# Help with Gerschgorin circle root localisation

Anonymous

I need help with the Gerschgorin circles root localisation.

Consider the following function:

• $f = x^5 - 13 x^3 - x^2 + 10 x + 170$

Visualize the roots using Gerschgorin-Circle. Include a plot of the function itself, complete with labels and title, in your report in a SageMath Jupyter notebook.

from sympy import symbols, Poly, solve
from sympy.plotting import plot

# Define the function f and the variable x
x = symbols('x')
f = x**5 - 13*x**3 - x**2 + 10*x + 170

# Convert f to a polynomial
f_poly = Poly(f, x)

# Find all the roots
roots = solve(f_poly, domain='CC')

# Print all roots
print("All roots:")
for root in roots:
root_approx = root.evalf()
print(root_approx)

# Filter real and complex roots
real_roots = [root for root in roots if root.is_real]
complex_roots = [root for root in roots if not root.is_real]

# Print real roots
print("Real roots:")
for root in real_roots:
root_approx = root.evalf()
print(root_approx)

# Print complex roots
print("Complex roots:")
for root in complex_roots:
root_approx = root.evalf()
print(root_approx)

# Print the number of real roots
num_real_roots = len(real_roots)
print("Number of real roots:", num_real_roots)

# Plot the real part of the function f
p1 = plot(f, (x, -5, 5), title='Graph of f(x)', real=True)

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