I am getting a warning <RuntimeWarning: invalid value encountered in sqrt>

Question

I am trying to run a quadratic equation in python. However, it keeps on giving me a warning

RuntimeWarning: invalid value encountered in sqrt

Here's my code:

import numpy as np


a = 0.75 + (1.25 - 0.75)*np.random.randn(10000)
print(a)
b = 8 + (12 - 8)*np.random.randn(10000)
print(b)
c = -12 + 2*np.random.randn(10000)
print(c)
x0 = (-b - np.sqrt(b**2 - (4*a*c)))/(2 * a)
print(x0)

Answer 1

This is not 100% Python related. You can't calculate the square root of a negative number (when dealing with real numbers that is).

You didn't take any precautions for when b**2 - (4*a*c) is a negative number.

>>> import numpy as np
>>>
>>> np.sqrt(4)
2.0
>>> np.sqrt(-4)
__main__:1: RuntimeWarning: invalid value encountered in sqrt
nan

Let's test if you have negative values:

>>> import numpy as np
>>> 
>>> a = 0.75 + (1.25 - 0.75) * np.random.randn(10000)
>>> b = 8 + (12 - 8) * np.random.randn(10000)
>>> c = -12 + 2 * np.random.randn(10000)
>>> 
>>> z = b ** 2 - (4 * a * c)
>>> print len([_ for _ in z if _ < 0])
71

Answer 2

Square roots are not defined for strictly negative real numbers, and numpy will produce nan for negative inputs of "real" dtype int, float, and it's two special values -np.inf and nan.

However, square roots are defined for all complex dtype:

dtype	e.g.x	np.sqrt(x)	RuntimeWarning
Positive float	1.	1.
Positive int	1	1
Positive complex	1+0J	1
Negative float	-1.	nan	⚠️
Negative int	-1	nan	⚠️
Negative complex	-1+0j	1j
Infinity	np.inf	np.inf
Negative infinity	-np.inf	nan	⚠️
NaN	np.nan	nan

Answer 3

If you're hoping to do complex analysis (working with imaginary numbers as defined by sqrt(-1)) you can import cmath and use cmath.sqrt(-1) instead of numpy.sqrt(-1).

For example, when I'm calculating the refractive index of materials from permittivity and permeability (by definition, j is involved), I'll write functions in python as such:

def n(f):
    y = cmath.sqrt(mu1f(f) - j*mu2f(f)) * (e1f(f) - j*e2f(f))
    return y.real

Where e1f etc. are previously defined interpolating functions, all of which are a function of incident frequency f. The y resultant is, in it of itself, a complex value, the complex index of refraction, but I'm oftentimes only interested in the real portion (the refractive index) so that is what is returned.

Hope this helps