ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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A diagonal positive-definite preconditioner can improve the relevant condition number

Example

Let

$A = diag (100, 1), M = diag (100, 1) .$

Then $M$ is diagonal and positive definite, and symmetric preconditioning turns the CG condition number from $100$ into $1$ .

Facts & Assumptions

Given: The displayed Hermitian positive-definite system and diagonal preconditioner.

[L1]

Symmetric positive-definite preconditioning preserves the Hermitian positive-definite CG problem and replaces the bound by the one for the transformed operator (Symmetric positive-definite preconditioning preserves a Hermitian positive-definite CG problem, and the CG bound uses the transformed condition number).

Verification

technique · direct calculation

1.1

The original spectral condition number is $κ_{2} (A) = \frac{100}{1} = 100.$ If $M = C C^{*}$ , then we may take $C = diag (10, 1)$ .

algebra

2.1

The symmetrically preconditioned operator is $C^{- 1} A C^{- *} = diag (1 / 10, 1) diag (100, 1) diag (1 / 10, 1) = I_{2},$ so $κ_{2} (C^{- 1} A C^{- *}) = 1.$ Therefore the relevant condition number has improved from $100$ to $1$ , exactly the transformed quantity singled out in [L1].

L1step 1.1algebra∎

Depends on

Symmetric positive-definite preconditioning preserves a Hermitian positive-definite CG problem, and the CG bound uses the transformed condition number

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

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