#!/usr/bin/env python
# coding: utf-8

# # Sensitivity of polynomial roots

# Consider the polynomial
# $$
# x^3 - 21 x^2 + 120 x - 100 = 0
# $$
# The roots are 1, 10, 10.

# In[18]:


from sympy import N, roots
from sympy import Integer as Int


# In[19]:


r = roots([1,-21,120,-100])
print('Roots = ',r)
for x in r:
    print(x)


# The output shows that the root 1 has multiplicity one and root 10 has multiplicity of two.
# 
# **Note**: The type of return value `r` is `dict`.

# Perturb the coefficient of $x^3$ 
# $$
# \frac{99}{100}x^3 - 21 x^2 + 120 x - 100 = 0
# $$

# In[20]:


r = roots([Int(99)/100,-21,120,-100])
for x in r:
    print(N(x))


# The double roots are very sensitive. Now perturb the coefficient in the other direction
# $$
# \frac{101}{100}x^3 - 21 x^2 + 120 x - 100 = 0
# $$

# In[21]:


r = roots([Int(101)/100,-21,120,-100])
for x in r:
    print(N(x))


# The double roots become complex, again showing they are very sensitive to the coefficient of $x^3$.