DefinitionDefinition: AI-adaptedProof: Not applicableaudited 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.

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

Definition

Let $A x = b$ be a linear system, let $x_{0}$ be an initial guess, and let

$r_{0} : = b - A x_{0} .$

For $m \geq 0$ , a GMRES iterate at step $m$ is any vector $x_{m} \in x_{0} + K_{m} (A, r_{0})$ such that

$∥ b - A x_{m} ∥_{2} = \min_{x \in x_{0} + K_{m} (A, r_{0})} ∥ b - A x ∥_{2} .$

Thus GMRES is defined by residual minimization over the affine Krylov space generated by the initial residual.

Depends on

Used by

Dependency tree · two levels

4 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 13: Krylov subspace methods; Introduction; Krylov subspaces; Gram-Schmidt process (review); The Arnoldi process; FOM and GMRES (standard reference, not scraped)