PropositionStatement: AI-adaptedProof: AI-generatedprecheck passaudited 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 residual has the form $r_{m} = p_{m} (A) r_{0}$ with $p_{m} (0) = 1$ and $\deg p_{m} \leq m$

Statement

Let $x_{m}$ be an $m$ th GMRES iterate for $A x = b$ with initial residual $r_{0} = b - A x_{0}$ , and let $r_{m} = b - A x_{m}$ . Then there is a polynomial $p_{m}$ such that

$r_{m} = p_{m} (A) r_{0}, p_{m} (0) = 1, \deg p_{m} \leq m .$

Moreover, among all polynomials $p$ with $\deg p \leq m$ and $p (0) = 1$ , GMRES chooses one minimizing $∥ p (A) r_{0} ∥_{2}$ .

Facts & Assumptions

Given: A GMRES iterate $x_{m}$ with residual $r_{m} = b - A x_{m}$ .

[L1]

Every vector in $K_{m} (A, r_{0})$ has the form $q (A) r_{0}$ with $\deg q < m$ (The Krylov subspace consists exactly of the vectors $p (A) b$ for zero polynomials or polynomials of degree less than $m$ ).

[L2]

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

Proof

technique · direct

1.1

Since $x_{m} - x_{0} \in K_{m} (A, r_{0})$ , [L1] gives a polynomial $q$ with $\deg q < m$ and $x_{m} = x_{0} + q (A) r_{0}$ . Therefore $r_{m} = b - A x_{m} = r_{0} - A q (A) r_{0} = p_{m} (A) r_{0},$ where $p_{m} (z) : = 1 - z q (z)$ . Then $p_{m} (0) = 1$ and $\deg p_{m} \leq m$ .

L1constructalgebra

2.1

Conversely, every polynomial $p$ with $\deg p \leq m$ and $p (0) = 1$ can be written as $p (z) = 1 - z q (z)$ with $\deg q < m$ . The corresponding vector $x = x_{0} + q (A) r_{0}$ lies in $x_{0} + K_{m} (A, r_{0})$ by [L1], and its residual is $p (A) r_{0}$ . Since [L2] makes $x_{m}$ minimize that residual norm over all such $x$ , the polynomial attached in step 1.1 minimizes $∥ p (A) r_{0} ∥_{2}$ among all admissible $p$ .

L1L2step 1.1∎

Depends on

Used by

For a diagonalizable matrix, the GMRES residual bound carries the eigenvector-conditioning factor κ(V) Proposition

Dependency tree · two levels

5 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)

The GMRES residual has the form rm=pm(A)r0rm​=pm​(A)r0​ with pm(0)=1pm​(0)=1 and deg⁡pm≤mdegpm​≤m