My goal is to write a small library for spectral finite elements in Python and to that purpose I tried extending python with a C++ library using Boost, with the hope that it would make my code faster.

class Quad {public:Quad(int, int);double integrate(boost::function<double(std::vector<double> const&)> const&);double integrate_wrapper(boost::python::object const&);std::vector< std::vector<double> > nodes;std::vector<double> weights;
};...namespace std {typedef std::vector< std::vector< std::vector<double> > > cube;typedef std::vector< std::vector<double> > mat;typedef std::vector<double> vec;
}...double Quad::integrate(boost::function<double(vec const&)> const& func) {double result = 0.;for (unsigned int i = 0; i < nodes.size(); ++i) {result += func(nodes[i]) * weights[i];}return result;
}// ---- PYTHON WRAPPER ----
double Quad::integrate_wrapper(boost::python::object const& func) {std::function<double(vec const&)> lambda;switch (this->nodes[0].size()) {case 1: lambda = [&func](vec const& v) -> double { return boost::python::extract<double>(func (v[0])); }; break;case 2: lambda = [&func](vec const& v) -> double { return boost::python::extract<double>(func(v[0], v[1])); }; break;case 3: lambda = [&func](vec const& v) -> double { return boost::python::extract<double>(func(v[0], v[1], v[2])); }; break;default: cout << "Dimension must be 1, 2, or 3" << endl; exit(0);}return integrate(lambda);
}// ---- EXPOSE TO PYTHON ----
BOOST_PYTHON_MODULE(hermite)
{using namespace boost::python;class_<std::vec>("double_vector").def(vector_indexing_suite<std::vec>());class_<std::mat>("double_mat").def(vector_indexing_suite<std::mat>());class_<Quad>("Quad", init<int,int>()).def("integrate", &Quad::integrate_wrapper).def_readonly("nodes", &Quad::nodes).def_readonly("weights", &Quad::weights);
}

I compared the performance of three different methods to calculate the integral of two functions. The two functions are:

• The function f1(x,y,z) = x*x
• A function that is more difficult to evaluate: f2(x,y,z) = np.cos(2*x+2*y+2*z) + x*y + np.exp(-z*z) +np.cos(2*x+2*y+2*z) + x*y + np.exp(-z*z) +np.cos(2*x+2*y+2*z) + x*y + np.exp(-z*z) +np.cos(2*x+2*y+2*z) + x*y + np.exp(-z*z) +np.cos(2*x+2*y+2*z) + x*y + np.exp(-z*z)

The methods used are:

1. Call the library from a C++ program:

   double func(vector<double> v) {return F1_OR_F2;
   }int main() {hermite::Quad quadrature(100, 3);double result = quadrature.integrate(func);cout << "Result = " << result << endl;
   }
   

2. Call the library from a Python script:

   import hermite
   def function(x, y, z): return F1_OR_F2
   my_quad = hermite.Quad(100, 3)
   result = my_quad.integrate(function)
   

3. Use a for loop in Python:

   import hermite
   def function(x, y, z): return F1_OR_F2
   my_quad = hermite.Quad(100, 3)
   weights = my_quad.weights
   nodes = my_quad.nodes
   result = 0.
   for i in range(len(weights)):result += weights[i] * function(nodes[i][0], nodes[i][1], nodes[i][2])
   

Here are the execution times of each of the method (The time was measured using the time command for method 1, and the python module time for methods 2 and 3, and the C++ code was compiled using Cmake and set (CMAKE_BUILD_TYPE Release))

• For f1:

  • Method 1: 0.07s user 0.01s system 99% cpu 0.083 total
  • Method 2: 0.19s
  • Method 3: 3.06s

• For f2:

  • Method 1: 0.28s user 0.01s system 99% cpu 0.289 total
  • Method 2: 12.47s
  • Method 3: 16.31s

Based on these results, my questions are the following:

• Why is the first method so much faster than the second?

• Could the python wrapper be improved to reach comparable performance between methods 1 and 2?

• Why is method 2 more sensitive than method 3 to the difficulty of the function to integrate?

EDIT: I also tried to define a function that accepts a string as argument, writes it to a file, and proceeds to compile the file and dynamically load the resulting .so file:

double Quad::integrate_from_string(string const& function_body) {// Write function to fileofstream helper_file;helper_file.open("/tmp/helper_function.cpp");helper_file << "#include <vector>\n#include <cmath>\n";helper_file << "extern \"C\" double toIntegrate(std::vector<double> v) {\n";helper_file << "    return " << function_body << ";\n}";helper_file.close();// Compile filesystem("c++ /tmp/helper_function.cpp -o /tmp/helper_function.so -shared -fPIC");// Load function dynamicallytypedef double (*vec_func)(vec);void *function_so = dlopen("/tmp/helper_function.so", RTLD_NOW);vec_func func = (vec_func) dlsym(function_so, "toIntegrate");double result = integrate(func);dlclose(function_so);return result;
}

It's quite dirty and probably not very portable, so I'd be happy to find a better solution, but it works well and plays nicely with the ccode function of sympy.

SECOND EDIT I have rewritten the function in pure Python Using Numpy.

import numpy as np
import numpy.polynomial.hermite_e as herm
import time
def integrate(function, degrees):dim = len(degrees)nodes_multidim = []weights_multidim = []for i in range(dim):nodes_1d, weights_1d = herm.hermegauss(degrees[i])nodes_multidim.append(nodes_1d)weights_multidim.append(weights_1d)grid_nodes = np.meshgrid(*nodes_multidim)grid_weights = np.meshgrid(*weights_multidim)nodes_flattened = []weights_flattened = []for i in range(dim):nodes_flattened.append(grid_nodes[i].flatten())weights_flattened.append(grid_weights[i].flatten())nodes = np.vstack(nodes_flattened)weights = np.prod(np.vstack(weights_flattened), axis=0)return np.dot(function(nodes), weights)def function(v): return F1_OR_F2
result = integrate(function, [100,100,100])
print("-> Result = " + str(result) + ", Time = " + str(end-start))

Somewhat surprisingly (at least to me), there is no significant difference in performance between this method and the pure C++ implementation. In particular, it takes 0.059s for f1 and 0.36s for f2.

Your functions take vectors by value, which involves copying the vector. integrate_wrapper does extra copies.

It also makes sense to accept boost::function by reference and capture func by reference in those lambdas.

Change these to (note the & and const& bits):

double integrate(boost::function<double(std::vector<double> const&)> const&);double Quad::integrate_wrapper(boost::python::object func) {std::function<double(vec const&)> lambda;switch (this->nodes[0].size()) {case 1: lambda = [&func](vec const& v) -> double { return boost::python::extract<double>(func (v[0])); }; break;case 2: lambda = [&func](vec const& v) -> double { return boost::python::extract<double>(func(v[0], v[1])); }; break;case 3: lambda = [&func](vec const& v) -> double { return boost::python::extract<double>(func(v[0], v[1], v[2])); }; break;default: cout << "Dimension must be 1, 2, or 3" << endl; exit(0);}return integrate(lambda);
}

Still though, calling a Python function from C++ is more expensive than calling a C++ function.

People normally use numpy for fast linear algebra in Python, it uses SIMD for many common operations. You should probably consider using numpy first before rolling out a C++ implementation. In C++ you would have to use Intel MKL on Eigen to vectorize.