ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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GMRES on a small nonsymmetric system becomes a two- or three-dimensional least-squares problem

Example

Consider

$A = (\begin{array}{cc} 2 & 1 \\ 0 & 1 \end{array}), b = e_{2}, x_{0} = 0.$

Then $r_{0} = e_{2}$ , so the one-step Arnoldi basis is $v_{1} = e_{2}$ , $v_{2} = e_{1}$ , and

${\overset{ˉ}{H}}_{1} = (\begin{array}{c} 1 \\ 1 \end{array}) .$

Thus one-step GMRES minimizes

$∥ e_{1} - {\overset{ˉ}{H}}_{1} y ∥_{2} = {∥ (\begin{array}{c} 1 - y \\ - y \end{array}) ∥}_{2},$

whose minimizer is $y = \frac{1}{2}$ . Hence

$x_{1} = \frac{1}{2} e_{2} .$

Facts & Assumptions

Given: The system $A x = b$ with the displayed $A, b, x_{0}$ .

[L1]

Arnoldi reduces GMRES to the least-squares problem $\min_{y} ∥ β e_{1} - {\overset{ˉ}{H}}_{m} y ∥_{2}$ (Arnoldi reduces GMRES to a least-squares problem for the small Hessenberg matrix).

Verification

technique · direct

1.1

Since $r_{0} = e_{2}$ , one has $β = 1$ and $v_{1} = e_{2}$ . Also $A v_{1} = (1, 1)^{T}$ , so $h_{11} = 1$ and the normalized residual is $v_{2} = e_{1}$ with $h_{21} = 1$ . Hence ${\overset{ˉ}{H}}_{1} = (\begin{array}{c} 1 \\ 1 \end{array})$ .

L1algebra

2.1

By [L1], one-step GMRES minimizes $∥ (1 - y, - y)^{T} ∥_{2}^{2} = (1 - y)^{2} + y^{2}$ , whose derivative is $- 2 + 4 y$ , so the minimizer is $y = \frac{1}{2}$ . Therefore $x_{1} = x_{0} + Q_{1} y = \frac{1}{2} e_{2}$ .

L1step 1.1algebra∎

Depends on

Arnoldi reduces GMRES to a least-squares problem for the small Hessenberg matrix

Used by

Nothing in the library uses this result yet.

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)