Question

I’m using right now the scipy.integrate.quad to successfully integrate some real integrands. Now a situation appeared that I need to integrate a complex integrand. quad seems not be able to do it, as the other scipy.integrate routines, so I ask: is there any way to integrate a complex integrand using scipy.integrate, without having to separate the integral in the real and the imaginary parts?

Answer 1

What’s wrong with just separating it out into real and imaginary parts? scipy.integrate.quad requires the integrated function return floats (aka real numbers) for the algorithm it uses.

import scipy
from scipy.integrate import quad

def complex_quadrature(func, a, b, **kwargs):
    def real_func(x):
        return scipy.real(func(x))
    def imag_func(x):
        return scipy.imag(func(x))
    real_integral = quad(real_func, a, b, **kwargs)
    imag_integral = quad(imag_func, a, b, **kwargs)
    return (real_integral[0] + 1j*imag_integral[0], real_integral[1:], imag_integral[1:])

E.g.,

>>> complex_quadrature(lambda x: (scipy.exp(1j*x)), 0,scipy.pi/2)
((0.99999999999999989+0.99999999999999989j),
 (1.1102230246251564e-14,),
 (1.1102230246251564e-14,))

which is what you expect to rounding error – integral of exp(i x) from 0, pi/2 is (1/i)(e^i pi/2 – e^0) = -i(i – 1) = 1 + i ~ (0.99999999999999989+0.99999999999999989j).

And for the record in case it isn’t 100% clear to everyone, integration is a linear functional, meaning that ∫ { f(x) + k g(x) } dx = ∫ f(x) dx + k ∫ g(x) dx (where k is a constant with respect to x). Or for our specific case ∫ z(x) dx = ∫ Re z(x) dx + i ∫ Im z(x) dx as z(x) = Re z(x) + i Im z(x).

If you are trying to do a integration over a path in the complex plane (other than along the real axis) or region in the complex plane, you’ll need a more sophisticated algorithm.

Note: Scipy.integrate will not directly handle complex integration. Why? It does the heavy lifting in the FORTRAN QUADPACK library, specifically in qagse.f which explicitly requires the functions/variables to be real before doing its “global adaptive quadrature based on 21-point Gauss–Kronrod quadrature within each subinterval, with acceleration by Peter Wynn’s epsilon algorithm.” So unless you want to try and modify the underlying FORTRAN to get it to handle complex numbers, compile it into a new library, you aren’t going to get it to work.

If you really want to do the Gauss-Kronrod method with complex numbers in exactly one integration, look at wikipedias page and implement directly as done below (using 15-pt, 7-pt rule). Note, I memoize’d function to repeat common calls to the common variables (assuming function calls are slow as if the function is very complex). Also only did 7-pt and 15-pt rule, since I didn’t feel like calculating the nodes/weights myself and those were the ones listed on wikipedia, but getting reasonable errors for test cases (~1e-14)

import scipy
from scipy import array

def quad_routine(func, a, b, x_list, w_list):
    c_1 = (b-a)/2.0
    c_2 = (b+a)/2.0
    eval_points = map(lambda x: c_1*x+c_2, x_list)
    func_evals = map(func, eval_points)
    return c_1 * sum(array(func_evals) * array(w_list))

def quad_gauss_7(func, a, b):
    x_gauss = [-0.949107912342759, -0.741531185599394, -0.405845151377397, 0, 0.405845151377397, 0.741531185599394, 0.949107912342759]
    w_gauss = array([0.129484966168870, 0.279705391489277, 0.381830050505119, 0.417959183673469, 0.381830050505119, 0.279705391489277,0.129484966168870])
    return quad_routine(func,a,b,x_gauss, w_gauss)

def quad_kronrod_15(func, a, b):
    x_kr = [-0.991455371120813,-0.949107912342759, -0.864864423359769, -0.741531185599394, -0.586087235467691,-0.405845151377397, -0.207784955007898, 0.0, 0.207784955007898,0.405845151377397, 0.586087235467691, 0.741531185599394, 0.864864423359769, 0.949107912342759, 0.991455371120813]
    w_kr = [0.022935322010529, 0.063092092629979, 0.104790010322250, 0.140653259715525, 0.169004726639267, 0.190350578064785, 0.204432940075298, 0.209482141084728, 0.204432940075298, 0.190350578064785, 0.169004726639267, 0.140653259715525,  0.104790010322250, 0.063092092629979, 0.022935322010529]
    return quad_routine(func,a,b,x_kr, w_kr)

class Memoize(object):
    def __init__(self, func):
        self.func = func
        self.eval_points = {}
    def __call__(self, *args):
        if args not in self.eval_points:
            self.eval_points[args] = self.func(*args)
        return self.eval_points[args]

def quad(func,a,b):
    ''' Output is the 15 point estimate; and the estimated error '''
    func = Memoize(func) #  Memoize function to skip repeated function calls.
    g7 = quad_gauss_7(func,a,b)
    k15 = quad_kronrod_15(func,a,b)
    # I don't have much faith in this error estimate taken from wikipedia
    # without incorporating how it should scale with changing limits
    return [k15, (200*scipy.absolute(g7-k15))**1.5]

Test case:

>>> quad(lambda x: scipy.exp(1j*x), 0,scipy.pi/2.0)
[(0.99999999999999711+0.99999999999999689j), 9.6120083407040365e-19]

I don’t trust the error estimate — I took something from wiki for recommended error estimate when integrating from [-1 to 1] and the values don’t seem reasonable to me. E.g., the error above compared with truth is ~5e-15 not ~1e-19. I’m sure if someone consulted num recipes, you could get a more accurate estimate. (Probably have to multiple by (a-b)/2 to some power or something similar).

Recall, the python version is less accurate than just calling scipy’s QUADPACK based integration twice. (You could improve upon it if desired).