---
title: "numerical-linear-algebra — Mova code model"
description: "Mova code model of models/numerical-linear-algebra.groupant: components with their labels, obernants and the model text."
url: "https://groupants.elamur.ai/free/en/mova/models/numerical-linear-algebra.groupant/"
---
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# numerical-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

- **kind:** groupant_group
- **status:** partial source bound group
- **bound sources:** 10
- **repository folder:** [`models/numerical-linear-algebra.groupant`](https://github.com/ElamurAI/mova-free/tree/main/models/numerical-linear-algebra.groupant)
- **repository commit:** `855da7bd4a7b`

> **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 | [checks](#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](#behavior-complement) |
| `numerical-linear-algebra-reverse-reconstruction` | reverse_reconstruction | partial source model | [reverse-reconstruction](#reverse-reconstruction) |
| `numerical-linear-algebra-factor-transport` | factorization_solve_relation | partial source model | [factor-transport](#factor-transport) |
| `numerical-linear-algebra-normal-equations` | least_squares_stationarity_relation | partial source model | [normal-equations](#normal-equations) |
| `numerical-linear-algebra-conditioning` | conditional_residual_error | partial source model | [conditioning](#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(m*n²), stationary O(k*n²); 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!=L*Lᵀ, triangular solutions with T*x!=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=T*x; 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=A*x+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(m*n²), stationary O(k*n²); 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](https://md.groupants.elamur.ai/free/en/mova/models/algorithm-library.groupant/#wave20-evidence) records actual receipts. Leaf oracles remain separate from cross-implementation agreement; family membership cannot verify unchecked callback premises or a shared error budget.