Eigenvalue bounds for preconditioned symmetric multiple saddle-point matrices

We develop eigenvalue bounds for symmetric, block tridiagonal multiple saddle-point linear systems, preconditioned with block diagonal matrices. We extend known results for 3 x 3 block systems [Bradley & Greif, IMA J.Numer. Anal. 43 (2023)] and for 4 x 4 systems [Pearson & Potschka, IMA J. Numer. Anal. 44 (2024)] to an arbitrary number of blocks. Moreover, our results generalize the bounds in [Sogn & Zulehner, IMA J. Numer. Anal. 39 (2018) (Thm. 2.6)], developed for an arbitrary number of blocks with null diagonal blocks. Extension to the bounds when the Schur complements are approximated is also provided, using perturbation arguments. Practical bounds are also obtained for the double saddle-point linear system. Numerical experiments validate our findings.