Complex Roots

Figure 6: Plot of $f(x)= x^3 + 2x^2 - x + 5$.

• Newton's method works with complex roots if we give it a complex value for the starting value.
• Use Newton's method on $f(x)= x^3 + 2x^2 - x + 5$. Figure 6 shows the graph of $f(x)$. It has, a real root at about $x = -3$, whereas the other two roots are complex because the x-axis is not crossed again.
• If we begin Newton's method with $x_0=1 + i$ (we used this in the lack of knowledge about the complex root), we get these successive iterates;

  1. 0.486238 + 1.04587i
  2. 0.448139 + 1.23665i
  3. 0.462720 + 1.22242i
  4. 0.462925 + l.22253i
  5. 0.462925 + 1.22253i
  

• Because the fourth and fifth iterates agree to six significant figures, we are sure that we have an estimate good to at least that many figures.
• The second complex root is the conjugate or this: 0.462925 - 1.22253i.
• If we begin with $x_0= 1 - i$, the method converges to the conjugate.
• If we begin with a real starting value-say, $x_0 = -3$-we get convergence to the root at $x=-2.92585$.