Spectral Analysis of Block Diagonally Preconditioned Multiple Saddle-Point Matrices with Inexact Schur Complements

We derive eigenvalue bounds for symmetric block-tridiagonal multiple saddle-point systems preconditioned with block-diagonal Schur complement matrices. This analysis applies to an arbitrary number of blocks and accounts for the case where the Schur complements are approximated, generalizing the findings in [Bergamaschi et al., Linear Algebra and its Applications, 2026]. Numerical experiments are carried out to validate the proposed estimates.

💡 Research Summary

The paper investigates the spectral properties of symmetric block‑tridiagonal multiple saddle‑point matrices when preconditioned with a block‑diagonal matrix built from (exact or inexact) Schur complements. The authors consider a system A x = b where A consists of N + 1 blocks: the leading block A₀ is symmetric positive definite, the remaining blocks A_k (k ≥ 1) are symmetric positive semidefinite, and the coupling blocks B_k have full rank. Such structures arise in constrained quadratic programming, magma dynamics, liquid‑crystal modeling, Stokes–Darcy coupling, and many multiphysics applications.

The ideal preconditioner P_D is block‑diagonal with exact Schur complements S₀ = A₀ and S_k = A_k + B_k S_{k‑1}^{‑1} B_kᵀ for k = 1,…,N. Computing S_k exactly is often prohibitive, especially for large N, so the authors introduce an inexact version P = blkdiag( \bar S₀,…,\bar S_N ) where each \bar S_k approximates the true S_k (or a perturbed version that accounts for previous approximations). The preconditioned matrix is Q = P^{‑1/2} A P^{‑1/2}.

A central contribution is the establishment of a one‑to‑one correspondence between the eigenvalues λ of Q and the zeros of a family of recursively defined polynomials U_k(λ; γ_E, γ_R). The parameters γ_E(k) and γ_R(k) are Rayleigh quotients of the scaled blocks E_k = \bar S_k^{‑1/2} A_k \bar S_k^{‑1/2} and R_k = \bar S_k^{‑1/2} B_k \bar S_{k‑1}^{‑1/2}, respectively. The recursion reads U₀ = 1, U₁ = λ − γ_E(0), U_{k+1} = (λ + (‑1)^k γ_E(k)) U_k − γ_R(k) U_{k‑1}, k ≥ 1. The authors prove that for any admissible choice of the γ‑parameters, each U_k has k distinct real roots, and the roots of successive polynomials interlace. Moreover, the sign pattern of U_k(0) follows a simple alternating rule, which leads to a precise description of how many positive and negative roots each polynomial possesses (Proposition 3).

Theorem 1 shows that every eigenvalue of P^{‑1}A lies either inside a specific interval I_k (defined recursively from the γ‑parameters) or coincides with a zero of U_k. Consequently, when the approximations \bar S_k are sufficiently accurate so that the γ‑parameters stay within prescribed bounds, all eigenvalues are confined to two symmetric intervals