Parasitic solutions¶

In [25]:

from sympy import *
init_printing()

Find $x$ such that $$ \exp(-x) = 99 x $$

In [26]:

x = symbols('x')
eq1 = exp(-x) - 99*x
r1 = solve(eq1,x)
for r in r1:
    print(N(r))

0.0100004983870833

Let us make a Taylor expansion of $\exp(-x)$ around $x=0$

In [27]:

series(exp(-x),x,0)

Out[27]:

$$1 - x + \frac{x^{2}}{2} - \frac{x^{3}}{6} + \frac{x^{4}}{24} - \frac{x^{5}}{120} + O\left(x^{6}\right)$$

We truncate this at the quadratic term and consider the equation $$ 1 - x + \frac{x^2}{2} = 99 x \qquad \Longrightarrow \qquad x^2 - 200 x + 2 = 0 $$

In [28]:

eq2 = x**2 - 200*x + 2
r2 = solve(eq2,x)
for r in r2:
    print(N(r))

0.0100005000500063
199.989999499950

The simplied problem has more solutions than the original problem. One of the solutions is the correct one while the other is a spurious or parasitic solution.