29.9. Numerical Methods (Brief Introduction)

Try these first

1. 1.

   What is the purpose of the Power Method in numerical computations?

   Hint

   Think about what 'dominant' means in this context.

2. 2.

   Name one advantage of using the QR Algorithm.

   Hint

   Consider the difference between finding just one eigenvalue versus all of them.

3. 3.

   What is the main purpose of the Power Method?

   • To find all eigenvalues
   • To find the dominant eigenvalue
   • To determine the matrix trace

   Hint

   Remember what 'dominant' refers to in a matrix context.

4. 4.

   The QR algorithm is best used for which of the following?

   • True
   • False

   Hint

   Think about the breadth of eigenvalue extraction.

5. 5.

   Given a large sparse matrix from a FEM model, explain how you would approach using the Power Method to estimate an eigenvalue, detailing each step.

   Hint

   Focus on the normalization process and how it stabilizes.

6. 6.

   How would the choice of the initial vector in the Power Method affect the outcome? Provide an example in civil engineering contexts.

   Hint

   Consider cases where poor choices of vectors led to inefficiency or inaccuracies.