CVP verified calculation capability

Numerical root 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 1.1.6 CVP VERIFIED CURRENT Core

Model

Newton

Newton: x ← x − f(x)/f′(x)
Bisection: midpoint of a sign-change bracket
Secant: x ← x₁ − f(x₁)(x₁−x₀)/(f(x₁)−f(x₀))

Modes: newton · newton raphson · newton-raphson · bisection · binary · secant

Inputs / Outputs

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`	Finite binary64 decimal; nonzero-to-zero underflow is NUMERIC_UNDERFLOW	—	input
`x1`	Finite binary64 decimal; nonzero-to-zero underflow is NUMERIC_UNDERFLOW	—	input
`a`	Finite binary64 decimal; nonzero-to-zero underflow is NUMERIC_UNDERFLOW	—	input
`b`	Finite binary64 decimal; nonzero-to-zero underflow is NUMERIC_UNDERFLOW	—	input
`tol`	Positive finite parsed value; default 1e-10	—	input / output
`max_iter`	Parsed value must be an integer from 1 to 100; default 50	—	input
`f`	Legacy alias of integrand; supplied aliases must agree	—	input
`guess`	Finite binary64 decimal; nonzero-to-zero underflow is NUMERIC_UNDERFLOW	—	input
`lo`	Finite binary64 decimal; nonzero-to-zero underflow is NUMERIC_UNDERFLOW	—	input
`hi`	Finite binary64 decimal; nonzero-to-zero underflow is NUMERIC_UNDERFLOW	—	input
`root`	—	—	output
`residual`	—	—	output
`scaled_residual`	For polynomial roots, \|p	—	output
`iterations`	—	—	output
`degree`	—	—	output
`stop_reason`	—	—	output

Trusted Math contract

Inspectable numeric contract for this Capability@Version. Not a rewrite of the engine.

Field	Declaration
`numeric_domain`	real
`exactness`	float64
`algorithm`	Newton / bisection / secant; polynomial values and residual comparisons use exact dyadic intermediates for binary64 inputs, then power-of-two solver scaling
`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 binary64 projections of signed residual and coefficient-scaled residual; exact dyadic residual stopping test; sin uses absolute residual; bisection may stop on bracket width; NO_CONVERGENCE coded
`conditioning`	multiple roots and flat residuals are ill-conditioned; residual convergence alone does not certify real-root existence or bound root-position error; a sign-change bracket for a continuous polynomial establishes root existence; 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)

Assumptions / Validity

Declared scope for this capability. Do not use it outside this domain.

Assumptions

The Result Card includes an engine-linked figure of f around the published root (marker locates the candidate at pack.root on y=0; its label includes the signed residual and does not certify a zero; Newton/secant seeds and the bisection bracket are read from inputs, not REST fields). Same fnValue as REST/MCP. Iterate history is not a REST field.
Not CAS / not complex roots listed
Not a system of equations
Not all-roots of a polynomial
Not ⁿ√a
Polynomial degree ≤ 8
Polynomial convergence uses |f(x)| divided by Σ|cᵢ||x|ⁱ; sin uses |sin x|. A small residual alone does not prove a nearby root for every polynomial.
Polynomial values, derivatives, and residual stopping tests use exact dyadic intermediates for the supplied binary64 coefficients and iterate. Solver values use power-of-two scaling; published signed and scaled residuals are projected to binary64. An underflowed solver value is not treated as an exact root.

Example

One numerical check. The calculator page holds worked steps and FAQ.

newton coeffs=-2,0,1 x0=1.5 → √2

Open this example in the calculator

Trust

CVP VERIFIED · 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.

CVP: CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published methods· View Manifest · CVP overview · Specification
Verified scope: all published methods
Assurance: Core
Declared partition coverage: PASS · 11/11 declared partitions (polynomial-degree-3, polynomial-degree-4, polynomial-degree-5, polynomial-degree-6, polynomial-degree-7, polynomial-degree-8, clustered-near-roots, newton, bisection, secant, invalid-domain)
Calculation version: 1.1.6
Verification: 39/39 calculation tests · 25/25 legacy golden · 7/7 legacy boundary · 7/7 legacy property
Production surface contract: 5/5
Versions: Calculation 1.1.6 · CVP protocol 1.0.0-proposed · Evidence 2026-10-28.metric-scope-v3
Verification revision: 2026-10-28.metric-scope-v3 · 2/2 property · digest 13de435e4844
Reference: O1 model · O2 expected_values · O2 numerical_behavior
Interfaces: PASS · UI (SSR) / REST / MCP
ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 11/11 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only. Raw HTTP REST/JSON-RPC transport rejection 5/5; raw-number source and conflicting pins are transport-only checks, not engine/SSR input-source parity.
CVP suite: 107/107 golden · 6/6 CVP boundary · 19/19 invalid · 2/2 property · 2/2 metamorphic · 21/21 cross-interface · 1/1 cross-calculator · 5/5 CVP contract
Supplemental domain review: Not performed
Named expert review: Not performed
Version lock: Supported (409 VERSION_MISMATCH)
Artifact consistency: Build verified · 1.1.6 (workspace artifacts agree)
Review-policy tier: Engine tested — catalog token engine_tested; supplemental; not CVP status

This calculator CURRENT · Public schema 1.1.6 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓

Call this capability

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":[-2,0,0,1],"x0":1.5,"x1":1,"a":1,"b":2,"max_iter":100},"calculation_version":"1.1.6"}'

{
  "inputs": {
    "method": "newton",
    "coeffs": [
      -2,
      0,
      0,
      1
    ],
    "x0": 1.5,
    "x1": 1,
    "a": 1,
    "b": 2,
    "max_iter": 100
  },
  "calculation_version": "1.1.6"
}

POST /api/v1/calc/numerical-root. This is a POST calculate endpoint — do not GET it.

Full REST contract: REST API docs.

Tool id: numerical-root
Engine: numerical-root
REST: POST /api/v1/calc/numerical-root
MCP: execute_calculation with capability_id: "math.numerical.root"
Agent input schema: Canonical (MCP / Agent)
Output schema: JSON Schema (latest)
Versioned input schema: Immutable pin (1.1.6)
Versioned output schema: Immutable pin (1.1.6)

References

Human-readable sources plus CVP reference evidence classes.

Sources

Reference evidence classes: O1 model · O2 expected_values · O2 numerical_behavior

Try with an agent

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":[-2,0,0,1],"x0":1.5,"x1":1,"a":1,"b":2,"max_iter":100},"calculation_version":"1.1.6"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'

Pin calculation_version (1.1.6) when the result must stay reproducible.

Inspect the trust tier and required assurance level before execution. Full protocol notes: MCP docs.