</a>, <a href="#n57">57</a>,
<a href="#n58">58</a>, <a href="#n59">59</a>, <a href="#n60">60</a>, <a href="#n61">61</a>, <a href="#n62">62</a>,
<a href="#n63">63</a>, <a href="#n64">64</a>
</div>
</td>
</tr>
</table>
<table class="tbl">
<thead>
<tr>
<td><i>i</i></td>
<td>weight - w<sub>i</sub></td>
<td>abscissa - <i>x</i><sub>i</sub></td>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>0.2491470458134028</td>
<td>-0.1252334085114689</td>
</tr>
<tr>
<td>2</td>
<td>0.2491470458134028</td>
<td>0.1252334085114689</td>
</tr>
<tr>
<td>3</td>
<td>0.2334925365383548</td>
<td>-0.3678314989981802</td>
</tr>
<tr>
<td>4</td>
<td>0.2334925365383548</td>
<td>0.3678314989981802</td>
</tr>
<tr>
<td>5</td>
<td>0.2031674267230659</td>
<td>-0.5873179542866175</td>
</tr>
<tr>
<td>6</td>
<td>0.2031674267230659</td>
<td>0.5873179542866175</td>
</tr>
<tr>
<td>7</td>
<td>0.1600783285433462</td>
<td>-0.7699026741943047</td>
</tr>
<tr>
<td>8</td>
<td>0.1600783285433462</td>
<td>0.7699026741943047</td>
</tr>
<tr>
<td>9</td>
<td>0.1069393259953184</td>
<td>-0.9041172563704749</td>
</tr>
<tr>
<td>10</td>
<td>0.1069393259953184</td>
<td>0.9041172563704749</td>
</tr>
<tr>
<td>11</td>
<td>0.0471753363865118</td>
<td>-0.9815606342467192</td>
</tr>
<tr>
<td>12</td>
<td>0.0471753363865118</td>
<td>0.9815606342467192</td>
</tr>
</tbody>
</table>
</div>
<div id="n13">
<a name="n13"></a>
<table>
<tr>
<td nowrap="nowrap" style="padding-right: 3em;"><h2>n = 13</h2></td>
<td>
<div class="anchors">
jump to n = <a href="#n2">2</a>, <a href="#n3">3</a>, <a href="#n4">4</a>, <a href="#n5">5</a>, <a href="#n6">6</a>,
<a href="#n7">7</a>, <a href="#n8">8</a>, <a href="#n9">9</a>, <a href="#n10">10</a>, <a href="#n11">11</a>, <a href="#n12">12</a>,
<a href="#n13">13</a>, <a href="#n14">14</a>, <a href="#n15">15</a>, <a href="#n16">16</a>, <a href="#n17">17</a>,
<a href="#n18">18</a>, <a href="#n19">19</a>, <a href="#n20">20</a>, <a href="#n21">21</a>, <a href="#n22">22</a>,
<a href="#n23">23</a>, <a href="#n24">24</a>, <a href="#n25">25</a>, <a href="#n26">26</a>, <a href="#n27">27</a>,
<a href="#n28">28</a>, <a href="#n29">29</a>, <a href="#n30">30</a>, <a href="#n31">31</a>, <a href="#n32">32</a>,
<a href="#n33">33</a>, <a href="#n34">34</a>, <a href="#n35">35</a>, <a href="#n36">36</a>, <a href="#n37">37</a>,
<a href="#n38">38</a>, <a href="#n39">39</a>, <a href="#n40">40</a>, <a href="#n41">41</a>, <a href="#n42">42</a>,
<a href="#n43">43</a>, <a href="#n44">44</a>, <a href="#n45">45</a>, <a href="#n46">46</a>, <a href="#n47">47</a>,
<a href="#n48">48</a>, <a href="#n49">49</a>, <a href="#n50">50</a>, <a href="#n51">51</a>, <a href="#n52">52</a>,
<a href="#n53">53</a>, <a href="#n54">54</a>, <a href="#n55">55</a>, <a href="#n56">56</a>, <a href="#n57">57</a>,
<a href="#n58">58</a>, <a href="#n59">59</a>, <a href="#n60">60</a>, <a href="#n61">61</a>, <a href="#n62">62</a>,
<a href="#n63">63</a>, <a href="#n64">64</a>
</div>
</td>
</tr>
</table>
<table class="tbl">
<thead>
<tr>
<td><i>i</i></td>
<td>weight - w<sub>i</sub></td>
<td>abscissa - <i>x</i><sub>i</sub></td>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>0.2325515532308739</td>
<td>0.0000000000000000</td>
</tr>
<tr>
<td>2</td>
<td>0.2262831802628972</td>
<td>-0.2304583159551348</td>
</tr>
<tr>
<td>3</td>
<td>0.2262831802628972</td>
<td>0.2304583159551348</td>
</tr>
<tr>
<td>4</td>
<td>0.2078160475368885</td>
<td>-0.4484927510364469</td>
</tr>
<tr>
<td>5</td>
<td>0.2078160475368885</td>
<td>0.4484927510364469</td>
</tr>
<tr>
<td>6</td>
<td>0.1781459807619457</td>
<td>-0.6423493394403402</td>
</tr>
<tr>
<td>7</td>
<td>0.1781459807619457</td>
<td>0.6423493394403402</td>