---
title: "algorithm-modified-gram-schmidt-tests — Mova code model"
description: "Mova code model of algorithms/src/modified_gram_schmidt/tests.rs: components with their labels, obernants and the model text."
url: "https://groupants.elamur.ai/free/en/mova/algorithms/src/modified_gram_schmidt/tests.rs.groupant/"
---
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# algorithm-modified-gram-schmidt-tests

The groupant beside algorithms/src/modified_gram_schmidt/tests.rs 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
- **status:** partial descriptive instance
- **source:** [`algorithms/src/modified_gram_schmidt/tests.rs`](https://github.com/ElamurAI/mova-free/blob/main/algorithms/src/modified_gram_schmidt/tests.rs)
- **source SHA-256:** `ba4ee8aafe3107c9…`
- **repository folder:** [`algorithms/src/modified_gram_schmidt/tests.rs.groupant`](https://github.com/ElamurAI/mova-free/tree/main/algorithms/src/modified_gram_schmidt/tests.rs.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 | [approximant](#approximant) |
| **ast** | partial | `syntax.json` |
| **descriptant** | present | [descriptant](#descriptant) |
| **invariant** | partial | [approximant](#approximant) |
| **tests** | finite checks | [tests](#tests) |

## 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 |
|---|---|---|---|
| `behavior_complement` | behavior_complement | partial relational specification | [behavior_complement](#behavior-complement) |
| `reverse_reconstruction` | reverse_reconstruction | partial relational specification | [reverse_reconstruction](#reverse-reconstruction) |
| `analytic_fixture_relation` | analytic_fixture_relation | partial relational specification | [analytic_fixture_relation](#analytic-fixture-relation) |
| `compiled_mutation_relation` | compiled_mutation_relation | partial relational specification | [compiled_mutation_relation](#compiled-mutation-relation) |
| `boundary_observation_relation` | boundary_observation_relation | partial relational specification | [boundary_observation_relation](#boundary-observation-relation) |

## Model text


### descriptant

#### One-pass modified Gram-Schmidt with reconstruction and orthogonality checks

Original std-only bounded Rust. Citations are in the implementation module. This file groupant preserves the prospective model, typed excluded-output and compatible-preimage relations, independent checks and exact source binding.

Epoch 2026-10-30: 31 new named tests; complete library 724 tests passes twice debug and twice release with identical nonempty normalized outcomes. Five production mutants fail named assertions after compilation; restoration reruns pass. Strict algorithm all-target Clippy passes. No timing was performed.

Model correction: the Jacobi implementation recomputes every original-matrix eigenpair residual each round, so its actual bound is O(b*n^3), stronger work than the prospective O(b*n^2+n^3) sketch. This is retained as a modeling correction, not a speed claim.

Gaps: no semantic AST or macro expansion, allocator exhaustion injection, general floating error proof, or calibrated approximate profiles.

### model

#### Mathematical combiner

Inputs: exact tests.rs, source-bound partial syntax, prospective recurrence, current forward/invariant, typed family and assertion outcomes. First validate the domain; compose ordered updates under dimension and finite-arithmetic guards; compare fresh observations with independent oracle relations. No logical implication from a finite pass to universal accuracy is allowed.

The shared helper dependency is algorithms/src/lu_decomposition/mod.rs: checked allocation, finite arithmetic, shape/control validation and vector operations. LU is the owner and the remaining modules import only crate-private helpers plus the shared public Error. Cross-file correctness still requires integration exports and tests; these leaf models do not prove wiring.

Epoch memory retains the original prospective math and its corrected Jacobi cost. Zero residual is an observed rounded equality; ambiguous preimages remain explicit.

### approximant

#### Forward checks and invariant

112 upper-triangular integer fixtures reconstructed independently; exact orthogonal and projection oracles; duplicate/zero columns, equality threshold, tiny scalar, dimensions, nonfinite values and near-dependent repeatability.

Domain: finite fixture generators and expected error/status cases named in tests.rs. Each assertion must hold for the observed output, not a surrogate implementation. Cost is the sum of bounded fixture calls; exhaustive small grids are finite and are not universal numerical proofs. Invariant: test execution does not mutate caller data or external state; no callbacks or random sampling. Allocation failure may abort fixture construction; production checked reservations are tested at overflow where applicable.

No executable 1/2/5/10% surrogate is attached. Residual and orthogonality norms are finite observations, not generic relative solution-error certificates. A meaningful percentage profile needs a separate domain, metric, zero-reference rule and analytical or finite evidence. No benchmark, production speedup or end-to-end Mova quality gain is claimed.

### models

## Typed obernant family

### behavior_complement

For these fixed fixtures and assertions, exclude a passing observation whenever any asserted factor, status, dimension, residual or independent oracle differs. Inputs outside the fixture generator are unconstrained by this finite relation; a pass excludes neither all numerical defects nor allocation-failure behavior. Specifically: 112 upper-triangular integer fixtures reconstructed independently; exact orthogonal and projection oracles; duplicate/zero columns, equality threshold, tiny scalar, dimensions, nonfinite values and near-dependent repeatability.

### reverse_reconstruction

A pass has as preimages all implementations whose observations satisfy the listed fixture predicates; infinitely many wrong implementations can agree on finite inputs. A named failure narrows the violated relation but cannot reconstruct the exact defect. Repeated identical outcomes recover determinism only on the exercised executions.

### analytic_fixture_relation

Independent expected observations: 112 upper-triangular integer fixtures reconstructed independently; exact orthogonal and projection oracles; duplicate/zero columns, equality threshold, tiny scalar, dimensions, nonfinite values and near-dependent repeatability. Production outputs are compared with arithmetic formulas or reconstruction, not only a second call to production.

### compiled_mutation_relation

A deliberate production edit is compiled and exercised by a named assertion: diagonal normalization doubled; orthogonal_oracle must fail. Restore source exactly and rerun the assertion. This establishes sensitivity to that edit only.

### boundary_observation_relation

Tests partition empty, scalar, ordinary, malformed and bounded resources where explicitly listed. They do not inject real allocator exhaustion. NaN and infinity are rejected; finite underflow is allowed and can change mathematical rank. No callbacks or input mutation are expected.

### tests

#### Checks

112 upper-triangular integer fixtures reconstructed independently; exact orthogonal and projection oracles; duplicate/zero columns, equality threshold, tiny scalar, dimensions, nonfinite values and near-dependent repeatability.

Run cargo test -p mova-algorithms modified_gram_schmidt::tests. Repeat debug and release outcomes. A compiled production mutant exercises a named assertion; exact source restoration must pass. Strict all-target Clippy and focused rustfmt complete static checks.

### prospective

#### Prospective linear algebra epoch (before implementation)

Domain: finite entries bounded by 1e100, dimensions 1..64 (MGS m>=n up to 128), tolerance finite 0..1e100, iteration budgets 0..4096. Empty matrices rejected. Exact symmetry for Jacobi eigen and Lanczos. Checked reservations and finite intermediate checks; failures precede successful reports. No callbacks, randomness, mutation of caller data, timing or accuracy-cap claims.

LU: partial pivot chooses earliest maximal absolute entry; swap permutation and completed L columns. L unit lower triangular, U upper triangular; PA=LU in reals. Reject exactly zero encountered pivot. Fresh Frobenius reconstruction. O(n^3) work O(n^2) space.
MGS: copy column v; sequential dot and projection against prior normalized columns; positive Rjj=norm(v); reject norm<=tolerance as dependent column. A=QR in reals, Q column orthonormal. Fresh reconstruction and orthogonality. O(m*n^2), O(m*n+n^2) space. Not numerical rank revelation.
Jacobi eigen: repeatedly annihilate earliest largest upper offdiagonal via stable bounded angle rotation, accumulate column eigenvectors; stop by fresh original-matrix residual or budget. Return unsorted diagonal approximations and canonical signs (first largest-magnitude coordinate nonnegative). A V=V D and V^T V=I in exact converged arithmetic. O(b*n^2+n^3), O(n^2). Offdiagonal decisions deterministic.
Power: normalize initial vector; Rayleigh quotient and fresh norm(Ax-lambda*x); check residual then budget; normalize Ax, classify zero image breakdown if encountered. Canonical signs. O(b*n^2), O(n^2+n). Residual only certifies observed approximate eigenpair; no unconditional dominant eigenvalue promise (start may miss dominant eigenspace).
Lanczos: symmetric three-term recurrence alpha=q^T A q, w=Aq-alpha*q-beta_prev*q_prev; two full MGS reorthogonalization passes against every current basis vector. Nonnegative beta=norm(w). Break down if beta<=tol, complete if n columns reached, budget otherwise. Report actual ||AQ-QT||F including terminal residual and dropped finite reorthogonalization corrections; orthogonality observed separately. O(k*n^2+n*k^2), O(n*k). No implicit eigenvalue accuracy guarantee; beta threshold is absolute. No explicit generic 1/2/5/10% surrogates justified.

Oracles planned: exact 2x2 LU with pivot, QR on orthogonal columns and reconstruction, 2x2 symmetric analytic eigenvalues, diagonal power start/missing-component and zero budget, diagonal Lanczos full span/early invariant subspace. Boundary validation and compiled mutants for each algorithm. Repeated debug/release normalized named outcomes. No performance prediction beyond operation bounds until exclusive measurement grant.

### validation

#### Validation epoch 2026-10-30

31 new tests (6 LU, 6 MGS, 6 Jacobi eigen, 6 power, 7 Lanczos); complete library 724 pass twice debug and twice release. Sorted nonempty test-name/outcome records agree in each mode. Strict cargo clippy -p mova-algorithms --all-targets -- -D warnings passes. Focused rustfmt passes.

Compiled mutant: diagonal normalization doubled; orthogonal_oracle must fail. Source restored byte-for-byte and named positive rerun passes. First exploratory LU mutation omitted the U row swap and failed by unwrap of an arithmetic error; it was not accepted as assertion evidence. The accepted LU mutation omits the permutation swap and fails assert_eq.

Receipts: $MOVA_DATA/runs/octopus-wave26-linear-2026-10-30/ contains prospective.md, debug-1/2.log, release-1/2.log, normalized records, controls.log, mutant-*.log, restored-*.log, clippy-verified.log, paired syntax JSON and generator log. Validation log paths are observations from this finite run, not universal certificates. No timing or allocator-exhaustion injection performed. Partial syntax is generated twice, compared exactly, then installed and SHA-bound.

Integration boundary: checking the full augmented temporary registry reports 21 diagnostics, all confined to parent algorithm-library/mova child lists and transitive source bindings that the integration owner will refresh. No new leaf diagnostics occur in that report. The full hierarchy is not yet declared valid. A single-node registry cannot validate this whole checkout because the checker discovers all sidecars.