Eigenvalue Bounds for Symmetric, Multiple Saddle-Point Matrices with SPD Preconditioners

We derive the eigenvalue bounds for symmetric block-tridiagonal multiple saddle-point systems preconditioned with the symmetric positive definite (SPD) preconditioner proposed by J. Pearson and A. Potschka in 2024 and further studied by L. Bergamaschi and coauthors, and for double saddle-point problems with inexact Schur complement matrices. The analysis applies to an arbitrary number of blocks. We validate the proposed estimates with both synthetic and realistic test problems, and show the good performance of the proposed preconditioner under the condition that the Schur complements are accurately approximated.