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 \(f'(r) = 0\)