DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Restarted GMRES $(m)$ as repeated GMRES cycles on fixed-size Krylov spaces

Definition

Fix a restart length $m \geq 1$ . Restarted GMRES $(m)$ produces iterates in cycles:

start from $x^{(0)} = x_{0}$ ;
at cycle $j$ , form the residual $r^{(j)} = b - A x^{(j)}$ ;
run ordinary GMRES for exactly $m$ steps with initial guess $x^{(j)}$ and initial residual $r^{(j)}$ ;
call the resulting residual minimizer $x^{(j + 1)}$ and restart from it.

Thus each cycle uses only the current residual and a fresh size- $m$ Krylov space.

Depends on

The GMRES iterate as the residual minimizer over an affine Krylov space

Used by

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Yousef Saad, CSCI 8314 Set 14: Krylov subspace methods (continued); Practical variants: restarting and truncating; Hermitian case: The Lanczos algorithm; Conjugate gradients (standard reference, not scraped)

Restarted GMRES(m)(m) as repeated GMRES cycles on fixed-size Krylov spaces