计算负实数值的 Gamma 函数（C++、Boost）

Question

以下无限严重需要计算非整数、负数、实数的阶乘：

（这是计算椭圆周长的一种方法，a和b是半长轴和半短轴，h定义为：
h = (ab)^2/(a+b)^2）

阶乘函数可以通过 Gamma 函数扩展到负值，该函数是为所有非负整数的实数定义的。

在编码严重时，我尝试了 boost::math::factorial 和 boost::math::tgamma ，它们给出的结果仅低至 -1 （不包括）-1.5 例如给出错误。

    #include <iostream>
    #include <boost/math/special_functions/factorials.hpp>

    int main()
    {
        double x;
        double f;
        double tg;

        x = -0.5;
        f = boost::math::factorial<double>(x);
        tg = boost::math::tgamma<double>(x);
        cout << "factorial of " << x << " = " << f << endl;
        cout << "tgamma of " << x << " = " << tg << endl << endl;

        x = -1.5;
        f = boost::math::factorial<double>(x);
        tg = boost::math::tgamma<double>(x);
        cout << "factorial of " << x << " = " << f << endl;
        cout << "tgamma of " << x << " = " << tg << endl << endl;
    
        return 0;
    }

输出：

-0.5 = 1
tgamma of -0.5 = -3.54491 的
阶乘在抛出 'boost::exception_detail::clone_implboost::exception_detail::error_info_injector<std::domain_error >' What() 实例后终止调用：函数 boost 中出错： :math::tgamma(long double)：以负整数 0 计算 tgamma。中止（核心转储）

增强阶乘： 增强阶乘
增强 tgamma： 增强 tgamma

我的问题：

1. 是否有替代 boost 的方法可以计算其负域的伽玛函数？
2. 我在上面链接的 boost 文档中找不到阶乘和 tgamma 函数的实现域。事实上，我可能只是错误地使用了它们。如何确定域/正确用法确实是什么？

谢谢。

Answer 1

我明白出了什么问题。该函数根据定义boost::math::factorial接受一个unsigned整数：

template <class T>
inline T factorial(unsigned i)
{
   return factorial<T>(i, policies::policy<>());
}

这意味着如果您使用 double 调用它，它将隐式转换为无符号。那不是你想要的。另外，factorial最终tgamma在内部使用，所以你得到这个：

#include <boost/math/special_functions/factorials.hpp>
#include <iostream>

void foo(long double x) {
    using namespace boost::math;
    try {
        auto f = factorial<long double>(x);
        std::cout << "factorial of " << static_cast<unsigned>(x) << " = " << f << "\n";
    } catch(std::exception const& e) {
        std::cout << "error at " << static_cast<unsigned>(x) << ": " << std::quoted(e.what()) << "\n";
    }
}

int main() {
    std::cout << std::unitbuf;
    foo(-2);
}

最终会这样做：

#0  boost::math::tgamma<long double, boost::math::policies::policy<boost::math::policies::default_policy, boost::math::policies::default_policy, boost::math::policies::default_policy, boost::math::policies::default_policy, boost::math::policies::default_policy, boost::math::policies::default_policy, boost::math::policies::default_policy, boost::math::policies::default_policy, boost::math::policies::default_policy, boost::math::policies::default_policy, boost::math::policies::default_policy, boost::math::policies::default_policy, boost::math::policies::default_policy> > (a=4294967295, z=...)
    at /home/sehe/custom/boost_1_73_0/boost/math/special_functions/gamma.hpp:1994
No locals.
#1  0x0000555555558eb3 in boost::math::factorial<long double, boost::math::policies::policy<boost::math::policies::default_policy, boost::math::policies::default_policy, boost::math::policies::default_policy, boost::math::policies::default_policy, boost::math::policies::default_policy, boost::math::policies::default_policy, boost::math::policies::default_policy, boost::math::policies::default_policy, boost::math::policies::default_policy, boost::math::policies::default_policy, boost::math::policies::default_policy, boost::math::policies::default_policy, boost::math::policies::default_policy> > (i=4294967294, pol=...)
    at /home/sehe/custom/boost_1_73_0/boost/math/special_functions/factorials.hpp:44
        result = -0.667762310955655363645
#2  0x0000555555558674 in boost::math::factorial<long double> (i=4294967294)
    at /home/sehe/custom/boost_1_73_0/boost/math/special_functions/factorials.hpp:53
No locals.
#3  0x0000555555557792 in foo (x=-2) at /home/sehe/Projects/stackoverflow/test.cpp:7
        f = <invalid float value>
#4  0x000055555555791f in main () at /home/sehe/Projects/stackoverflow/test.cpp:16
No locals.

所以它试图给你boost::math::factorial<long double> (i=4294967294)

使固定

请勿用于factorials非负整数以外的其他整数。

实时编译器资源管理器

#include <boost/math/special_functions/factorials.hpp>
#include <iostream>

void foo(long double x) {
    using namespace boost::math;
    try {
        auto tg = tgamma<long double>(x);
        std::cout << "tgamma    of " << x << " = " << tg << "\n" << std::endl;
    } catch(std::exception const& e) {
        std::cout << "error at " << x << ": " << std::quoted(e.what()) << std::endl;
    }
}

int main() {
    for (auto x : { 1., 2., 3., 4., 5., -.2, -2., -.5, -1.5 })
        foo(x);
}

印刷：

tgamma    of 1 = 1

tgamma    of 2 = 1

tgamma    of 3 = 2

tgamma    of 4 = 6

tgamma    of 5 = 24

tgamma    of -0.2 = -5.82115

error at -2: "Error in function boost::math::tgamma<long double>(long double): Evaluation of tgamma at a negative integer -2."
tgamma    of -0.5 = -3.54491

tgamma    of -1.5 = 2.36327

可以理解，它在 处溢出-2，但这是正确的。