On the sensitivity of Lanczos recursions to the spectrum

We obtain novel, explicit formulas for the sensitivity of Jacobi matrices to small perturbations of their spectra. Our derivation is based on the connection between Lanczos's algorithm and the discrete Gel'fand-Levitan inverse spectral method. We prove uniform stability of Lanczos recursions in discrete primitive norms, for perturbations of the eigenvalues relative to their separations. A stronger, l1 norm stability bound is also derived, under additional assumptions of rate of decay of the perturbations of the spectrum, which arise naturally for Sturm-Liouville operators.