5.1.5. Stopping Criteria

Try these first

1. 1.

   What is a primary reason for using stopping criteria in numerical methods?

   Hint

   Think about wasting time on iterations.

2. 2.

   State one stopping criterion related to the function value.

   Hint

   The function value must be close to zero.

3. 3.

   What does the criterion |f(xₙ)| < 𝜖 signify in numerical methods?

   • The function has a solution
   • The function value is near zero
   • There are no solutions

   Hint

   Think about the definition of a root.

4. 4.

   True or False: The maximum number of iterations is a valid stopping criterion.

   • True
   • False

   Hint

   Consider what would happen without a limit.

5. 5.

   You are solving an equation using the Newton-Raphson method. Your current approximation xₙ = 3.5, the next approximation xₙ₊₁ comes out to be 3.100. Analyze whether to stop based on |xₙ - xₙ₋₁| < 𝜖 if 𝜖 = 0.1.

   Hint

   Check the difference carefully.

6. 6.

   Consider using stopping criteria. You set a maximum of 50 iterations for an equation where each iteration feels like it gives you closer answers. However, after 49 iterations, the change between answers is becoming minimal, what would you do?

   Hint

   Weigh the trade-off between efficiency and accuracy.