CVP verified calculation capability
Solve Ax=b for order 2–4, plus det, inverse, rank, LU, Cholesky (SPD A=LLᵀ), QR and SVD (square or rectangular m×n with m,n∈[2,4]), least squares (same rectangular shape), RREF, condition number, nullspace, column space, rowspace, left-nullspace, eigen, multiply, transpose. Same engine as /workspace/matrix. Discovery pages share this capability. Not a CAS. Not Statistics regression.
Math Capability ID math.linear_algebra version 1.14.1 CVP VERIFIED STALE Core
2×2 determinant
det = a₁₁a₂₂ − a₁₂a₂₁PA = LU, then Ly = Pb, Ux = yModes: solve · det · inverse · rank · multiply · transpose · eigen · lu · chol · qr · svd · lstsq · rref · condition · nullspace · columnspace · rowspace · leftnullspace
Canonical symbols for this capability. Agent schemas remain the contract.
| Symbol | Quantity | Unit | Role |
|---|---|---|---|
order | — | — | input / output |
rows | — | — | input / output |
cols | — | — | input / output |
a11 | — | — | input |
a12 | — | — | input |
a13 | — | — | input |
a21 | — | — | input |
a22 | — | — | input |
a23 | — | — | input |
a31 | — | — | input |
a32 | — | — | input |
a33 | — | — | input |
b1 | — | — | input |
b2 | — | — | input |
b3 | — | — | input |
A | — | — | input |
x | — | — | output |
y | — | — | output |
z | — | — | output |
residual_2 | — | — | output |
backward_error | — | — | output |
rref | — | — | output |
pivot_columns | — | — | output |
det | — | — | output |
rank | — | — | output |
method | — | — | output |
algorithm | — | — | output |
numeric_backend | — | — | output |
condition_number | — | — | output |
basis | — | — | output |
dimension | — | — | output |
singular_values | — | — | output |
rank_numeric | — | — | output |
S | — | — | output |
VT | — | — | output |
eigenvalues | — | — | output |
L | — | — | output |
Q | — | — | output |
inv | — | — | output |
reconstruction_residual | — | — | output |
orthogonality_residual | — | — | output |
residual_max | — | — | output |
singular | — | — | output |
eigenvectors | — | — | output |
U | — | — | output |
R | — | — | output |
Inspectable numeric contract for this Capability@Version. Not a rewrite of the engine.
| Field | Declaration |
|---|---|
numeric_domain | real |
exactness | float64 |
algorithm | small-square seed: LU+partial-pivot solve (not explicit inverse), Gauss–Jordan inverse/RREF, Cholesky LLᵀ (SPD), Householder QR (square or rectangular), one-sided Jacobi SVD (square or rectangular; does not form AᵀA), Faddeev–LeVerrier eigen (P2 vs Hessenberg+QR), QR/SVD lstsq (rectangular m×n), 1-norm κ; scale-aware singularity ε·max(m,n)·‖A‖_max |
valid_domain | real square order 2–4 for solve/det/inverse/LU/Cholesky; Cholesky requires SPD; qr, svd, and lstsq also A ∈ R^{m×n} with m,n ∈ [2,4] |
singularity_policy | SINGULAR_MATRIX on inverse of a singular A; κ₁ → infinite (null) when A is singular; Cholesky NOT_SYMMETRIC / NOT_POSITIVE_DEFINITE; pivot/rank tests are scale-aware (ε·max(m,n)·‖A‖_max), not a fixed 1e-15 floor |
branch_policy | not applicable — real matrices; eigen may report conjugate pairs |
tolerance | note: scale-aware pivot / SVD rank vs ε·max(m,n)·‖A‖_max (or σ_max); LINEAR_ALGEBRA_SINGULARITY_THRESHOLD is the O(1) documented floor, not the runtime test |
convergence | not applicable — dense ≤4×4 direct factorizations; Jacobi SVD is the bounded iterate |
conditioning | κ₁ = ‖A‖₁‖A⁻¹‖₁; ill_conditioned warning when κ₁ ≥ engine threshold; unique solve publishes residual_2 = ‖Ax−b‖₂ and η = ‖Ax−b‖₂ / (‖A‖₁‖x‖₂ + ‖b‖₂) (denominator 0 → η = residual_2). REST exposes algorithm, numeric_backend, condition_number, warnings. Engine also returns singularity_threshold. |
rounding | ieee754_round_ties_to_even |
overflow | ieee754_nonfinite_or_coded_error |
reproducibility | deterministic + Capability@Version |
independent_oracle | cvp:math.linear_algebra O2 (linear-algebra-reference) / Evidence published cases |
golden_vectors | Evidence golden + boundary + property (and published_reference_case when listed) |
Declared scope for this capability. Do not use it outside this domain.
Assumptions
One numerical check. The calculator page holds worked steps and FAQ.
2x+y=5, x+y=3 → x=2, y=1
CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Core assurance · all published modes. Computational verification is defined by CVP. Supplemental domain review (internal) and named expert review are defined by the Review policy and do not by themselves determine CVP conformance.
Human page, REST, and MCP share the same engine and calculation_version.
Build verified hashes workspace artifacts only. After deploy, attest:production GETs canonical URLs twice — the public/cached view an Agent actually hits, and origin — including every /capabilities/{id} HTML page. Public Capability HTML version must match catalog before production attestation passes. Live Trust rows above fetch /.well-known/calculatorx-production-attestation.json
and the output schema const. Build hashes: release integrity. That proves currently published page / engine / schema / evidence / MCP agree — not that a historical version is independently immutable. Reference Calculation Standard — Production surface contract 5/5.
curl -sS -X POST 'https://www.calculatorx.com/api/v1/calc/linear-system' \
-H 'Content-Type: application/json' \
-d '{"inputs":{"mode":"solve","order":2,"a11":2,"a12":1,"a21":1,"a22":1,"b1":5,"b2":3},"calculation_version":"1.14.1"}'{
"inputs": {
"mode": "solve",
"order": 2,
"a11": 2,
"a12": 1,
"a21": 1,
"a22": 1,
"b1": 5,
"b2": 3
},
"calculation_version": "1.14.1"
}POST /api/v1/calc/linear-system. This is a POST calculate endpoint — do not GET it.
Full REST contract: REST API docs.
Human-readable sources plus CVP reference evidence classes.
Sources
Reference evidence classes: O1 model · O2 expected_values · O2 numerical_behavior
Inspect, then execute. Same engine as REST and the human calculator.
curl -sS -X POST 'https://www.calculatorx.com/mcp' \
-H 'Content-Type: application/json' \
-H 'MCP-Protocol-Version: 2026-07-28' \
-H 'Mcp-Method: tools/call' \
-H 'Mcp-Name: inspect_capability' \
-d '{"jsonrpc":"2.0","id":1,"method":"tools/call","params":{"name":"inspect_capability","arguments":{"capability_id":"math.linear_algebra"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'curl -sS -X POST 'https://www.calculatorx.com/mcp' \
-H 'Content-Type: application/json' \
-H 'MCP-Protocol-Version: 2026-07-28' \
-H 'Mcp-Method: tools/call' \
-H 'Mcp-Name: execute_calculation' \
-d '{"jsonrpc":"2.0","id":2,"method":"tools/call","params":{"name":"execute_calculation","arguments":{"capability_id":"math.linear_algebra","inputs":{"mode":"solve","order":2,"a11":2,"a12":1,"a21":1,"a22":1,"b1":5,"b2":3},"calculation_version":"1.14.1"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'Pin calculation_version (1.14.1) when the result must stay reproducible.
Inspect the trust tier and required assurance level before execution. Full protocol notes: MCP docs.