The rootfinding problem

• Cannot be solved in finite number of operations
• Condition number of a root is magnitude of derivative of the inverse
  • When $|f'|$ is small at the root (flatter), harder to find root
• The backward error in root estimate is equal to the residual
• Multiplicity: $f(r) = 0 = f'(r) = \cdots = f^{(m-1)}(r) = 0$, but $f^{(m)}(r) \neq 0$
• Simple root: if $f(r) = 0$ and $f'(r) = 0$