使用来自 boost::math 的 Gauss-Kronrod 求积对复杂函数进行积分

Question

我正在尝试使用 Boost 提供的 Gauss-Kronrod 求积法在 C++ 中以数值方式在域 [0, 5] 上集成一个具有实参和复数返回值的函数，例如 f(x) = 1 + i。

我已经使用 boost 来解决真正有效的被积函数，因此该库已正确安装。 documentation 表示“Gauss-Kronrod 正交支持在实线上定义的被积函数并返回复数值。”

#include <iostream>
#include <cmath>
#include <complex>
#include <boost/math/quadrature/gauss_kronrod.hpp>

using namespace boost::math::quadrature;
using complex = std::complex<double>;


complex f(double t){
    return complex{1, 1};
};


int main() {
    complex error;
    complex a{0};
    complex b{5};
    unsigned int max_depth = 0;
    complex tolerance = 0;
    complex Q = gauss_kronrod<complex, 61>::integrate(f, a, b, max_depth, tolerance, &error);
    std::cout << Q << ", " << error << "\n";
    return 0;
}

我的预期结果是 5 + 5i。我得到编译器错误

/usr/include/boost/math/quadrature/gauss_kronrod.hpp:1871:17: error: no match for 'operator<=' (operand types are 'std::complex<double>' and 'std::complex<double>')

意味着没有为 std::complex 定义一些运算符。我是否误读了文档并且无法进行复杂的集成？

编辑： 将我的代码更改为user14717 提供的版本会给我一个新的编译器错误：

/usr/include/boost/math/quadrature/gauss_kronrod.hpp: In instantiation of 'static boost::math::quadrature::gauss_kronrod<Real, N, Policy>::value_type boost::math::quadrature::gauss_kronrod<Real, N, Policy>::integrate(F, Real, Real, unsigned int, Real, Real*, Real*) [with F = std::complex<double> (*)(double); Real = double; unsigned int N = 61; Policy = 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::quadrature::gauss_kronrod<Real, N, Policy>::value_type = double]':
/home/olaf/testing numerical integration/boost/main.cpp:21:91:   required from here
/usr/include/boost/math/quadrature/gauss_kronrod.hpp:1877:47: error: cannot convert 'std::complex<double>' to 'double' in return
                return f(t*inv)*(1 + t_sq)*inv*inv;
                                               ^~~
/usr/include/boost/math/quadrature/gauss_kronrod.hpp:1890:32: error: cannot convert ?std::complex<double>? to 'double' in return
                return f(arg)*z*z;
                                ^
/usr/include/boost/math/quadrature/gauss_kronrod.hpp:1907:40: error: cannot convert 'std::complex<double>' to 'double' in return
                return f(b - arg) * z * z;
                                        ^

Answer 1

解决方法如下：

#include <iostream>
#include <cmath>
#include <complex>
#include <boost/math/quadrature/gauss_kronrod.hpp>

using namespace boost::math::quadrature;
using complex = std::complex<double>;


complex f(double t){
    return complex{1, 1};
};


int main() {
    double error;
    double a{0};
    double b{5};
    unsigned int max_depth = 0;
    double tolerance = 0;
    complex Q = gauss_kronrod<double, 61>::integrate(f, a, b, max_depth, tolerance, &error);
    std::cout << Q << ", " << error << "\n";
    return 0;
}

您的错误是您在复杂类型上进行了模板化。模板参数必须是真实类型。

我注意到文档没有在此处提供示例，因此您确实无法知道这一点。我会为此提交一个补丁。 . .