Convergence Properties

There is a beautiful theory, based on the properties of orthogonal polynomials, that describes when eigenvalues converge. The characteristic polynomials of the tridiagonal matrices $T_{j}$ are orthogonal polynomials with respect to a scalar product, defined by the expansion of the starting vector $v$ as a sum of eigenvectors. One gets bounds on differences $\vert\theta_i^{(j)}-\lambda_i\vert$ between the eigenvalues $\theta_i^{(j)}$ of $T_{j}$ and $\lambda_i$ of $A$ by replacing these unknown orthogonal polynomials with the well-known Chebyshev polynomials, the so-called Kaniel-Paige-Saad bounds; see [353].

This theory says that we get convergence to those eigenvalues that are represented in the starting vector and faster convergence to those in the ends of the spectrum. The better separated these are from the rest of the eigenvalues, the faster will they converge.

In practical cases, we are often interested just in the lowest eigenvalues, and fortuitously these are among the first to converge. On the other hand, the relative separation of the lowest eigenvalues is often poor, since the separation is relative to the whole spread of the spectrum, not to the distance to the origin.