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ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

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A nearly noiseless weakly scaled observation amplifies innovation-solve residual

Example

Take a scalar predicted variance P−=1, observation coefficient H=10−3, and measurement variance R=10−12. For innovation v=1, suppose the computed triangular solve has residual L11z^−v=10−9, with no other mean-update error.

Facts & Assumptions

Given: The scalar data and the exact factor mean update of Finite-horizon square-root Kalman filter by rectangular factors and QR.

[F1]

The innovation variance is S=HP−HT+R, the cross covariance is B=P−HT, and the exact gain is BS−1. The block QR measurement update gives the canonical Kalman update

[F2]

The finite-horizon bound includes the forward-solve residual multiplied by an upper bound for the gain norm. The square-root filter has an explicit conditioned finite-horizon error bound

Proof

Given: The scalar inputs in the Example.

1.1F1givenalgebra

The signal part of the observation variance is HP−HT=10−6, so R/(HP−HT)=10−6 and the observation noise is small relative to the predicted signal variance. The innovation variance and cross covariance are S=10−6+10−12=10000011012,B=10−3. Thus the exact gain is K=BS=1091000001.

2.1F1F2step 1.1algebra∎

In the scalar block QR, L112=S and L21L11=B, so L21/L11=K. If L11z^−v=ρ=10−9, then z^=(v+ρ)/L11. The induced mean-update error is therefore L21z^−Kv=Kρ=11000001≈9.99999000001×10−7. An absolute solve residual of 10−9 has produced an update error about 10−6 because the gain is about 103. The K#ρ term in [F2] provides a certified upper-bound counterpart when only a gain norm bound is available.

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