Convergence Theory for Iterative Eigensolvers
summary
Projection methods for computing m eigenvalues rely on two explicit constants (a starting-vector orientation c1 and an eigenvector‑conditioning c2) times a polynomial‑approximation factor; bounds require building Krylov subspaces of size at least 2m. Practical fixes that work are restarting with polynomial filters, shift‑invert and polynomial preconditioning (minres‑constructed π), and inverting differential operators to restore superlinear convergence. Experiments include 10,000 Ritz values for 3×3 Jordan blocks showing kidney‑shaped loci inside the numerical range.
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chapters (15)
- 0:00Start
- 2:22Why Is the Non Symmetric Eigenvalue Problem Relevant
- 3:49Overview of Projection Based eigen Solvers
- 5:59Restarting Krylov Methods
- 9:14The Containment Gap
- 9:50Typical Convergence Curves Look like
- 12:53Ground Rules
- 13:53Happy Breakdown
- 18:16Eigenvalues
- 18:48Potential Theory
- 21:50Super Linear Convergence
- 22:27Polynomial Preconditioning
- 23:22Eigenvectors
- 26:26Approximate Eigenvalues
- 31:00Closing Thoughts
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