Linear Systems and Eigenvalue Problems — Problem 2

Definitions and mathematical statement

A parameterized linear system can be written schematically as \[ A(\theta)x=b(\theta),\qquad \theta\in\Theta, \] where $\theta$ denotes the parameters, $A(\theta)$ the coefficient matrix, $b(\theta)$ the right-hand side, and $x$ the unknown vector. This notation is explanatory: the supplied record does not specify $\Theta$, the scalar domain, matrix dimensions, or admissible coefficient and right-hand-side families.

Problem (source formulation). For each class of parameterized benchmark linear systems, develop algorithms that are fast both theoretically and practically.

Unresolved qualifications. The benchmark classes over which "for each" quantifies are not identified in the supplied record. Neither the required output nor the meaning of theoretical or practical speed is specified. In particular, the record supplies no assumptions concerning consistency, uniqueness, conditioning, exact versus approximate computation, accuracy, computational model, input size, preprocessing, or reuse across parameter values. Thus the displayed system is only a schematic representation, and the source formulation does not determine a precise algorithmic specification or complexity bound.

Short English statement

For every intended class of parameterized benchmark linear systems, develop algorithms with strong theoretical and practical performance. The supplied record does not identify those classes or define the required outputs and performance criteria, so the research objective remains underspecified.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $800493$, source problem code catalogue entry, titled "Linear Systems and Eigenvalue Problems --- Problem 2," in the AMR Open Problem Lists collection. The registry attributes the problem to Amsel, Baumann et al. (2026) and labels it open; this is a supplied status label, not an independently verified mathematical or literature assessment.
• Supplied source reference: Amsel, Baumann et al., Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop (2026), Problem 2, https://arxiv.org/abs/2602.05394. The record reports access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance details; the source has not been independently accessed or verified for this draft.