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<title>std::laguerre, std::laguerref, std::laguerrel</title>
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<h1 id="firstHeading" class="firstHeading"><span style="font-size:0.7em; line-height:130%">std::</span>laguerre, <span style="font-size:0.7em; line-height:130%">std::</span>laguerref, <span style="font-size:0.7em; line-height:130%">std::</span>laguerrel</h1>
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<div id="contentSub"><span class="subpages">&lt; <a href="cpp.html">cpp</a>&lrm; | <a href="cpp-numeric.html">numeric</a>&lrm; | <a href="cpp-numeric-special_functions.html">special functions</a></span></div>
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<div>&#23450;&#20041;&#20110;&#22836;&#25991;&#20214; <code><a href="cpp-header-cmath.html">&lt;cmath&gt;</a></code></div>
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<div><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span> &nbsp; &nbsp; &nbsp;laguerre<span class="br0">(</span> <span class="kw4">unsigned</span> <span class="kw4">int</span> n, <span class="kw4">double</span> x <span class="br0">)</span><span class="sy4">;</span><br /></span>
<p><span class="kw4">float</span> &nbsp; &nbsp; &nbsp; laguerre<span class="br0">(</span> <span class="kw4">unsigned</span> <span class="kw4">int</span> n, <span class="kw4">float</span> x <span class="br0">)</span><span class="sy4">;</span><br />
<span class="kw4">long</span> <span class="kw4">double</span> laguerre<span class="br0">(</span> <span class="kw4">unsigned</span> <span class="kw4">int</span> n, <span class="kw4">long</span> <span class="kw4">double</span> x <span class="br0">)</span><span class="sy4">;</span><br />
<span class="kw4">float</span> &nbsp; &nbsp; &nbsp; laguerref<span class="br0">(</span> <span class="kw4">unsigned</span> <span class="kw4">int</span> n, <span class="kw4">float</span> x <span class="br0">)</span><span class="sy4">;</span><br /></p>
<span class="kw4">long</span> <span class="kw4">double</span> laguerrel<span class="br0">(</span> <span class="kw4">unsigned</span> <span class="kw4">int</span> n, <span class="kw4">long</span> <span class="kw4">double</span> x <span class="br0">)</span><span class="sy4">;</span></div>
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<td>(1)</td>
<td><span class="t-mark-rev t-since-cxx17">(C++17 &#36215;)</span></td>
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<div><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span> &nbsp; &nbsp; &nbsp;laguerre<span class="br0">(</span> <span class="kw4">unsigned</span> <span class="kw4">int</span> n, IntegralType x <span class="br0">)</span><span class="sy4">;</span></span></div>
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<td>(2)</td>
<td><span class="t-mark-rev t-since-cxx17">(C++17 &#36215;)</span></td>
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<div class="t-li1"><span class="t-li">1)</span> &#20197;&#21442;&#25968; <code>x</code> &#35745;&#31639; <code>n</code> &#27425;&#38750;&#20851;&#32852;<a href="https://en.wikipedia.org/wiki/Laguerre_polynomials" class="extiw" title="enwiki:Laguerre polynomials">&#25289;&#30422;&#23572;&#22810;&#39033;&#24335;</a></div>
<div class="t-li1"><span class="t-li">2)</span> &#25509;&#21463;&#20219;&#20309;<a href="cpp-types-is_integral.html">&#25972;&#25968;&#31867;&#22411;</a>&#21442;&#25968;&#30340;&#37325;&#36733;&#38598;&#25110;&#20989;&#25968;&#27169;&#26495;&#12290;&#31561;&#20215;&#20110;&#23558;&#21442;&#25968;&#36716;&#22411;&#21040; <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span></span></span> &#21518;&#30340; <span class="t-v">(1)</span> &#12290;</div>
<h3><span class="mw-headline" id=".E5.8F.82.E6.95.B0">&#21442;&#25968;</span></h3>
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<td>n</td>
<td>-</td>
<td>&#22810;&#39033;&#24335;&#30340;&#27425;&#25968;&#65292;&#26080;&#31526;&#21495;&#25972;&#25968;&#31867;&#22411;&#30340;&#20540;</td>
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<td>x</td>
<td>-</td>
<td>&#21442;&#25968;&#65292;&#28014;&#28857;&#25110;&#25972;&#25968;&#31867;&#22411;&#30340;&#20540;</td>
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<h3><span class="mw-headline" id=".E8.BF.94.E5.9B.9E.E5.80.BC">&#36820;&#22238;&#20540;</span></h3>
&#33509;&#26080;&#38169;&#35823;&#21457;&#29983;&#65292;&#21017;&#36820;&#22238; <code>x</code> &#30340; <code>n</code> &#38454;&#38750;&#20851;&#32852;&#25289;&#30422;&#23572;&#22810;&#39033;&#24335;&#30340;&#20540;&#65292;&#21363; <span class="texhtml" style="white-space: nowrap;"><span class="t-mfrac"></span></span>
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<td><i>e</i><sup>x</sup></td>
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<td>n!</td>
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<span class="t-mfrac"></span>
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<td>d<span class="t-su">n<br /></span></td>
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<td>dx<span class="t-su">n<br /></span></td>
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(x<span class="t-su">n<br /></span><i>e</i><sup>-x</sup>) &#12290;
<h3><span class="mw-headline" id=".E9.94.99.E8.AF.AF.E5.A4.84.E7.90.86">&#38169;&#35823;&#22788;&#29702;</span></h3>
<p>&#21487;&#33021;&#25253;&#21578; <a href="cpp-numeric-math-math_errhandling.html"><tt>math_errhandling</tt></a> &#20013;&#25351;&#23450;&#30340;&#38169;&#35823;</p>
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<li>&#33509;&#21442;&#25968;&#26159; NaN &#65292;&#21017;&#36820;&#22238; NaN &#19988;&#19981;&#25253;&#21578;&#23450;&#20041;&#22495;&#38169;&#35823;</li>
<li>&#33509; <code>x</code> &#20026;&#36127;&#65292;&#21017;&#21487;&#33021;&#21457;&#29983;&#23450;&#20041;&#22495;&#38169;&#35823;</li>
<li>&#33509; <code>n</code> &#22823;&#20110;&#25110;&#31561;&#20110; 128 &#65292;&#21017;&#34892;&#20026;&#26159;&#23454;&#29616;&#23450;&#20041;&#30340;</li>
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<h3><span class="mw-headline" id=".E6.B3.A8.E8.A7.A3">&#27880;&#35299;</span></h3>
<p>&#19981;&#25903;&#25345; C++17 &#65292;&#20294;&#25903;&#25345; <a href="cpp-experimental-special_functions.html">ISO 29124:2010</a> &#30340;&#23454;&#29616;&#20250;&#25552;&#20379;&#27492;&#20989;&#25968;&#65292;&#33509;&#23454;&#29616;&#23450;&#20041;&#20102; <code>__STDCPP_MATH_SPEC_FUNCS__</code> &#20026;&#33267;&#23569; 201003L &#30340;&#20540;&#65292;&#19988;&#29992;&#25143;&#22312;&#21253;&#21547;&#20219;&#20309;&#26631;&#20934;&#24211;&#22836;&#25991;&#20214;&#21069;&#23450;&#20041;&#20102; <code>__STDCPP_WANT_MATH_SPEC_FUNCS__</code> &#12290;</p>
<p>&#19981;&#25903;&#25345; ISO 29124:2010 &#20294;&#25903;&#25345; TR 19768:2007 (TR1) &#30340;&#23454;&#29616;&#65292;&#22312;&#22836;&#25991;&#20214; <code>tr1/cmath</code> &#21450;&#21629;&#21517;&#31354;&#38388; <code>std::tr1</code> &#20013;&#25552;&#20379;&#27492;&#20989;&#25968;&#12290;</p>
<p>&#27492;&#20989;&#25968;&#30340;&#19968;&#31181;&#23454;&#29616;&#20134;<a rel="nofollow" class="external text" href="http://www.boost.org/doc/libs/release/libs/math/doc/html/math_toolkit/sf_poly/laguerre.html">&#21487;&#29992;&#20110; boost.math</a></p>
<p>&#25289;&#30422;&#23572;&#22810;&#39033;&#24335;&#26159;&#26041;&#31243; <span class="texhtml" style="white-space: nowrap;">xy<span class="t-su">,,<br /></span>+(1-x)y<span class="t-su">,<br /></span>+ny = 0</span> &#30340;&#22810;&#39033;&#24335;&#35299;</p>
<p>&#21069;&#20960;&#20010;&#35299;&#26159;&#65306;</p>
<ul>
<li>laguerre(0, x) = 1</li>
<li>laguerre(1, x) = <span class="texhtml" style="white-space: nowrap;">-x + 1</span></li>
<li>laguerre(2, x) = <span class="texhtml" style="white-space: nowrap;"><span class="t-mfrac"></span></span>
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<td>1</td>
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<td>2</td>
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[x<span class="t-su">2<br /></span>-4x+2]</li>
<li>laguerre(3, x) = <span class="texhtml" style="white-space: nowrap;"><span class="t-mfrac"></span></span>
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<td>1</td>
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<td>6</td>
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[-x<span class="t-su">3<br /></span>-9x<span class="t-su">2<br /></span>-18x+6]</li>
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<h3><span class="mw-headline" id=".E7.A4.BA.E4.BE.8B">&#31034;&#20363;</span></h3>
<div class="t-example">
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<div class="cpp source-cpp">
<pre class="de1"><span class="co2">#define __STDCPP_WANT_MATH_SPEC_FUNCS__ 1</span>
<span class="co2">#include &lt;cmath&gt;</span>
<span class="co2">#include &lt;iostream&gt;</span>
<span class="kw4">double</span> L1<span class="br0">(</span><span class="kw4">double</span> x<span class="br0">)</span> <span class="br0">{</span> <span class="kw1">return</span> <span class="sy2">-</span>x <span class="sy2">+</span> <span class="nu0">1</span><span class="sy4">;</span> <span class="br0">}</span>
<span class="kw4">double</span> L2<span class="br0">(</span><span class="kw4">double</span> x<span class="br0">)</span> <span class="br0">{</span> <span class="kw1">return</span> <span class="nu16">0.5</span><span class="sy2">*</span><span class="br0">(</span>x<span class="sy2">*</span>x<span class="sy2">-</span><span class="nu0">4</span><span class="sy2">*</span>x<span class="sy2">+</span><span class="nu0">2</span><span class="br0">)</span><span class="sy4">;</span> <span class="br0">}</span>
<span class="kw4">int</span> main<span class="br0">(</span><span class="br0">)</span>
<span class="br0">{</span>
<span class="co1">// &#28857;&#26816;&#26597;</span>
<a href="cpp-io-cout.html"><span class="kw1758">std::<span class="me2">cout</span></span></a> <span class="sy1">&lt;&lt;</span> <a href="cpp-numeric-special_math-laguerre.html"><span class="kw1435">std::<span class="me2">laguerre</span></span></a><span class="br0">(</span><span class="nu0">1</span>, <span class="nu16">0.5</span><span class="br0">)</span> <span class="sy1">&lt;&lt;</span> <span class="st0">&#39;=&#39;</span> <span class="sy1">&lt;&lt;</span> L1<span class="br0">(</span><span class="nu16">0.5</span><span class="br0">)</span> <span class="sy1">&lt;&lt;</span> <span class="st0">&#39;<span class="es1">\n</span>&#39;</span>
<span class="sy1">&lt;&lt;</span> <a href="cpp-numeric-special_math-laguerre.html"><span class="kw1435">std::<span class="me2">laguerre</span></span></a><span class="br0">(</span><span class="nu0">2</span>, <span class="nu16">0.5</span><span class="br0">)</span> <span class="sy1">&lt;&lt;</span> <span class="st0">&#39;=&#39;</span> <span class="sy1">&lt;&lt;</span> L2<span class="br0">(</span><span class="nu16">0.5</span><span class="br0">)</span> <span class="sy1">&lt;&lt;</span> <span class="st0">&#39;<span class="es1">\n</span>&#39;</span><span class="sy4">;</span>
<span class="br0">}</span></pre></div>
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<p>&#36755;&#20986;&#65306;</p>
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<pre class="de1">0.5=0.5
0.125=0.125</pre></div>
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<h3><span class="mw-headline" id=".E5.8F.82.E9.98.85">&#21442;&#38405;</span></h3>
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<div><a href="cpp-numeric-special_functions-assoc_laguerre.html"><span class="t-lines"><span>assoc_laguerre</span><span>assoc_laguerref</span><span>assoc_laguerrel</span></span></a></div>
<div><span class="t-lines"><span><span class="t-mark-rev t-since-cxx17">(C++17)</span></span><span><span class="t-mark-rev t-since-cxx17">(C++17)</span></span><span><span class="t-mark-rev t-since-cxx17">(C++17)</span></span></span></div>
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<td>&#36830;&#24102;&#25289;&#30422;&#23572;&#22810;&#39033;&#24335;<br />
<span class="t-mark">(&#20989;&#25968;)</span></td>
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<h3><span class="mw-headline" id=".E5.A4.96.E9.83.A8.E9.93.BE.E6.8E.A5">&#22806;&#37096;&#38142;&#25509;</span></h3>
<p><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/LaguerrePolynomial.html">Weisstein, Eric W. &ldquo;&#25289;&#30422;&#23572;&#22810;&#39033;&#24335;&rdquo;</a>&#26469;&#33258; MathWorld--A Wolfram Web Resource &#12290;</p>
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