Example of Newton's Method with Deflation and Horn
Problem: Find all solutions of
Solution : We’ll start with Newton’s method with p0 = 3. Set f(x) = x4+3x3−12x−16 and evaluate both this function and its derivative via synthetic division.
We now have another approximate solution and another factorization. Namely,
We’ll use the quadratic equation to find the last two zeros.
The two solution are −1.4929856 ± 1.3255958i
Errors accumulate in the coefficients during the deflation
process. We’ll use
Newton’s method with the original coefficients to refine the last three solutions.
Again we use Horner’s rule for the calculations . The results are summarized