CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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GMRES minimizes the residual norm over the affine Krylov space $x_{0} + K_{m} (A, r_{0})$

Statement

If $y_{m}$ minimizes $∥ β e_{1} - {\overset{ˉ}{H}}_{m} y ∥_{2}$ in the Arnoldi coordinates of the previous theorem, then

$x_{m} : = x_{0} + Q_{m} y_{m}$

is a GMRES iterate. Equivalently, GMRES minimizes the residual norm over the affine Krylov space $x_{0} + K_{m} (A, r_{0})$ .

Facts & Assumptions

Given: The Arnoldi parametrization from the GMRES reduction theorem and a vector $y_{m}$ minimizing $∥ β e_{1} - {\overset{ˉ}{H}}_{m} y ∥_{2}$ .

[F1]

A GMRES iterate is, by definition, a residual minimizer over $x_{0} + K_{m} (A, r_{0})$ (The GMRES iterate as the residual minimizer over an affine Krylov space).

[L1]

Arnoldi identifies the residual norm on $x_{0} + K_{m} (A, r_{0})$ with the small least-squares norm $∥ β e_{1} - {\overset{ˉ}{H}}_{m} y ∥_{2}$ (Arnoldi reduces GMRES to a least-squares problem for the small Hessenberg matrix).

Proof

technique · direct

1.1

By [L1], every $x \in x_{0} + K_{m} (A, r_{0})$ has the form $x = x_{0} + Q_{m} y$ , and for that $x$ one has $∥ b - A x ∥_{2} = ∥ β e_{1} - {\overset{ˉ}{H}}_{m} y ∥_{2} .$

2.1

Since $y_{m}$ minimizes the right-hand side, the vector $x_{m} = x_{0} + Q_{m} y_{m}$ minimizes the left-hand side over the whole affine Krylov space. By [F1], that is exactly the GMRES condition.

F1step 1.1∎

Depends on

Used by

Dependency tree · two levels

6 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

Lloyd N. Trefethen and David Bau III, Numerical Linear Algebra (standard reference, not scraped)

GMRES minimizes the residual norm over the affine Krylov space x0+Km(A,r0)x0​+Km​(A,r0​)