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See Review policy for supplemental domain assurance." }, "internal_review": { "status": "not_applicable", "reviewer": null }, "page": "https://www.calculatorx.com/calc/math/polynomial-roots", "api": "https://www.calculatorx.com/api/v1/calc/polynomial-roots", "schema_input": "https://www.calculatorx.com/schemas/math.polynomial.roots.input.json", "schema_output": "https://www.calculatorx.com/schemas/math.polynomial.roots.output.json", "assumptions": [ "The Result Card includes an engine-linked figure of p (markers are published real roots on y=0; complex roots stay off the real plane). Same Horner nesting as REST/MCP. Curve samples are not a REST field.", "Real coefficients", "Leading coefficient nonzero", "Degree ≤ 8", "Not a second quadratic page", "Not Ferrari quartic", "Not CAS factoring" ], "limitations": [ "Finite real constant-first coefficients,degree1–8,leadingnonzero. Arrayorcomma/semicolon/space-separated text;numeric tokens≤512chars,totaltext≤4608. Strict decimal/scientific;nonzero text→zero rejectsNUMERIC_UNDERFLOW;unknown/conflicting aliases andinapplicablex reject. Cubic a,b,c,d forms mustagreewithcoefficients;b/c/d omitteddefaultzero.", "all_roots acceptsall/roots/durand_kerner/durand/weierstrass/dk aliases;cardano accepts cubic/cubic_equation/cubic_formula;evaluate accepts eval/horner/polynomial_evaluate/polynomial_evaluation. Method/mode aliases agree;case/space/hyphennormalized. Presetpages locktheir method.", "Exact acceptedbinary64 rational arithmetic bounded16384bits;Horner projects once. Exact GCD square-free decomposition preservesmultiplicity. Stablequadratic/Cardano orscaledDK withtwo2048iterationattempts;exactSturm classifiesrealroots (512levels/4096steps).", "Every square-free rootset mustpass exactrational Weierstrass/Gershgorin diskdisjointness and2e-10relativeL1 inclusion-error budget,plusnormalizedL1residual≤2e-12. Unresolved/overlappingdisks,reals oriteration failNON_CONVERGENCE;normalizationrange losescoefficient→NUMERICAL_RANGE_ERROR;unrepresentable root/value rejects RESULT_OVERFLOW/UNDERFLOW. Not a universalfinite-input success guarantee.", "residual_max isunscaled Euclidean |p(z)|max,nullablewhenoutside finitebinary64;normalized_residual ismax (|Re p|+|Im p|)/sum |ci|(|re|+|im|)^i roundedbinary64. A zero displayedresidual isnot proof ofexactroots. Cubic delta usesexactdepresseddiscriminant sign,cubic_case true evenwhendelta isnull on range.", "Independent Fraction-factor/Decimal250/750 fixedvectors checkcomplete multisets inclmultiplicity with4e-10relativeL2+2subnormalquanta;zero roots exact,Horner≤1ULP. IndependentPython recomputesdiagnosticsfromreturnedroots. No whole-domain orcross-runtime bitwiseguarantee;nearclustersandcoeffperturbations areillconditioned.", "Display never snapsnonzero roots/coefficients. Real curve usesexactHorner;complexroots arelistedoffrealplane. Graphomittedwhenfinitecoordinates/labels cannotbedistinguished. NativeCSV/JSON preservefullpack/rawinputs. Typinglocal;sharedURLs andREST/MCP sendinputs andmayenterlogs." ], "boundary_conditions": [ "degree 0 or one coefficient → INVALID_INPUT", "leading coefficient 0 → INVALID_INPUT", "degree > 8 → VALUE_OUT_OF_RANGE", "method quadratic / cas / ferrari / newton / eigen / factor → INVALID_MODE" ], "sources": [ { "title": "ISO 80000-2:2019 Quantities and units — Mathematics", "url": "https://www.iso.org/standard/64973.html", "supports": "Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.", "kind": "standard", "publisher": "International Organization for Standardization", "reference_scope": "all" }, { "title": "NIST DLMF / Cardano cubic and Durand–Kerner simultaneous iteration", "supports": "Closed form for degree 1–3; Weierstrass–Durand–Kerner iteration for degree 4–8; Horner evaluation", "kind": "reference", "reference_scope": "model", "oracle_class": "O1" }, { "title": "NIST DLMF §3.8 — Nonlinear Equations", "section": "§3.8", "url": "https://dlmf.nist.gov/3.8", "supports": "Iterative root finding, convergence considerations, and polynomial-root methods.", "kind": "reference", "publisher": "National Institute of Standards and Technology" }, { "title": "NumPy — numpy.polynomial.polynomial.polyval", "url": "https://numpy.org/doc/stable/reference/generated/numpy.polynomial.polynomial.polyval.html", "supports": "Polynomial evaluation using Horner’s method; NumPy’s documented coefficients are in ascending degree order. CalculatorX’s input order must follow its own specification. Mathematical reference only; not validation or endorsement of this implementation.", "kind": "reference", "publisher": "NumPy Developers" } ], "methods": [], "fingerprint": "sha256:3abd0fcd292b44b441ce3ee81bf5a4c08088ba45a935f040daba543514e46afd", "tier": "engine_tested", "verification": { "tests": { "passed": 21, "total": 21, "golden": 7, "boundary": 6, "property": 8 }, "contract_checks": { "passed": 6, "total": 6 } }, "evidence_core_generated_at": "2026-10-27T09:16:21.351Z", "rendered_at": "2026-10-29T17:04:55.776Z", "release_integrity": { "status": "verified", "scope": "workspace", "production_attestation_required": true, "calculation_version": "1.2.2", "artifacts": { "engine": "1.2.2", "page": "1.2.2", "capability": "1.2.2", "evidence": "1.2.2", "input_schema": "1.2.2", "output_schema": "1.2.2", "mcp_contract": "1.2.2" }, "digests": { "input_schema_sha256": "59ff6f0d9c8221359f8e7b8b047f2641a66de48e301090643cbbbc0ec64b21d3", "output_schema_sha256": "adde668f2cd7988cb2e3c5cc4469cdf4b0fec92d089c6cccae78fe663b741568", "evidence_sha256": "9248af5629070d1b201999c581d30b9f604225c4b23f533fe8b3eec1b0f871f7", "evidence_core_sha256": "9248af5629070d1b201999c581d30b9f604225c4b23f533fe8b3eec1b0f871f7", "capability_sha256": "6018de205de94dedd92ef318af11aba46c02ab31c9818bfeb0367f24a1b0ed5b", "engine_sha256": "c6ebd8436f08b413e1ac6317b2d48550f2e6639a90dff254244e3aa562158b8d", "mcp_contract_sha256": "1cafec71298a0deb89bd361494db4322a690d14b61e801243407475112bdacbd" }, "contract_checks": { "total": 6, "passed": 6, "failed": 0, "items": [ { "id": "release-artifacts-aligned", "status": "pass", "detail": "1.2.2" }, { "id": "schema-output-required-declared", "status": "pass" }, { "id": "schema-valid-input-executes", "status": "pass" }, { "id": "engine-success-output-validates-schema", "status": "pass" }, { "id": "schema-rejects-incomplete-input", "status": "pass" }, { "id": "rest-success-payload-validates-schema", "status": "pass" } ] }, "report": "https://www.calculatorx.com/.well-known/calculatorx-release-integrity.json" }, "tests": [], "disclosure": { "policy": "calculatorx-public-evidence-v1", "full_validation": "private", "representation": "summary" } }