LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck pass

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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.

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

The scaled complex Householder wrapper terminates and has a unitary representative on success

Statement

For every admitted finite input of Scaled complex Householder reflector step with explicit range status, the wrapper terminates with its specified success, identity, breakdown or range-failure return. On a zero input column it uses the identity and copies the separate block exactly, with zero target and separate-block errors and without evaluating $P (0)$ . On a successful nonzero input it has $v \neq 0$ , and $P (v) = I - 2 v v^{*} / (v^{*} v)$ is unitary. Under the hypotheses of Scaled complex Householder and Givens steps have explicit local backward errors, the corresponding target-overwrite or separate-block error bounds apply to the actual application. No unconditional range success is asserted.

Facts & Assumptions

Given: The finite stored input dimensions and canonical operationwise arithmetic realization.

[F1]

Scaled real and complex Householder and Givens update arithmetic fixes every scaled norm, phase, update and failure operation.

[F2]

Scaled complex Householder and Givens steps have explicit local backward errors supplies the exact unitary representative and local error coefficients for successful calls.

Proof

1.1givenF1choose

Every loop in [F1] is over a finite input dimension. Its zero branch returns immediately; its nonzero branch either terminates at its first range/invalid operation or completes the finite sequence. The wrapper preserves these results with the stated breakdown label.

2.1F1F2step 1.1algebra∎

In the nonzero-input formation branch, success requires positive stored-vector squared norm. If $v = 0$ , its squares and sum are exactly zero, contradicting success in that branch. Thus $v \neq 0$ . Put $K = v v^{*} / (v^{*} v)$ ; then $K^{*} = K$ and $K^{2} = K$ , so $(I - 2 K)^{*} (I - 2 K) = I$ . Under its stated hypotheses, [F2] supplies the corresponding target/separate-block error bounds. In the zero-input branch the representative is $I$ , the block is copied exactly, and both target and separate-block errors are zero; no expression $P (0)$ is evaluated.

Depends on

Used by

Nothing in the library uses this result yet.

Cited to discharge well-definedness by Scaled complex Householder reflector step with explicit range status.

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

James Demmel, Math 221 Lecture 16: Hessenberg QR iteration (standard reference, not scraped)
David Bindel, CS 6210 Matrix Computations, Hessenberg QR iteration (standard reference, not scraped)