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TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)
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The interval block LDL certifier returns the exact inertia on success

Statement

If Outward interval block LDL inertia certifier with pivot-inconclusive status returns certified-inertia with (n+,n−,n0), every Hermitian matrix in its input family has exactly that inertia. A pivot-inconclusive return does not imply that any family member is singular.

Facts & Assumptions

Given: A nonempty Hermitian family with conjugate-symmetric interval enclosure, one fixed permutation and block plan, and a certified-inertia return.

[F1]

Outward interval DAG operations enclose exact sums, products, reciprocals and Schur updates (Outward-rounded operations are inclusion-isotone and enclose DAG evaluations).

[F2]

Each nonsingular Hermitian pivot block and its Schur complement contribute additively to inertia (Block LDL congruence adds the Hermitian pivot inertias).

[F3]

Nonsingular congruence, including a permutation congruence, preserves complex Hermitian inertia (Complex Hermitian inertia is invariant under nonsingular congruence).

[F4]

On success the algorithm's scalar and two-by-two sign tests certify the indicated nonzero pivot signs for every member of the current family enclosure (Outward interval block LDL inertia certifier with pivot-inconclusive status).

[F5]

For a two-by-two Hermitian pivot, a negative determinant gives inertia (1,1,0); a positive determinant and trace sign give respectively (2,0,0) or (0,2,0) (A Hermitian two-by-two pivot inertia is determined by determinant and trace signs).

Proof

1.1givenF1F4induction

Induct over the pivot blocks. Initially the active interval matrix is the input family enclosure, so it encloses each original family member. If this holds before a block, [F1] makes the outward inverse and update enclose that member's exact pivot inverse and Schur complement; [F4] ensures the pivot is nonsingular throughout the family. Hence the enclosure invariant holds at the next block. The midpoint run only selects the candidate block plan; this interval induction is independent of its pivots.

2.1F2F3F4F5step 1.1algebra∎

A certified one-by-one pivot has the sign of its real interval. By [F5], each accepted two-by-two pivot has exactly its reported inertia. Applying [F2] successively to each pivot and its Schur complement, using congruence invariance [F3], proves that the sum returned at completion is the inertia of every original family member. If a test is unresolved the algorithm returns no inertia claim, and failure alone supplies no singularity conclusion.

Depends on

Used by

Cited to discharge well-definedness by Outward interval block LDL inertia certifier with pivot-inconclusive status.

Dependency tree · two levels

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Sources