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

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CG on a $2 \times 2$ Hermitian positive-definite system reaches the solution in at most two steps

Example

Take

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

Then $A$ is Hermitian positive definite and the exact solution is

$x_{*} = (\binom{1}{1}) .$

CG reaches $x_{*}$ by the second step.

Facts & Assumptions

Given: The displayed system and the exact-arithmetic CG recurrence started from $x_{0} = 0$ .

[L1]

CG residuals are mutually orthogonal and search directions are $A$ -conjugate (In exact arithmetic, CG residuals are mutually orthogonal and the search directions are $A$ -conjugate).

[L2]

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

Verification

technique · direct calculation

1.1

Here $r_{0} = p_{0} = b = (2, 1)^{T}$ , so $α_{0} = \frac{r_{0}^{T} r_{0}}{p_{0}^{T} A p_{0}} = \frac{5}{9}, x_{1} = (\binom{10 / 9}{5 / 9}), r_{1} = b - A x_{1} = (\binom{- 2 / 9}{4 / 9}) .$ Then $β_{0} = \frac{r_{1}^{T} r_{1}}{r_{0}^{T} r_{0}} = \frac{4}{81}, p_{1} = r_{1} + β_{0} p_{0} = (\binom{- 10 / 81}{40 / 81}) .$ One checks $r_{0}^{T} r_{1} = 2 (- 2 / 9) + 1 (4 / 9) = 0, p_{0}^{T} A p_{1} = (2, 1) \cdot (\binom{- 20 / 81}{40 / 81}) = 0,$ exactly as [L1] predicts.

L1algebra

2.1

Next $α_{1} = \frac{r_{1}^{T} r_{1}}{p_{1}^{T} A p_{1}} = \frac{20 / 81}{200 / 729} = \frac{9}{10},$ so $x_{2} = x_{1} + α_{1} p_{1} = (\binom{10 / 9}{5 / 9}) + \frac{9}{10} (\binom{- 10 / 81}{40 / 81}) = (\binom{1}{1}) = x_{*} .$ Thus the method terminates in two steps, which matches the $n = 2$ upper bound in [L2].

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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)

CG on a 2×22×2 Hermitian positive-definite system reaches the solution in at most two steps