18.3.1. Zero-Stability (for Multistep Methods)

Try these first

1. 1.

   What is zero-stability?

   Hint

   Think about how errors affect numerical solutions.

2. 2.

   Why is the characteristic equation important?

   Hint

   Consider what the roots indicate for stability.

3. 3.

   What is zero-stability?

   Hint

   Focus on the relationship between errors and solution accuracy.

4. 4.

   True or False: For zero-stability, roots of the characteristic equation can lie outside the unit circle.

   • True
   • False

   Hint

   Think about how this affects solution growth.

5. 5.

   Given the characteristic polynomial z^3 - 1 = 0, assess the zero-stability of the corresponding method.

   Hint

   Graph the roots on the complex plane to visualize their location.

6. 6.

   Consider the characteristic equation z^3 - 2z^2 + z - 1 = 0. Are these roots zero-stable? Provide your reasoning.

   Hint

   Use the quadratic formula if necessary and evaluate each solution.