ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

An ill conditioned design with large forward sensitivity

Example

Let A=diag(1,2^-40), with condition number 2^40. For b=(1,0)^T the unique solution is x=(1,0)^T; for b-tilde=(1,2^-40)^T it is x-tilde=(1,1)^T. The relative data change is 2^-40 while the relative solution change is 1. Thus backward stability does not imply small forward error for an ill-conditioned design.

Facts & Assumptions

Given: the diagonal design and right-hand sides.

[F1]

Reduced QR over the reals solves full-column-rank least squares without squaring the condition number: full-column-rank least squares has a unique minimizer.

[F2]

Computed reduced QR least squares is backward stable for full column rank: its guarantee controls perturbations of data under its hypotheses.

[F3]

The condition number kappa_p(A) = ||A||_p ||A^{-1}||_p of a nonsingular linear system: $κ_{2} (A) = ∥ A ∥_{2} ∥ A^{- 1} ∥_{2}$ .

[F4]

The matrix norm induced by a published vector p-norm: $∥ M ∥_{2} = \sup_{∥ z ∥_{2} = 1} ∥ M z ∥_{2}$ .

Verification

1.1givenF1algebra

Direct multiplication by A^-1=diag(1,2^40) gives x=A^-1b=(1,0)^T and x-tilde=A^-1b-tilde=(1,1)^T; uniqueness follows from [F1].

2.1step 1.1algebra

The input difference and x norm are both measured against unit vectors: ||b-tilde-b||_2/||b||_2=2^-40 and ||x-tilde-x||_2/||x||_2=1.

3.1step 1.1step 2.1F2F3F4algebra∎

For every $z$ , $∥ A z ∥_{2}^{2} = z_{1}^{2} + 2^{- 80} z_{2}^{2} \leq ∥ z ∥_{2}^{2}$ , with equality at $e_{1}$ , so [F4] gives $∥ A ∥_{2} = 1$ . Likewise $∥ A^{- 1} z ∥_{2}^{2} = z_{1}^{2} + 2^{80} z_{2}^{2} \leq 2^{80} ∥ z ∥_{2}^{2}$ , with equality at $e_{2}$ , hence $∥ A^{- 1} ∥_{2} = 2^{40}$ . Thus [F3] gives $κ_{2} (A) = 2^{40}$ . The exact data calculation proves order-one forward change under a relative $2^{- 40}$ right-hand-side perturbation; backward stability [F2] alone is not a forward-accuracy guarantee.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

21 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

Fundamentals of Numerical Computation, QR factorization and least squares (standard reference, not scraped)
Fundamentals of Numerical Computation, normal equations (standard reference, not scraped)
Stuart and Voss, Matrix Analysis and Algorithms, Warwick lecture notes (standard reference, not scraped)