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CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • 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.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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A preconditioner can worsen the condition number that actually controls CG

Statement refuted

Every invertible, or even every symmetric positive-definite, preconditioner improves the condition number that governs CG.

Facts & Assumptions

Given: The residual-and-error maps for preconditioning and the symmetric positive-definite CG transform.

[F1]

Symmetric preconditioning replaces A by the transformed operator C−1AC−∗ (Invertible preconditioners give equivalent linear systems, with the transformed residuals and errors written explicitly).

[L1]

The CG bound is governed by the spectral condition number of the transformed operator (Symmetric positive-definite preconditioning preserves a Hermitian positive-definite CG problem, and the CG bound uses the transformed condition number).

Counterexample

technique · direct calculation
1.1

Take A=I2,M=diag⁡(1,1/100). Then A is already Hermitian positive definite with κ2(A)=1. Since M=CC∗ with C=diag⁡(1,1/10), [F1] gives the transformed operator C−1AC−∗=diag⁡(1,10)I2diag⁡(1,10)=diag⁡(1,100).

F1algebra
2.1

The transformed spectral condition number is therefore κ2(C−1AC−∗)=100>1=κ2(A). So this symmetric positive-definite preconditioner worsens the condition number that actually appears in the CG bound from [L1]. The refuted statement is false.

L1step 1.1algebra∎

Depends on

Used by

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Dependency tree · two levels

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Sources