Spectral disk for a Radau stage preconditioner
Let $q \ge 2$ and let $A_q \in \mathbb{R}^{q \times q}$ be the Butcher matrix of the $q$-stage Radau IIA method: $(A_q)_{ij} = \int_0^{c_i} \ell_j(t)\,dt$, where $0 < c_1 < \dots < c_q = 1$ are the zeros of $P_q(2t - 1) - P_{q-1}(2t - 1)$, $P_j$ is the Legendre polynomial of degree $j$ with $P_j(1) = 1$, and $\ell_1, \dots, \ell_q$ are the Lagrange basis polynomials of the nodes. Let $A_q^{-1} = L_qU_q$ be its LU factorisation without pivoting, with $L_q$ lower triangular and $U_q$ unit upper triangular, and let $\widehat U_q = U_q - I_q$. For $n \ge 1$ let $M, K \in \mathbb{R}^{n \times n}$ be symmetric positive definite mass and stiffness matrices from a finite-element discretisation of a coercive self-adjoint second-order elliptic operator, let $\tau > 0$, and let $$\mathcal{A} = A_q^{-1} \otimes M + \tau I_q \otimes K, \qquad \mathcal{P}_L = L_q \otimes M + \tau I_q \otimes K.$$ Here $\mathcal{A}$ is the matrix of the Radau IIA stage equations for $Mu' + Ku = f$ with time step $\tau$, and $\mathcal{P}_L$ is Axelsson, Dravins and Neytcheva's preconditioner for it. They proved the bound below for $q = 2$, showed that at least $n$ of the eigenvalues $\nu$ equal $1$, and tabulated $\|\widehat U_q\|_2$ for $q \le 12$: $0.33$ for $q = 2$ and $0.41$ for $q = 3$, rising to $0.69$ for $q = 12$. The eigenvalues $\nu$ are those of the $q \times q$ matrices $I_q + (I_q + \mu L_q^{-1})^{-1}\widehat U_q$, for $\mu$ ranging over the eigenvalues of $\tau M^{-1}K$, which are positive (Outrata, 2026). Dravins, Serra-Capizzano and Neytcheva sharpened the bound for $q = 2$, placing the eigenvalues other than $1$ in $[1 - r_2^*, 1)$ with $r_2^* = 3\sqrt6/(11\sqrt6 + 24) \approx 0.144$, and showed for $q = 3$ that all the eigenvalues lie in a disk about $1$ of radius $r_3^*$, which they evaluate numerically as about $0.206$, below $\|\widehat U_3\|_2$.
Problem. For every $q \ge 2$ with $\|\widehat U_q\|_2 < 1$, every $n \ge 1$, all such $M$ and $K$, and every $\tau > 0$, does every generalised eigenvalue $\nu \in \mathbb{C}$ of $\mathcal{A}x = \nu\mathcal{P}_Lx$ satisfy $$|\nu - 1| \le \|\widehat U_q\|_2?$$
Source: Axelsson, Dravins and Neytcheva (2024), Conjecture 1, for positive definite finite-element matrices M and K; OpenProblemsInNLA states the special case of symmetric positive definite M and K from a coercive self-adjoint operator, the setting of Outrata (2026).
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- NA — Numerical Analysis
- Keywords
- eigenvalue bounds, OpenProblemsInNLA, preconditioners, Runge–Kutta methods
- Literature checked
- Oct. 8, 2026 by clemens-admin
References
@article{axelsson2024,
author = {Axelsson, Owe and Dravins, Ivo and Neytcheva, Maya},
title = {Stage-parallel preconditioners for implicit {R}unge--{K}utta methods of arbitrarily high order, linear problems},
journal = {Numer. Linear Algebra Appl.},
volume = {31},
number = {1},
year = {2024},
pages = {e2532},
doi = {10.1002/nla.2532},
}
@article{dravins2024,
author = {Dravins, Ivo and Serra-Capizzano, Stefano and Neytcheva, Maya},
title = {Spectral Analysis of Preconditioned Matrices Arising from Stage-Parallel Implicit {R}unge--{K}utta Methods of Arbitrarily High Order},
journal = {SIAM J. Matrix Anal. Appl.},
volume = {45},
number = {2},
year = {2024},
pages = {1007--1034},
doi = {10.1137/23M1552498},
eprint = {2302.04657},
archivePrefix = {arXiv},
}
@article{outrata2026,
author = {Outrata, Michal},
title = {On Recent Advances of Spectral Analysis for Systems Arising From Fully-Implicit {RK} Methods},
journal = {Proc. Appl. Math. Mech.},
volume = {26},
number = {1},
year = {2026},
pages = {e70082},
doi = {10.1002/pamm.70082},
eprint = {2510.21241},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-ie27,
author = {Townsend, Alex},
title = {{IE-27}: Spectral disk for a Radau stage preconditioner},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/linear-systems-and-elimination/IE-27/README.md},
note = {Read at commit eb7c9d6, 2026-10-07},
}
Resolution claims
For the $q$-stage Radau IIA matrix with $A_q^{-1}=L_qU_q$, $B_q=L_q^{-1}$ and $N_q=U_q-I_q$, the spectral disk for the stage preconditioner reduces to $\rho((I+\mu B_q)^{-1}N_q)\le\|N_q\|_2$ for every real $\mu>0$. For every stage number this report proves the inequality when $0<\mu\le1/(q\|B_q\|_2)$ or $\mu\ge2\|L_q\|_2$, gives a rank-two identity for the diagonally scaled inverse, and gives a single matrix inequality in $L_q$ and $N_q$ that is …
For the $q$-stage Radau IIA collocation matrix we give a self-contained differentiation formula for its inverse, prove that every pivot of the no-pivot factorization $A_q^{-1}=L_qU_q$ is positive, reduce the spatial pencil exactly to the stage matrices $I_q+(I_q+\mu L_q^{-1})^{-1}(U_q-I_q)$ with $\mu>0$, and prove a common-energy sufficient condition for the disk $|\nu-1|\le\|U_q-I_q\|_2$. We supply a closed-form two-stage bound and an explicit three-stage …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-08 | Open → Partially resolved | The registry's curators | Continued analysis and extensions: the spectral disk for a Radau stage preconditioner v1 recorded from OpenProblemsInNLA @ eb7c9d6, IE-27/report.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00193, "Spectral disk for a Radau stage preconditioner", clopen.solutions.