List of Figures

1. Damped, vibrating mass-spring system.
2. Residual estimates, L-shaped membrane matrix.
3. Ritz values, L-shaped membrane matrix.
4. Residual estimates, Medline SVD matrix
5. Residual estimates, shift-and-invert L-shaped membrane matrix.
6. Jacobi-Davidson for exterior eigenvalues with several strategies for solving the correction equation.
7. Jacobi-Davidson for exterior eigenvalues (top) and interior eigenvalues (bottom). The correction equations have been solved with 5 steps of plain GMRES (left) and with 5 steps of preconditioned GMRES (right).
8. Residual estimates, Lanczos with shift-and-invert, L-membrane 9-point finite difference approximation.
9. Relative accuracy of eigenvalues computed with and without direct balancing for the QH$768$ and TOLOSA matrices.
10. Relative accuracy of eigenvalues computed with and without Krylov balancing for the QH$768$ and TOLOSA matrices.
11. Comparison of the relative accuracy of the largest and smallest (in magnitude) eigenvalues of the QH$768$ matrix, with different Krylov-based balancing algorithms, using the default settings of five iterations and a cutoff value of $10^{-8}$.
12. Comparison of the relative accuracy of the largest and smallest (in magnitude) eigenvalues of the TOLOSA matrix, with different Krylov-based balancing algorithms, using the default settings of five iterations and a cutoff value of $10^{-8}$.
13. Jacobi-Davidson for exterior eigenvalues (left side) and interior eigenvalues (right side).
14. Convergence history for BFW$782$.
15. Procrustes problem
16. Jordan problem
17. Trace minimization problem
18. LDA toy problem
19. Simultaneous Schur problem
20. Simultaneous diagonalization problem
21. The unconstrained differential of $F(Y)$ can be projected to the tangent space to obtain the covariant gradient, $G$, of $F$.
22. In a flat space, comparing vectors at nearby points is not problematic since all vectors lie in the same tangent space.
23. In a curved manifold, comparing vectors at nearby points can result in vectors which do not lie in the tangent space.
24. Profile of a nonsymmetric skyline or variable-band matrix.
25. Logarithmic plots of residual norms of the inexact rational Krylov method for the Olmstead problem. The circles denote $\Vert f_j\Vert$ and the bullets $\Vert r_j\Vert$.
26. Conjugate gradient versus steepest ascent, $\delta _1 / \delta _0=10$
27. Conjugate gradient versus steepest ascent, $\delta _1 / \delta _0=100$
28. Conjugate gradient versus steepest ascent, $\delta _1 / \delta _0=1000$