ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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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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Reduced QR avoids the condition-number squaring seen in the normal equations

Example

Let $A = (\begin{array}{cc} 1 & 0 \\ 0 & 10^{- 4} \\ 0 & 0 \end{array}), b = (\begin{array}{c} 1 \\ 1 \\ 0 \end{array}) .$ Then $A$ already has reduced QR factorisation $A = \hat{Q} \hat{R}$ with $\hat{Q} = (e_{1}, e_{2})$ and $\hat{R} = diag (1, 10^{- 4})$ . The least-squares solution is $x = (1, 10^{4})^{T}$ , while $κ_{2} (\hat{R}) = 10^{4}$ and $κ_{2} (A^{T} A) = 10^{8}$ .

Facts & Assumptions

Given: The displayed matrix $A$ , vector $b$ , and reduced QR factorisation.

[L1]

Reduced QR solves full-column-rank least squares through $\hat{R} x = {\hat{Q}}^{*} b$ and avoids the condition-number square of the normal equations (Reduced QR over the reals solves full-column-rank least squares without squaring the condition number).

Verification

technique · direct

1.1

The columns of $A$ are already orthogonal, so $\hat{Q} = (e_{1}, e_{2})$ and $\hat{R} = diag (1, 10^{- 4})$ form a reduced QR factorisation. Also ${\hat{Q}}^{*} b = (1, 1)^{T}$ .

givenalgebra

2.1

Solving $\hat{R} x = {\hat{Q}}^{*} b$ gives $x_{1} = 1$ and $10^{- 4} x_{2} = 1$ , hence $x = (1, 10^{4})^{T}$ . The factor $\hat{R}$ has spectral condition number $10^{4}$ , whereas $A^{T} A = diag (1, 10^{- 8})$ has spectral condition number $10^{8}$ . This is exactly the contrast described in [L1].

step 1.1L1algebra

3.1

Steps 1.1-2.1 verify the example.

step 1.1step 2.1∎

Depends on

Reduced QR over the reals solves full-column-rank least squares without squaring the condition number

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, Section 3.3.3 and Example 3.3.2 (standard reference, not scraped)