Comparative numerical laboratory
Eigenvalue Methods: Power Iteration and Shifted Inverse Iteration — Chapter 6
Approximate eigenvalues of a symmetric \(2\times2\) matrix and compare convergence behavior.
\[
A=\begin{pmatrix}2&1\\1&3\end{pmatrix}, Av=\lambda v.
\]
Methods compared
| Method | Convenience | Inconvenience / warning |
|---|---|---|
| Power iteration | Simple dominant-eigenvalue method. | Needs eigenvalue separation and targets the dominant eigenpair. |
| Shifted inverse iteration | Targets eigenvalues near the shift. | Requires linear solves and a good shift. |
| Rayleigh quotient | Natural eigenvalue estimate from a vector. | Can be misleading if the vector is poor. |
Illustrative plot / animation
Numerical diagnostics
What the student must compare
- Rayleigh quotient
- residual \(\|Av-\lambda v\|\)
- dominance/separation
- shift effect