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Numerical Root Calculator

Find one approximate real zero of a polynomial or sin x using Newton, bisection, or secant. Inspect the residual and stopping reason. Calculate locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published methods
Input interpretation
Enter values to calculate.
Result
—
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) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
One approximate real zero of a polynomial or sin x by Newton, bisection, or secant, with an engine-linked figure of f(x)=0.
Scope
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.
Verification
Engine tested · Source checked · v1.1.6 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published methods· View Manifest · CVP overview · Specification
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
Legacy regression
39/39 tests · Production surface contract 5/5
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
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.
Supplemental domain review
Not performed
Named expert review
Not performed
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 · Manifest
Sources
Sources
Evidence
25 legacy golden · 7 legacy boundary · legacy regression suite · 107/107 oracle-backed golden · 19/19 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.1.6 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Newtonx ← x − f(x)/f′(x)
Bisectionmidpoint of a sign-change bracket
Secantx ← x₁ − f(x₁)(x₁−x₀)/(f(x₁)−f(x₀))
iPolynomial coefficients are constant-first (c0,c1,…), degree ≤ 8. For poly, stopping uses a coefficient-scaled residual; bisection may instead stop on bracket width. eˣ has no finite real root. This is not ⁿ√a — that remains /calc/math/root.

How to use

1

Enter f and a method

Poly needs coeffs (constant first). x² − 2 is −2,0,1. Newton needs x0; bisection needs a and b with opposite signs.

2

Read the approximate root

The main value is a rounded display approximation; result details and REST retain the full binary64 root. Signed residual f(x), coefficient-scaled residual, and tolerance are separate. Iterations count solver updates after the initial guess (iteration 0); bisection counts midpoint evaluations.

Example calculations

Common configurations with formula and result.

ϟ

√2

x² − 2 = 0, Newton from 1.5

x ← x − (x²−2)/(2x)
1.414213…
ϟ

Linear

x − 4 = 0

coeffs −4,1
4
ϟ

sin x = 0

Newton from 3

x ← x − sin x / cos x
π

Numerical Root calculator specification

Version 1.1.6 · Engine tested

Calculation status

Review policy · Evidence

Definition
A real root of f(x)=0 is a number r with f(r)=0. This calculator approximates a zero of a polynomial or sin x using Newton, bisection, or secant. eˣ has no finite real zero.
What it calculates
One approximate real zero of a polynomial or sin x by Newton, bisection, or secant, with an engine-linked figure of f(x)=0.
Inputs
  • method
  • integrand?
  • coeffs?
  • x0?
  • x1?
  • a?
  • b?
  • tol?
  • max_iter?
Outputs
  • root
  • residual
  • scaled_residual?
  • iterations
  • tol
  • stop_reason
Formula
f(x) = 0
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.
Units
  • dimensionless
Boundary conditions
  • Nonzero decimal input underflow → NUMERIC_UNDERFLOW; no false zero input
  • Conflicting aliases → AMBIGUOUS_INPUT; unknown machine fields → UNKNOWN_INPUT
  • max_iter must be an integer from 1 to 100; omitted tol defaults to 1e-10
  • no sign change → INVALID_INPUT
  • Bisection requires opposite endpoint signs even for tiny finite function values → INVALID_INPUT
  • all-zero coefficients or an underflowed bisection endpoint sign → INVALID_INPUT
  • an underflowed bisection midpoint sign → NO_CONVERGENCE
  • unknown method (including cas, halley) → INVALID_MODE
  • no convergence → NO_CONVERGENCE
  • eˣ=0 has no finite real root → NO_CONVERGENCE
Example
newton coeffs=-2,0,1 x0=1.5 → √2
Validation cases

3 published on this page · 39/39 tests · Production surface contract 5/5 · View evidence

  • method=newton coeffs=-2,0,1 x0=1.5 → root≈1.414213562
  • method=halley coeffs=-2,0,1 x0=1 → INVALID_MODE
  • method=newton integrand=exp x0=0 → NO_CONVERGENCE
Sources
Calculation version
1.1.6

Background

Interpretation and common distinctions.

What does this numerical root calculator compute?

A real root of f(x)=0 is a number r with f(r)=0. This calculator approximates a zero of a polynomial or sin x using Newton, bisection, or secant. eˣ has no finite real zero.

The published formula or method is f(x) = 0. The following references support the stated mathematics or method:

Which inputs and limits should you check?

Convergence, the signed residual, the stopping criterion, and the method’s required interval or starting values must be interpreted together. A reported root is conditional on the numerical contract.

The worked example uses x² − 2 = 0, Newton from 1.5 with x ← x − (x²−2)/(2x). Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.numerical.root v1.1.6 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

How should I check an approximate root?

Check the reported stopping reason before interpreting the displayed root. For a polynomial, this calculator compares the coefficient-scaled residual, |f(x)| / Σ|cᵢ||x|ⁱ, with the tolerance; bisection can also stop when its bracket is narrow enough. The signed residual is the exact polynomial value at the supplied binary64 coefficients and returned root, rounded to binary64. The scaled residual is also displayed as binary64, while the stopping comparison uses exact intermediate values. These details do not guarantee the error in the root location, especially for multiple roots. A small residual alone does not bound the error in x: NIST DLMF §3.8(vi) explains how sensitive polynomial zeros can be to coefficient changes. The plot helps locate the reported approximation but is not an additional accuracy test.

Which method and record can I cite?

The NIST descriptions of Newton's rule and bisection and secant support the method formulas; they do not certify this implementation or a particular computed result. For the calculator's tested behavior, cite the version 1.1.6 Evidence with its calculation version and verification status. A URL with input parameters reproduces a particular calculation, while the canonical URL identifies the calculator itself. Read the current production status alongside the Evidence before describing a result as production verified.

Does meeting the residual tolerance prove that a real root exists?

Check real-root existence separately when using Newton or secant. For example, x² − 2x + 1.0000000000000009 has a negative discriminant and no real root, yet its small positive residual at x=1 meets the default residual tolerance. This is a tolerance-based candidate, not a certificate of a real zero. A valid sign-change bracket for a continuous polynomial establishes that a root exists; its stopping reason still determines how to interpret the reported position.

How are text inputs and share links interpreted?

Inputs use finite binary64 decimal values, including scientific notation. A nonzero decimal that rounds to zero is rejected as NUMERIC_UNDERFLOW; other finite binary64 rounding remains possible. Coefficients are constant first, separated by commas, semicolons, or whitespace. Maximum iterations must be an integer from 1 to 100. Conflicting aliases are rejected. Empty fields in shared links remain empty; defaults apply only to absent fields. Clean pages use a polynomial example for Numerical Root and a sine example for Nonlinear Equation. The numerical model is shared.

Ordinary interactive calculations run locally. Opening or sharing a URL sends its query parameters to the server and they may appear in server or third-party logs; REST and MCP requests also send their inputs to the server. Avoid putting sensitive inputs in share links.

Other calculators in this family: Brent Method Calculator, Definite Integral Calculator, Euler Method Calculator, Gauss Quadrature Calculator, Golden Section Search Calculator, Nonlinear Equation Calculator, Numerical Derivative Calculator, Numerical Interpolation Calculator . Explore all Numerical Calculus.

→ Brent Method Calculator Brent’s method (golden section plus inverse parabolic interpolation) for a minimum of a polynomial, sin, or exp on [a, b]. Same engine as One-Dimensional Optimization. Not root-finding Brent. Runs locally. › → Definite Integral Calculator Composite trapezoid, Simpson 1/3, or Gauss–Legendre of a polynomial, sin, or exp on [a, b]. Not indefinite symbolic integrate. Runs locally. › → Euler Method Calculator Forward Euler for a scalar first-order IVP y′=f(t,y). Same engine as Ordinary Differential Equation. Not RK4, not rk45, not a system. Runs locally. › → Gauss Quadrature Calculator n-point Gauss–Legendre quadrature of a polynomial, sin, or exp on [a, b]. Same engine as Definite Integral. Not indefinite CAS. Runs locally. › → Golden Section Search Calculator Golden-section search for a minimum of a polynomial, sin, or exp on [a, b]. Same engine as One-Dimensional Optimization. Not Brent, not multivariate. Runs locally. › → Nonlinear Equation Calculator Solve a polynomial or sin x for one approximate real zero using Newton, bisection, or secant. Inspect the residual and stopping reason. Calculate locally. › → Numerical Derivative Calculator Finite-difference derivative of a polynomial, sin, or exp, plus a numeric two-sided limit. Central, forward, backward, or limit. Discovery /two-sided-limit. Not symbolic d/dx. Runs locally. › → Numerical Interpolation Calculator Piecewise-linear, Lagrange, or natural cubic spline of tabulated (x, y). Not regression, not CAS. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this the nth-root calculator?

No. ⁿ√a is /calc/math/root. This page approximates one real zero of a polynomial or sin x. eˣ=0 has no finite real solution.

Is the nonlinear equation page a second engine?

No. /calc/math/nonlinear-equation is the same math.numerical.root engine. All n polynomial roots stay at /polynomial-roots.

Why not three pages for Newton, bisection, and secant?

They are methods of one model. Switching method does not change the capability.