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

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Clustered eigenvalues give a visibly better CG condition-number bound than equally sized spread spectra

Example

Compare the Hermitian positive-definite matrices

$A_{c l} = diag (9, 10, 11), A_{s p} = diag (1, 10, 100) .$

Both are $3 \times 3$ , but the clustered spectrum of $A_{c l}$ gives a much sharper CG bound than the spread spectrum of $A_{s p}$ .

Facts & Assumptions

Given: The two displayed Hermitian positive-definite matrices.

[L1]

The CG error bound is $∥ e_{m} ∥_{A} \leq 2 {(\frac{\sqrt{κ_{2} (A)} - 1}{\sqrt{κ_{2} (A)} + 1})}^{m} ∥ e_{0} ∥_{A}$ (CG obeys the Chebyshev $A$ -norm bound in terms of the spectral condition number $κ_{2} (A)$ ).

Verification

technique · direct calculation

1.1

The spectral condition numbers are $κ_{2} (A_{c l}) = \frac{11}{9}, κ_{2} (A_{s p}) = 100.$ So the contraction factors in [L1] are $θ_{c l} = \frac{\sqrt{11 / 9} - 1}{\sqrt{11 / 9} + 1} = \frac{\sqrt{11} - 3}{\sqrt{11} + 3},$ and $θ_{s p} = \frac{\sqrt{100} - 1}{\sqrt{100} + 1} = \frac{9}{11} .$ Moreover $\sqrt{11} < 10 / 3$ , so $θ_{c l} < 1 / 18$ .

L1algebra

2.1

At $m = 2$ , [L1] yields $∥ e_{2} ∥_{A_{c l}} \leq 2 θ_{c l}^{2} ∥ e_{0} ∥_{A_{c l}} < \frac{1}{162} ∥ e_{0} ∥_{A_{c l}},$ whereas $∥ e_{2} ∥_{A_{s p}} \leq 2 θ_{s p}^{2} ∥ e_{0} ∥_{A_{s p}} = \frac{162}{121} ∥ e_{0} ∥_{A_{s p}} .$ Thus the clustered eigenvalues give a visibly sharper theoretical CG estimate than the spread spectrum of the same size.

L1step 1.1algebra∎

Depends on

CG obeys the Chebyshev $A$-norm bound in terms of the spectral condition number $\kappa_2(A)$

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

Jonathan Richard Shewchuk, An Introduction to the Conjugate Gradient Method Without the Agonizing Pain (standard reference, not scraped)