Polynomial roots

Polynomial roots are nice because we can solve for them directly. Polynomials of degree \(n\) have exactly \(n\) roots, including multiplicity (coincident roots). The roots may be complex, even in polynomials with real coefficients. If the root is complex (\(a+bi\)), then its complex conjugate (\(a-bi\)) is also a root.

The roots of polynomials of order \(\le 4\) can be solved analytically.

E.g.:

\(a x^2 +b x + c = 0\)

has roots:

\(x = \frac{-b \pm \sqrt{b^2-4ac}} {2a}\)

per the quadratic formula.

Certaint polynomials of order \(\gt 4\) may have analytically solvable roots but there is no such general formula per the Abel-Ruffini theorem.

All the roots of polynomials may be found numerically by solving for the eigenvalues of the polynomial companion matrix. This is the basis for modern numerical methods which we will cover when we get to matrix eigenvalue calculations.

Example usage of numerical root finding tools.

# prompt: Give me a 10th order polynomial with integer coefficients,, print the polynomial, its companion matrix to 2 decimal places, and then list the roots

import numpy as np
from scipy.linalg import companion

# Define the coefficients of the polynomial
coefficients = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]

# Print the polynomial
print("Polynomial:")
print(np.poly1d(coefficients))

# Calculate the companion matrix
companion_matrix = companion(coefficients)

# Print the companion matrix
print("\nCompanion Matrix:")
print(np.round(companion_matrix, 2))

# Find the eigenvalues (roots) of the companion matrix
roots = np.linalg.eigvals(companion_matrix)

# Print the roots
print("\nRoots from eigenvalues of companion matrix:")
print(np.round(roots, 2))

print("\nRoots from 'roots' function:")
print(np.round(np.roots(coefficients),2))

Polynomial:
   10     9     8     7     6     5     4     3     2
1 x  + 2 x + 3 x + 4 x + 5 x + 6 x + 7 x + 8 x + 9 x + 10 x + 11

Companion Matrix:
[[ -2.  -3.  -4.  -5.  -6.  -7.  -8.  -9. -10. -11.]
 [  1.   0.   0.   0.   0.   0.   0.   0.   0.   0.]
 [  0.   1.   0.   0.   0.   0.   0.   0.   0.   0.]
 [  0.   0.   1.   0.   0.   0.   0.   0.   0.   0.]
 [  0.   0.   0.   1.   0.   0.   0.   0.   0.   0.]
 [  0.   0.   0.   0.   1.   0.   0.   0.   0.   0.]
 [  0.   0.   0.   0.   0.   1.   0.   0.   0.   0.]
 [  0.   0.   0.   0.   0.   0.   1.   0.   0.   0.]
 [  0.   0.   0.   0.   0.   0.   0.   1.   0.   0.]
 [  0.   0.   0.   0.   0.   0.   0.   0.   1.   0.]]

Roots from eigenvalues of companion matrix:
[-1.26+0.36j -1.26-0.36j -0.88+0.96j -0.88-0.96j -0.25+1.26j -0.25-1.26j
  0.44+1.17j  0.44-1.17j  0.95+0.73j  0.95-0.73j]

Roots from 'roots' function:
[-1.26+0.36j -1.26-0.36j -0.88+0.96j -0.88-0.96j -0.25+1.26j -0.25-1.26j
  0.44+1.17j  0.44-1.17j  0.95+0.73j  0.95-0.73j]

Note: in many software environments, \(j\) is used as the imaginary number instead of \(i\).