--- title: "spectral-linear-algebra — Mova code model" description: "Mova code model of models/spectral-linear-algebra.groupant: components with their labels, obernants and the model text." url: "https://groupants.elamur.ai/free/en/mova/models/spectral-linear-algebra.groupant/" --- Elamur: [Mova](https://md.mova.elamur.ai/free/en/) · [Dictionary](https://md.dictionary.elamur.ai/free/en/) · [Research](https://md.research.elamur.ai/free/en/) · **Groupants** This page as HTML: # spectral-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 - **kind:** groupant_group - **status:** partial source bound group - **bound sources:** 10 - **repository folder:** [`models/spectral-linear-algebra.groupant`](https://github.com/ElamurAI/mova-free/tree/main/models/spectral-linear-algebra.groupant) - **repository commit:** `48f2e6f74146` > **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](#forward) | | **ast** | not materialized | | | **composition** | partial source model | [composition](#composition) | | **cost** | partial source model | [cost](#cost) | | **descriptant** | partial source model | [epoch-memory](#epoch-memory) | | **invariant** | partial source model | [invariant](#invariant) | | **tests** | bounded executed | [wave26-evidence](#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](#behavior-complement) | | `spectral-linear-algebra-reverse-reconstruction` | reverse_reconstruction | partial relational specification | [reverse-reconstruction](#reverse-reconstruction) | | `spectral-linear-algebra-permutation_coordinate` | permutation_coordinate | partial relational specification | [permutation_coordinate](#permutation-coordinate) | | `spectral-linear-algebra-basis_projection` | basis_projection | partial relational specification | [basis_projection](#basis-projection) | | `spectral-linear-algebra-spectral_multiplicity` | spectral_multiplicity | partial relational specification | [spectral_multiplicity](#spectral-multiplicity) | | `spectral-linear-algebra-residual_conditioning` | residual_conditioning | partial relational specification | [residual_conditioning](#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](https://md.groupants.elamur.ai/free/en/mova/models/algorithm-library.groupant/#wave26-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](https://md.groupants.elamur.ai/free/en/mova/models/algorithm-library.groupant/#wave26-evidence) for scopes and limitations. These tagged APIs are not an invented serial pipeline.