Elamur Groupantsnumerical-linear-algebra
The groupant beside models/numerical-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 | checks |
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 |
|---|---|---|---|
numerical-linear-algebra-behavior-complement | behavior_complement | partial source model | behavior-complement |
numerical-linear-algebra-reverse-reconstruction | reverse_reconstruction | partial source model | reverse-reconstruction |
numerical-linear-algebra-factor-transport | factorization_solve_relation | partial source model | factor-transport |
numerical-linear-algebra-normal-equations | least_squares_stationarity_relation | partial source model | normal-equations |
numerical-linear-algebra-conditioning | conditional_residual_error | partial source model | conditioning |
Model text
model
numerical-linear-algebra
forward
Factor bounded symmetric systems, solve triangular systems, minimize linear least squares and iterate linear solves with fresh residuals.
invariant
Dimensions and coordinates agree; residual, reconstruction residual and normal residual are different observations; explicit errors/statuses retained.
cost
Cholesky O(n³), triangular O(n²), QR O(mn²), stationary O(kn²); O(m*n+n²) inputs/work; n<=64,m<=128,iterations<=10000.
composition
Cholesky A=L*Lᵀ transports b through lower and upper solves. QR normal equations transport least squares to a quadratic. Stationary solvers share Ax=b but not trajectories.
epoch-memory
Wave20, 2026-10-30. Models retain real-arithmetic assumptions separately from native finite checks. Kernels are library tools, not automatically adopted into training or inference.
checks
["factor_triangular_and_stationary_solvers_join_spd_system","qr_and_constrained_optimizers_join_known_quadratic"]
limitations
Positive pivots in floating arithmetic do not certify spectral conditioning. Unpivoted QR rejects exact zero encountered pivots only. Stationary convergence requires matrix properties not inferred from finite success.
catalog
Typed family relations
Each relation keeps its own domain and assurance.
behavior-complement
Type: behavior_complement.
At the real-model level, exclude nontriangular Cholesky factors or A!=LLᵀ, triangular solutions with Tx!=b, and least-squares candidates with Aᵀ(A*x-b)!=0. These exact exclusions describe targets, not an exact-equality acceptance test for f64 outputs. The native relation excludes reports differing from rounded factor/solve transitions, incorrect recomputed reconstruction/primal/normal residuals, or a stationary Converged status when its computed residual exceeds tolerance. A small native residual does not exclude large coordinate error on ill-conditioned inputs; numerical rank is not certified by unpivoted QR.
reverse-reconstruction
Type: reverse_reconstruction.
For fixed real triangular T and observed x, compatible right sides satisfy b=Tx; allowing T to vary gives many systems with the same solution. For fixed full-column-rank A, the real QR least-squares fiber for x is b=Ax+z with Aᵀz=0, so nonzero orthogonal residuals remain compatible. A real positive-diagonal Cholesky factor uniquely determines A=L*Lᵀ, while rounded native factor/report fibers can include neighboring matrices. Stationary output plus residual/status leaves many initial vectors, budgets and systems compatible; native antecedents must reproduce the rounded execution, not merely these real identities.
factor-transport
Type: factorization_solve_relation.
For real SPD A=L*Lᵀ, solving Ly=b then Lᵀx=y solves Ax=b. Native fixtures test independent [1,2] coordinates, with finite factor and rounding residuals retained.
normal-equations
Type: least_squares_stationarity_relation.
Aᵀ(Ax-b)=0 characterizes real least squares; unique x additionally requires full column rank. Forming AᵀA can square condition number; the harness's fixed well-conditioned design does not certify arbitrary data.
conditioning
Type: conditional_residual_error.
For nonsingular A, ||x-x*||<=||A^-1||*||Ax-b|| in real arithmetic. Without certified inverse norm and roundoff accounting, reported residuals give no universal solution-error percentage.
wave20-evidence
Family evidence — 2026-10-30
This family binds ten implementation/test sources. Named joins ["factor_triangular_and_stationary_solvers_join_spd_system","qr_and_constrained_optimizers_join_known_quadratic"] in algorithms/tests/composition_wave20.rs pass four times within the 695-check algorithm suite.
Forward: Factor bounded symmetric systems, solve triangular systems, minimize linear least squares and iterate linear solves with fresh residuals.
Invariant: Dimensions and coordinates agree; residual, reconstruction residual and normal residual are different observations; explicit errors/statuses retained.
Cost: Cholesky O(n³), triangular O(n²), QR O(mn²), stationary O(kn²); O(m*n+n²) inputs/work; n<=64,m<=128,iterations<=10000.
Assumptions: Positive pivots in floating arithmetic do not certify spectral conditioning. Unpivoted QR rejects exact zero encountered pivots only. Stationary convergence requires matrix properties not inferred from finite success.
Integrated evidence records actual receipts. Leaf oracles remain separate from cross-implementation agreement; family membership cannot verify unchecked callback premises or a shared error budget.
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