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.

A rounded two by two Cholesky factor and its residual

Example

In binary64 round-to-nearest, ties-to-even, let $A = (\begin{array}{cc} 1 & 1 / 2 \\ 1 / 2 & 1 \end{array}) .$ For every nonzero real $v = (v_{1}, v_{2})^{T}$ , $v^{T} A v = (v_{1} + v_{2} / 2)^{2} + 3 v_{2}^{2} / 4 > 0,$ so $A$ is positive definite. The rounded recurrence gives ${\hat{l}}_{11} = 1$ , ${\hat{l}}_{21} = 1 / 2$ , $p_{2} = 3 / 4$ exactly, and ${\hat{l}}_{22} = fl (\sqrt{3 / 4})$ . Since $\sqrt{3} / 2$ is irrational and every binary64 number is rational, ${\hat{l}}_{22}^{2} \neq 3 / 4$ . Therefore $E = \hat{L} {\hat{L}}^{T} - A = (\begin{array}{cc} 0 & 0 \\ 0 & {\hat{l}}_{22}^{2} - 3 / 4 \end{array}) \neq 0.$ The successful factor satisfies the backward bound, but its residual is not zero.

Facts & Assumptions

Given: the real symmetric matrix and binary64 RNE arithmetic.

[F1]

Failure-aware rounded Cholesky factorization: successful factorization uses the rounded pivot and square-root recurrence.

[F2]

A successful rounded Cholesky factorization has componentwise and normwise backward error bounds: success gives $E = \hat{L} {\hat{L}}^{T} - A$ with the stated bound.

[F3]

Operationwise real and complex floating point arithmetic with rounded square roots and fused multiply-add: positive square roots and basic operations are rounded to nearest, ties to even.

Verification

1.1givenF1F3algebra

The displayed quadratic form proves positive definiteness. The first pivot is $1$ , ${\hat{l}}_{21} = (1 / 2) / 1 = 1 / 2$ , and the second pivot is $1 - (1 / 2)^{2} = 3 / 4$ exactly, so the procedure succeeds through the square-root step.

2.1step 1.1F2algebra∎

Every binary64 value is rational, whereas sqrt(3/4)=sqrt(3)/2 is irrational; hence l-hat_22^2!=3/4. Multiplication of the displayed factor gives exactly the stated nonzero E, to which [F2] applies.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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, Cholesky factorization (standard reference, not scraped)
Stuart and Voss, Matrix Analysis and Algorithms, Warwick lecture notes (standard reference, not scraped)