165 lines
14 KiB
HTML
165 lines
14 KiB
HTML
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<title>std::assoc_laguerre, std::assoc_laguerref, std::assoc_laguerrel</title>
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<body class="mediawiki ltr sitedir-ltr ns-0 ns-subject page-cpp_numeric_special_functions_assoc_laguerre skin-cppreference2 action-view cpp-navbar">
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<h1 id="firstHeading" class="firstHeading"><span style="font-size:0.7em; line-height:130%">std::</span>assoc_laguerre, <span style="font-size:0.7em; line-height:130%">std::</span>assoc_laguerref, <span style="font-size:0.7em; line-height:130%">std::</span>assoc_laguerrel</h1>
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<div id="contentSub"><span class="subpages">< <a href="cpp.html">cpp</a>‎ | <a href="cpp-numeric.html">numeric</a>‎ | <a href="cpp-numeric-special_functions.html">special functions</a></span></div>
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<div>定义于头文件 <code><a href="cpp-header-cmath.html"><cmath></a></code></div>
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<td></td>
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<div><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span> assoc_laguerre<span class="br0">(</span> <span class="kw4">unsigned</span> <span class="kw4">int</span> n, <span class="kw4">unsigned</span> <span class="kw4">int</span> m, <span class="kw4">double</span> x <span class="br0">)</span><span class="sy4">;</span><br /></span>
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<p><span class="kw4">float</span> assoc_laguerre<span class="br0">(</span> <span class="kw4">unsigned</span> <span class="kw4">int</span> n, <span class="kw4">unsigned</span> <span class="kw4">int</span> m, <span class="kw4">float</span> x <span class="br0">)</span><span class="sy4">;</span><br />
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<span class="kw4">long</span> <span class="kw4">double</span> assoc_laguerre<span class="br0">(</span> <span class="kw4">unsigned</span> <span class="kw4">int</span> n, <span class="kw4">unsigned</span> <span class="kw4">int</span> m, <span class="kw4">long</span> <span class="kw4">double</span> x <span class="br0">)</span><span class="sy4">;</span><br />
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<span class="kw4">float</span> assoc_laguerref<span class="br0">(</span> <span class="kw4">unsigned</span> <span class="kw4">int</span> n, <span class="kw4">unsigned</span> <span class="kw4">int</span> m, <span class="kw4">float</span> x <span class="br0">)</span><span class="sy4">;</span><br /></p>
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<span class="kw4">long</span> <span class="kw4">double</span> assoc_laguerrel<span class="br0">(</span> <span class="kw4">unsigned</span> <span class="kw4">int</span> n, <span class="kw4">unsigned</span> <span class="kw4">int</span> m, <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>
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<td><span class="t-mark-rev t-since-cxx17">(C++17 起)</span></td>
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<div><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span> assoc_laguerre<span class="br0">(</span> <span class="kw4">unsigned</span> <span class="kw4">int</span> n, <span class="kw4">unsigned</span> <span class="kw4">int</span> m, IntegralType x <span class="br0">)</span><span class="sy4">;</span></span></div>
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<td>(2)</td>
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<td><span class="t-mark-rev t-since-cxx17">(C++17 起)</span></td>
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<div class="t-li1"><span class="t-li">1)</span> 以参数 <code>x</code> 计算 <code>n</code> 次 <code>m</code> 阶<a href="https://en.wikipedia.org/wiki/Laguerre_polynomials#Generalized_Laguerre_polynomials" class="extiw" title="enwiki:Laguerre polynomials">关联拉盖尔多项式</a></div>
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<div class="t-li1"><span class="t-li">2)</span> 接受任意<a href="cpp-types-is_integral.html">整数类型</a>参数的重载集或函数模板。等价于将参数转型到 <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span></span></span> 后的 <span class="t-v">(1)</span> 。</div>
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<h3><span class="mw-headline" id=".E5.8F.82.E6.95.B0">参数</span></h3>
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<table class="t-par-begin">
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<td>n</td>
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<td>-</td>
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<td>多项式的次数，无符号整数类型值</td>
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<tr class="t-par">
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<td>m</td>
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<td>-</td>
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<td>多项式的阶数，无符号整数类型值</td>
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<tr class="t-par">
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<td>x</td>
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<td>-</td>
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<td>参数，浮点或整数类型值</td>
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</tr>
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<h3><span class="mw-headline" id=".E8.BF.94.E5.9B.9E.E5.80.BC">返回值</span></h3>
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若无错误发生，则返回 <code>x</code> 的关联拉盖尔多项式的值，即 <span class="mjax" style="display:none">\((-1)^m \: \frac{ \mathsf{d} ^ m}{ \mathsf{d}x ^ m} \, \mathsf{L}_{n+m}(x)\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">(-1)<span class="t-su">m<br /></span><span class="t-mfrac"></span></span>
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<td>d<span class="t-su">m<br /></span></td>
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<td>dx<span class="t-su">m<br /></span></td>
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L<span class="t-su t-su-b"><br />
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n+m</span>(x) （其中 <span class="mjax" style="display:none">\(\mathsf{L}_{n+m}(x)\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">L<span class="t-su t-su-b"><br />
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n+m</span>(x)</span> 是非关联拉盖尔多项式， <span class="t-c"><span class="mw-geshi cpp source-cpp"><a href="cpp-numeric-special_math-laguerre.html"><span class="kw1435">std::<span class="me2">laguerre</span></span></a><span class="br0">(</span>n<span class="sy2">+</span>m, x<span class="br0">)</span></span></span> ）。
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<h3><span class="mw-headline" id=".E9.94.99.E8.AF.AF.E5.A4.84.E7.90.86">错误处理</span></h3>
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<p>可能报告 <a href="cpp-numeric-math-math_errhandling.html"><tt>math_errhandling</tt></a> 中指定的错误。</p>
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<ul>
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<li>若参数是 NaN ，则返回 NaN 且不报告定义域错误</li>
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<li>若 <code>x</code> 为负，则可能出现定义域错误</li>
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<li>若 <code>n</code> 或 <code>m</code> 大于或等于 128 ，则行为是实现定义的。</li>
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</ul>
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<h3><span class="mw-headline" id=".E6.B3.A8.E8.A7.A3">注解</span></h3>
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<p>不支持 C++17 ，但支持 <a href="cpp-experimental-special_functions.html">ISO 29124:2010</a> 的实现会提供此函数，若实现定义了 <code>__STDCPP_MATH_SPEC_FUNCS__</code> 为至少 201003L 的值，且用户在包含任何标准库头文件前定义了 <code>__STDCPP_WANT_MATH_SPEC_FUNCS__</code> 。</p>
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<p>不支持 ISO 29124:2010 但支持 TR 19768:2007 (TR1) 的实现，在头文件 <code>tr1/cmath</code> 及命名空间 <code>std::tr1</code> 中提供此函数。</p>
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<p>此函数的一种实现亦<a rel="nofollow" class="external text" href="http://www.boost.org/doc/libs/release/libs/math/doc/html/math_toolkit/sf_poly/laguerre.html">可用于 boost.math</a></p>
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<p>关联拉盖尔多项式是方程 <span class="mjax" style="display:none">\(x\ddot{y} + (m+1-x)\dot{y} + ny = 0\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">xy<span class="t-su">,,<br /></span>+(m+1-x)y<span class="t-su">,<br /></span>+ny = 0</span> 的多项式解</p>
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<p>前几个解是：</p>
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<ul>
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<li>assoc_laguerre(0, m, x) = 1</li>
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<li>assoc_laguerre(1, m, x) = <span class="texhtml" style="white-space: nowrap;">-x + m + 1</span></li>
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<li>assoc_laguerre(2, m, x) = <span class="texhtml" style="white-space: nowrap;"><span class="t-mfrac"></span></span>
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<table>
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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>-2(m+2)x+(m+1)(m+2)]</li>
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<li>assoc_laguerre(3, m, x) = <span class="texhtml" style="white-space: nowrap;"><span class="t-mfrac"></span></span>
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<table>
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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>-3(m+3)x<span class="t-su">2<br /></span>-3(m+2)(m+3)x+(m+1)(m+2)(m+3)]</li>
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</ul>
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<h3><span class="mw-headline" id=".E7.A4.BA.E4.BE.8B">示例</span></h3>
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<div class="t-example">
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<div class="t-example-live-link"></div>
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<div dir="ltr" class="mw-geshi" style="text-align: left;">
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<div class="cpp source-cpp">
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<pre class="de1"><span class="co2">#include <cmath></span>
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<span class="co2">#include <iostream></span>
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<span class="kw4">double</span> L1<span class="br0">(</span><span class="kw4">unsigned</span> m, <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> m <span class="sy2">+</span> <span class="nu0">1</span><span class="sy4">;</span> <span class="br0">}</span>
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<span class="kw4">double</span> L2<span class="br0">(</span><span class="kw4">unsigned</span> m, <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">2</span><span class="sy2">*</span><span class="br0">(</span>m<span class="sy2">+</span><span class="nu0">2</span><span class="br0">)</span><span class="sy2">*</span>x<span class="sy2">+</span><span class="br0">(</span>m<span class="sy2">+</span><span class="nu0">1</span><span class="br0">)</span><span class="sy2">*</span><span class="br0">(</span>m<span class="sy2">+</span><span class="nu0">2</span><span class="br0">)</span><span class="br0">)</span><span class="sy4">;</span> <span class="br0">}</span>
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<span class="kw4">int</span> main<span class="br0">(</span><span class="br0">)</span>
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<span class="br0">{</span>
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<span class="co1">// 点检查</span>
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<a href="cpp-io-cout.html"><span class="kw1758">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> <a href="cpp-numeric-special_math-assoc_laguerre.html"><span class="kw1387">std::<span class="me2">assoc_laguerre</span></span></a><span class="br0">(</span><span class="nu0">1</span>, <span class="nu0">10</span>, <span class="nu16">0.5</span><span class="br0">)</span> <span class="sy1"><<</span> <span class="st0">'='</span> <span class="sy1"><<</span> L1<span class="br0">(</span><span class="nu0">10</span>, <span class="nu16">0.5</span><span class="br0">)</span> <span class="sy1"><<</span> <span class="st0">'<span class="es1">\n</span>'</span>
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<span class="sy1"><<</span> <a href="cpp-numeric-special_math-assoc_laguerre.html"><span class="kw1387">std::<span class="me2">assoc_laguerre</span></span></a><span class="br0">(</span><span class="nu0">2</span>, <span class="nu0">10</span>, <span class="nu16">0.5</span><span class="br0">)</span> <span class="sy1"><<</span> <span class="st0">'='</span> <span class="sy1"><<</span> L2<span class="br0">(</span><span class="nu0">10</span>, <span class="nu16">0.5</span><span class="br0">)</span> <span class="sy1"><<</span> <span class="st0">'<span class="es1">\n</span>'</span><span class="sy4">;</span>
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<span class="br0">}</span></pre></div>
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</div>
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<p>输出：</p>
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<div dir="ltr" class="mw-geshi" style="text-align: left;">
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<div class="text source-text">
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<pre class="de1">10.5=10.5
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60.125=60.125</pre></div>
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</div>
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</div>
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<h3><span class="mw-headline" id=".E5.8F.82.E9.98.85">参阅</span></h3>
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<table class="t-dsc-begin">
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<td>
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<div class="t-dsc-member-div">
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<div><a href="cpp-numeric-special_functions-laguerre.html"><span class="t-lines"><span>laguerre</span><span>laguerref</span><span>laguerrel</span></span></a></div>
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<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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</div>
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</td>
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<td>拉盖尔多项式<br />
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<span class="t-mark">(函数)</span></td>
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</tr>
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</table>
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<h3><span class="mw-headline" id=".E5.A4.96.E9.83.A8.E9.93.BE.E6.8E.A5">外部链接</span></h3>
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<p><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html">Weisstein, Eric W. “关联拉盖尔多项式”</a>来自 MathWorld--A Wolfram Web Resource 。</p>
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</div>
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<div class="visualClear"></div>
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