Convergence of Lanczos inner eigenvalues

We can extend the previous result for ${\lambda }_1$.

With $A\to -A$, the same result applies to ${\lambda }_n$.

There is a similar result for “inner” eigenvalues, but the convergence is slower.

$$\begin{array}{c}{\lambda }_i\ge {\lambda }_i(T_k)\ge {\lambda }_i-({\lambda }_1-{\lambda }_n)(\frac{{\kappa }_i\tan \varphi }{T_{k-i}^{\mathrm{cheb}}(1+2{\rho }_i)}{)}^2\\ {\rho }_i=\frac{{\lambda }_i-{\lambda }_{i+1}}{{\lambda }_{i+1}-{\lambda }_n},{\kappa }_i=\prod _{j=1}^{i-1}\frac{{\lambda }_j(T_k)-{\lambda }_n}{{\lambda }_j(T_k)-{\lambda }_i}\end{array}$$

For ${\lambda }_i(T_k)$, the convergence slows down with increasing $i$.

Lanczos process, Convergence of Lanczos eigenvalues for symmetric matrices