Elamur Groupantsspectral-linear-algebra
The groupant beside models/spectral-linear-algebra.groupant in the Mova repository: what the code computes, what it never does, how to go back from its outputs, its invariants, cost and checks.
Facts
Implementation material. A Mova code model describes this code at one source revision: partial models, finite checks and their stated scope. It is not a reviewed general exposition, and its labels are shown as written, never raised.
Components
| component | label | where |
|---|---|---|
| approximant | partial source model | forward |
| ast | not materialized | |
| composition | partial source model | composition |
| cost | partial source model | cost |
| descriptant | partial source model | epoch-memory |
| invariant | partial source model | invariant |
| tests | bounded executed | wave26-evidence |
Obernants
The typed relations around the forward model: the behaviour complement (what the code never does), the reverse reconstruction (every input compatible with an output) and specialised views.
| id | type | label | where |
|---|---|---|---|
spectral-linear-algebra-behavior-complement | behavior_complement | partial relational specification | behavior-complement |
spectral-linear-algebra-reverse-reconstruction | reverse_reconstruction | partial relational specification | reverse-reconstruction |
spectral-linear-algebra-permutation_coordinate | permutation_coordinate | partial relational specification | permutation_coordinate |
spectral-linear-algebra-basis_projection | basis_projection | partial relational specification | basis_projection |
spectral-linear-algebra-spectral_multiplicity | spectral_multiplicity | partial relational specification | spectral_multiplicity |
spectral-linear-algebra-residual_conditioning | residual_conditioning | partial relational specification | residual_conditioning |
Model text
model
spectral-linear-algebra
forward
Tagged dense matrix factorization and spectral observations: pivoted LU, symmetric Jacobi eigenpairs, modified Gram-Schmidt basis, dominant-eigen iteration and Lanczos tridiagonal projection. Reconstruction/residual observations differ from eigenvector accuracy. Real identities: PA=LU; A=QR and Q^TQ=I; AV=VD,V^TV=I; T=Q^TAQ. Native residuals measure departures.
invariant
Dimensions, finite values, symmetry/rank conditions, sign/tie conventions and statuses remain leaf-specific. Permutation and basis coordinates must travel with results; a small residual does not by itself identify a unique eigenvector.
cost
LU O(n³), O(n²) storage. Jacobi actually recomputes ||AV-VD||F via n matrix-vector products per rotation candidate; budget b costs O((b+1)n³), with repeated allocations. MGS m-by-n costs O(m n²); power O((b+1)n²); k-vector Lanczos O(k n²+n k²) with reorthogonalization and O(nk+k²) storage. Leaf caps/statuses remain binding.
composition
LU solves require the returned permutation and nonsingularity policy. MGS Q supplies orthonormal coordinates only on its admitted rank domain. Lanczos T is a projection Q^T A Q, not the entire matrix; Ritz vectors require lifting by Q. Comparing Jacobi eigenvalues to power iteration requires matching selected eigenvalue and convergence status, especially magnitude ties and multiplicity. Floating residuals need conditioning and spectral-gap assumptions before any solution-error interpretation. Power dominance additionally requires a magnitude gap and nonzero initial dominant-eigenspace projection; small residual alone allows a nondominant eigenpair.
limitations
Successful allocations and each leaf's finite domain/error precedence. These are tagged relations, not automatic production adoption. Floating residuals are not solution-error guarantees. No generic 1/2/5/10% composition-wide profile; finite evidence remains separate from analytical claims.
epoch-memory
Wave26, 2026-10-30. Tagged library tools; no automatic training adoption, source-volume quality claim or generic numerical profile.
checks
["cargo test -p mova-algorithms lu_decomposition::tests","cargo test -p mova-algorithms jacobi_eigen::tests","cargo test -p mova-algorithms modified_gram_schmidt::tests","cargo test -p mova-algorithms power_iteration::tests","cargo test -p mova-algorithms lanczos::tests"]
wave26-executed-checks
spectral-linear-algebra: wave26 checks
31 source-local test declarations across five modules. Every module has a compiled failing production mutation followed by exact restoration. The final whole library passed 877 checks twice debug and twice release. See integrated evidence for scopes and limitations. These tagged APIs are not an invented serial pipeline.
catalog
Typed family relations
behavior-complement
Type: behavior_complement.
In the tagged family relation, exclude outputs violating the selected leaf's invariant, domain/error precedence or coordinate convention. In particular: Dimensions, finite values, symmetry/rank conditions, sign/tie conventions and statuses remain leaf-specific. Permutation and basis coordinates must travel with results; a small residual does not by itself identify a unique eigenvector. Cross-family equality without the adapter assumptions in composition is excluded.
reverse-reconstruction
Type: reverse_reconstruction.
Exact complete LU reconstructs A=P^-1LU and full QR reconstructs A=QR. Eigenvector signs and repeated eigenspaces are ambiguous. Partial Q,T determines Q^TAQ but leaves action outside span(Q) undetermined. Rounded factors admit tolerance fibers.
permutation_coordinate
Type: permutation_coordinate.
Ignoring LU permutation changes the solved equation; reconstructed coordinates require the exact pivot convention.
basis_projection
Type: basis_projection.
MGS and Lanczos bases define coordinates; Q^T A Q can agree while omitted orthogonal directions differ.
spectral_multiplicity
Type: spectral_multiplicity.
Power has no dominance guarantee without a gap and nonzero initial projection; signs and repeated-eigenvalue bases are nonunique.
residual_conditioning
Type: residual_conditioning.
Residual thresholds are stopping observations. Eigenvalue gaps and conditioning are extra assumptions, not automatic guarantees.
wave26-evidence
spectral-linear-algebra: wave26 checks
31 source-local test declarations across five modules. Every module has a compiled failing production mutation followed by exact restoration. The final whole library passed 877 checks twice debug and twice release. See integrated evidence for scopes and limitations. These tagged APIs are not an invented serial pipeline.
This page in Markdown → https://md.groupants.elamur.ai/free/en/mova/models/spectral-linear-algebra.groupant/