Binomial expansion with fractional powers in Python

Question

Is there a quick method of expanding and solving binomials raised to fractional powers in Scypy/numpy?

For example, I wish to solve the following equation

y * (1 + x)^4.8 = x^4.5

where y is known (e.g. 1.03).

This requires the binomial expansion of (1 + x)^4.8.

I wish to do this for millions of y values and so I'm after a nice and quick method to solve this.

I've tried the sympy expand (and simplification) but it seems not to like the fractional exponent. I'm also struggling with the scipy fsolve module.

Any pointers in the right direction would be appreciated.

EDIT:

By far the simplest solution I found was to generate a truth table (https://en.wikipedia.org/wiki/Truth_table) for assumed values of x (and known y). This allows fast interpolation of the 'true' x values.

y_true = np.linspace(7,12, 1e6)
x = np.linspace(10,15, 1e6)

a = 4.5
b = 4.8
y = x**(a+b) / (1 + x)**b

x_true = np.interp(y_true, y, x)

Answer 1

EDIT: Upon comparing output with that of Woldfram alpha for y=1.03, it looks like fsolve will not return complex roots. https://stackoverflow.com/a/15213699/3456127 is a similar question that may be of more help.

Rearrange your equation: y = x^4.5 / (1+x)^4.8. Scipy.optimize.fsolve() requires a function as its first argument.

Either:

from scipy.optimize import fsolve
import math
def theFunction(x):   
    return math.pow(x, 4.5) / math.pow( (1+x) , 4.8)

for y in millions_of_values:
    fsolve(theFunction, y)

Or using lambda (anonymous function construct):

from scipy.optimize import fsolve
import math
for y in millions_of_values:
    fsolve((lambda x: (math.pow(x, 4.5) / math.pow((1+x), 4.8))), y)