ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-01

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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.
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CG can terminate at a relative grade strictly smaller than the ambient dimension

Example

Let

$A = diag (1, 1, 2), b = (\begin{array}{c} 1 \\ 1 \\ 0 \end{array}), x_{0} = (\begin{array}{c} 0 \\ 0 \\ 0 \end{array}) .$

Then the initial residual is $r_{0} = b$ , its grade is $1$ , and CG reaches the exact solution in one step although the ambient dimension is $3$ .

Facts & Assumptions

Given: The displayed Hermitian positive-definite system.

[F1]

The grade $ν (A, r_{0})$ is the degree of the relative minimal polynomial of $r_{0}$ (The grade of a start vector and its relative minimal polynomial).

[L1]

Exact-arithmetic CG terminates no later than the relative grade (In exact arithmetic, CG terminates no later than the relative grade and hence in at most $n$ steps).

Verification

technique · direct calculation

1.1

Since $A r_{0} = r_{0}$ , the polynomial $q (z) = z - 1$ annihilates $r_{0}$ . The vector $r_{0}$ is nonzero, so no constant polynomial can annihilate it. Therefore $q_{A, r_{0}} (z) = z - 1$ and $ν (A, r_{0}) = 1$ by [F1]. In particular, $1 < 3 = \dim R^{3}$ .

F1algebra

2.1

The exact solution is $x_{*} = A^{- 1} b = (1, 1, 0)^{T} = r_{0}$ . Because $p_{0} = r_{0}$ and $α_{0} = \frac{r_{0}^{T} r_{0}}{p_{0}^{T} A p_{0}} = \frac{2}{2} = 1,$ the first CG update gives $x_{1} = x_{0} + α_{0} p_{0} = r_{0} = x_{*}$ . Thus the method stops at step $1 = ν (A, r_{0})$ , exactly as [L1] allows.

L1step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

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

Magnus R. Hestenes and Eduard Stiefel, Methods of Conjugate Gradients for Solving Linear Systems (standard reference, not scraped)