165 lines
13 KiB
HTML
165 lines
13 KiB
HTML
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<title>std::beta, std::betaf, std::betal</title>
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<body class="mediawiki ltr sitedir-ltr ns-0 ns-subject page-cpp_numeric_special_functions_beta 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>beta, <span style="font-size:0.7em; line-height:130%">std::</span>betaf, <span style="font-size:0.7em; line-height:130%">std::</span>betal</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> beta<span class="br0">(</span> <span class="kw4">double</span> x, <span class="kw4">double</span> y <span class="br0">)</span><span class="sy4">;</span><br /></span>
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<p><span class="kw4">float</span> betaf<span class="br0">(</span> <span class="kw4">float</span> x, <span class="kw4">float</span> y <span class="br0">)</span><span class="sy4">;</span><br /></p>
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<span class="kw4">long</span> <span class="kw4">double</span> betal<span class="br0">(</span> <span class="kw4">long</span> <span class="kw4">double</span> x, <span class="kw4">long</span> <span class="kw4">double</span> y <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">Promoted beta<span class="br0">(</span> Arithmetic x, Arithmetic y <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> 计算 <code>x</code> 与 <code>y</code> 的 <a href="https://en.wikipedia.org/wiki/Beta_function" class="extiw" title="enwiki:Beta function">Β 函数</a>。</div>
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<div class="t-li1"><span class="t-li">2)</span> 所有 <span class="t-v">(1)</span> 所不覆盖的算术类型参数组合的重载集或函数模板。若任何参数拥有<a href="cpp-types-is_integral.html">整数类型</a>，则它被转型到 <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span></span></span> 。若任何参数为 <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">long</span> <span class="kw4">double</span></span></span> ，则返回类型 <code>Promoted</code> 亦为 <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">long</span> <span class="kw4">double</span></span></span> ，否则返回类型始终是 <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span></span></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>x, y</td>
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<td>-</td>
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<td>浮点或整数类型值</td>
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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> 与 <code>y</code> 的 beta 函数值，即 <span class="mjax" style="display:none">\(\int_{0}^{1}{ {t}^{x-1}{(1-t)}^{y-1}\mathsf{d}t}\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;"><span style="font-size: 1.3em; vertical-align:-0.08em; margin-right: 0.08em;">∫</span><span class="t-su t-su-b">1<br />
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0</span>t<span class="t-su">x-1<br /></span>(1-t)<span class="t-su">(y-1)<br /></span>d<i>t</i></span> ，或等价地为 <span class="mjax" style="display:none">\(\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;"><span class="t-mfrac"></span></span>
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<table>
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<td>Γ(x)Γ(y)</td>
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<td>Γ(x+y)</td>
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</tr>
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</table>
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。
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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>若任一参数是 NaN ，则返回 NaN 且不报告定义域错误</li>
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<li>仅要求函数在 <code>x</code> 与 <code>y</code> 均大于零的情况有定义，并且允许其他情况下报告定义域错误。</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_beta/beta_function.html">可用于 boost.math</a></p>
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<p><span class="t-c"><span class="mw-geshi cpp source-cpp">beta<span class="br0">(</span>x, y<span class="br0">)</span></span></span> 等于 <span class="t-c"><span class="mw-geshi cpp source-cpp">beta<span class="br0">(</span>y, x<span class="br0">)</span></span></span></p>
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当 <code>x</code> 与 <code>y</code> 为正整数时， beta(x,y) 等于 <span class="mjax" style="display:none">\(\frac{(x-1)!(y-1)!}{(x+y-1)!}\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;"><span class="t-mfrac"></span></span>
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<table>
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<td>(x-1)!(y-1)!</td>
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</tr>
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<td>(x+y-1)!</td>
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</table>
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。
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<p><br /></p>
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二项式系数可用 Β 函数表示： <span class="mjax" style="display:none">\(\binom{n}{k} = \frac{1}{(n+1)B(n-k+1,k+1)}\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;"><span class="t-mparen"><span style="font-size: 1em">⎛<br />
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⎜<br />
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⎝</span><span style="">n<br />
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k</span><span style="font-size: 1em">⎞<br />
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⎟<br />
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⎠</span></span>=<span class="t-mfrac"></span></span>
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<table>
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<td>1</td>
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</tr>
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<tr>
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<td>(n+1)Β(n-k+1,k+1)</td>
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</tr>
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</table>
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。
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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 <string></span>
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<span class="co2">#include <iostream></span>
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<span class="co2">#include <iomanip></span>
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<span class="kw4">double</span> binom<span class="br0">(</span><span class="kw4">int</span> n, <span class="kw4">int</span> k<span class="br0">)</span> <span class="br0">{</span> <span class="kw1">return</span> <span class="nu0">1</span><span class="sy2">/</span><span class="br0">(</span><span class="br0">(</span>n<span class="sy2">+</span><span class="nu0">1</span><span class="br0">)</span><span class="sy2">*</span><a href="cpp-numeric-special_math-beta.html"><span class="kw1393">std::<span class="me2">beta</span></span></a><span class="br0">(</span>n<span class="sy2">-</span>k<span class="sy2">+</span><span class="nu0">1</span>,k<span class="sy2">+</span><span class="nu0">1</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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<a href="cpp-io-cout.html"><span class="kw1758">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> <span class="st0">"Pascal's triangle:<span class="es1">\n</span>"</span><span class="sy4">;</span>
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<span class="kw1">for</span><span class="br0">(</span><span class="kw4">int</span> n <span class="sy1">=</span> <span class="nu0">1</span><span class="sy4">;</span> n <span class="sy1"><</span> <span class="nu0">10</span><span class="sy4">;</span> <span class="sy2">++</span>n<span class="br0">)</span> <span class="br0">{</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-string-basic_string.html"><span class="kw1230">std::<span class="me2">string</span></span></a><span class="br0">(</span><span class="nu0">20</span><span class="sy2">-</span>n<span class="sy2">*</span><span class="nu0">2</span>, <span class="st0">' '</span><span class="br0">)</span><span class="sy4">;</span>
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<span class="kw1">for</span><span class="br0">(</span><span class="kw4">int</span> k <span class="sy1">=</span> <span class="nu0">1</span><span class="sy4">;</span> k <span class="sy1"><</span> n<span class="sy4">;</span> <span class="sy2">++</span>k<span class="br0">)</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-io-manip-setw.html"><span class="kw1797">std::<span class="me2">setw</span></span></a><span class="br0">(</span><span class="nu0">3</span><span class="br0">)</span> <span class="sy1"><<</span> binom<span class="br0">(</span>n,k<span class="br0">)</span> <span class="sy1"><<</span> <span class="st0">' '</span><span class="sy4">;</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> <span class="st0">'<span class="es1">\n</span>'</span><span class="sy4">;</span>
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<span class="br0">}</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">Pascal's triangle:
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2
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3 3
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4 6 4
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5 10 10 5
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6 15 20 15 6
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7 21 35 35 21 7
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8 28 56 70 56 28 8
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9 36 84 126 126 84 36 9</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-math-tgamma.html"><span class="t-lines"><span>tgamma</span><span>tgammaf</span><span>tgammal</span></span></a></div>
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<div><span class="t-lines"><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span></span></div>
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</div>
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</td>
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<td>gamma 函数<br />
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<span class="t-mark">(函数)</span></td>
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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/BetaFunction.html">Weisstein, Eric W. “ Β 函数。”</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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