[FIXED] when doing multi-integral, 'takes 0 positional arguments but 1 was given'

Issue

I am doing a multi integral with 4 variables, among them 2 have limits as functions. However the error appears on one of my constant-limit variable. Really cannot figure our why. Many thanks for your advice!

from numpy import sqrt, sin, cos, pi, arcsin, maximum
from sympy.functions.special.delta_functions import Heaviside
from scipy.integrate import nquad

def bmax(x):
    return 1.14*10**9/sin(x/2)**(9/7)

def thetal(x,y,z):
    return arcsin(3.7*10**15*sqrt(cos(x/2)**2/10**6-1.23*10**10/z+0.003*sin(x/2)**2*(2.51*10**63/sin(x/2)**9/y**7-1))/(z*sin(x/2)**2*cos(x/2)*(2.51*10**63/sin(x/2)**9/y**7-1)))

def rt(x,y):
    return 3.69*10**12/(2.5*10**63/sin(x/2)**7*y**7-sin(x/2)**2)

def rd(x,y):
    return maximum(1.23*10**10,rt(x,y))

def rl(x,y):
    return rd(x,y)*(sqrt(1+5.04*10**16/(rd(x,y)*cos(x/2)**2))-1)/2

def wbound():
    return [1.23*10**10,3.1*10**16]

def zbound():
    return [10**(-10),pi-10**(-10)]

def ybound(z):
    return [0,bmax(z)-10**(-10)]

def xbound(z,y,w):
    return [thetal(z,y,w),pi-thetal(z,y,w)]

def f(x,y,z,w):
    return [5.77/10**30*sin(z)*sin(z/2)*y*sin(x)*Heaviside(w-rl(z,y))*Heaviside(w-rd(z,y))/w**2]

result = nquad(f, [xbound, ybound,zbound,wbound])

Solution

The reason for that error is that although you don't want these bounds to depend on the variables, nquad still passes the variables to the functions you provide to it. So the bound functions have to take the right number of variables:

def wbound():
    return [1.23*10**10,3.1*10**16]

def zbound(w_foo):
    return [10**(-10),pi-10**(-10)]

def ybound(z, w_foo):
    return [0,bmax(z)-10**(-10)]

def xbound(z,y,w): 
    return [thetal(z,y,w),pi-thetal(z,y,w)]

Now the functions zbound and ybound accept the extra variables but simply ignore them.

I'm not sure about the last bound, xbound(...): Do you want the variables y and z to be flipped? The supposedly correct ordering according to the definition of scipy.integrate.nquad would be

def xbound(y,z,w):
    ... 

Edit: As kazemakase pointed out, the function f should return a float instead of a list so the brackets [...] in the return statement should be removed.

Answered By - obachtos