CVP verified calculation capability
Newton, bisection, or secant for an approximate real zero of a polynomial or sin. exp has no finite real zero and returns NO_CONVERGENCE. Discovery /nonlinear-equation. Not the nth-root page, not CAS, not a system.
Math · Numerical Capability ID math.numerical.root version 1.1.3 CVP VERIFIED STALE Core
Newton
x ← x − f(x)/f′(x)midpoint of a sign-change bracketx ← x₁ − f(x₁)(x₁−x₀)/(f(x₁)−f(x₀))Canonical symbols for this capability. Agent schemas remain the contract.
| Symbol | Quantity | Unit | Role |
|---|---|---|---|
method | — | — | input / output |
integrand | — | — | input / output |
coeffs | c0,c1,…,cn constant first; not all zero | — | input / output |
coefficients | Alias of coeffs; c0,c1,…,cn constant first; not all zero | — | input |
x0 | — | — | input |
x1 | — | — | input |
a | — | — | input |
b | — | — | input |
tol | Default 1e-10 | — | input / output |
max_iter | Maximum updates or midpoint evaluations; default 50 | — | input |
root | — | — | output |
residual | — | — | output |
scaled_residual | For polynomial roots, |p | — | output |
iterations | — | — | output |
degree | — | — | output |
stop_reason | — | — | output |
Inspectable numeric contract for this Capability@Version. Not a rewrite of the engine.
| Field | Declaration |
|---|---|
numeric_domain | real |
exactness | float64 |
algorithm | Newton / bisection / secant on a published 1-D residual |
valid_domain | bracket / guess as published; finite real zeros of poly / sin; exp has no finite real zero |
singularity_policy | INVALID_INPUT on a degenerate or same-sign bracket (endpoint signs compared without multiplying values); NO_CONVERGENCE when the iterate fails |
branch_policy | not applicable — real iterate |
tolerance | note: default 1e-10; polynomial coefficient-scaled residual, sin absolute residual, or bisection bracket half-width |
convergence | max_iter: default 50, maximum 100; criterion: poly reports signed raw residual and absolute coefficient-scaled residual; sin uses absolute residual; bisection may stop on bracket width; NO_CONVERGENCE coded |
conditioning | multiple roots and flat residuals are ill-conditioned; not a CAS |
rounding | ieee754_round_ties_to_even |
overflow | ieee754_nonfinite_or_coded_error |
reproducibility | deterministic algorithm + Capability@Version |
independent_oracle | cvp:math.numerical.root O2 / 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.
newton coeffs=-2,0,1 x0=1.5 → √2
CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Core assurance · all published methods. 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/numerical-root' \
-H 'Content-Type: application/json' \
-d '{"inputs":{"method":"newton","coeffs":[-2,0,1],"x0":1.5},"calculation_version":"1.1.3"}'{
"inputs": {
"method": "newton",
"coeffs": [
-2,
0,
1
],
"x0": 1.5
},
"calculation_version": "1.1.3"
}POST /api/v1/calc/numerical-root. 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.numerical.root"},"_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.numerical.root","inputs":{"method":"newton","coeffs":[-2,0,1],"x0":1.5},"calculation_version":"1.1.3"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'Pin calculation_version (1.1.3) when the result must stay reproducible.
Inspect the trust tier and required assurance level before execution. Full protocol notes: MCP docs.