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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.

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; seeds/bracket 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))
iSame engine as /numerical-root. Not ⁿ√a. Not all-roots of a polynomial (/polynomial-roots). Not symbolic solve. Not a system. Degree ≤ 8 for poly.

How to use

1

Choose f and a method

Poly needs coeffs (constant first). sin means f(x)=sin x. Newton needs x0; bisection needs a sign-change bracket. eˣ=0 has no finite real root.

2

Read one real root

All n polynomial roots stay at /polynomial-roots. The main value is a rounded display approximation; details and REST retain the full binary64 root. For polynomials, signed and scaled residuals are distinct. Iterations count updates after the initial guess (iteration 0), or bisection midpoint evaluations.

Example calculations

Common configurations with formula and result.

ϟ

sin x = 0

Newton from 3

x ← x − sin x / cos x
π
ϟ

√2

x² − 2 = 0

coeffs −2,0,1
1.414213…

Nonlinear Equation 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 page uses math.numerical.root for a polynomial or sin x. 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
  • x0?
  • x1?
  • a?
  • b?
  • coeffs?
  • tol?
  • max_iter?
Outputs
  • root
  • residual
  • scaled_residual?
  • iterations
  • tol
  • stop_reason
Formula
x ← x − f(x)/f′(x) or a sign-change bracket
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; seeds/bracket from inputs, not REST fields). Same fnValue as REST/MCP. Iterate history is not a REST field.
  • Not symbolic CAS solve
  • Not a system of equations
  • Not all-roots of a polynomial
  • Not ⁿ√a
  • Polynomial degree ≤ 8
  • eˣ=0 has no finite real root
  • Polynomial evaluation and residual stopping use exact dyadic intermediates for binary64 inputs; power-of-two solver scaling and binary64 residual output prevent cancellation from creating a false 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 rejects same-sign endpoints even when finite function values are tiny → INVALID_INPUT
  • all-zero polynomial coefficients or an underflowed endpoint sign → INVALID_INPUT
  • unknown method (including cas, halley, multivariable) → INVALID_MODE
  • no convergence → NO_CONVERGENCE
  • eˣ=0 has no finite real root → NO_CONVERGENCE
Example
newton integrand=sin x0=3 → π
Validation cases

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

  • method=newton integrand=sin x0=3 → root≈3.14159265359
  • method=cas integrand=sin x0=1 → INVALID_MODE
Sources
Calculation version
1.1.6

Background

Interpretation and common distinctions.

What does this nonlinear equation calculator compute?

A real root of f(x)=0 is a number r with f(r)=0. This page uses math.numerical.root for a polynomial or sin x. eˣ has no finite real zero.

The published formula or method is x ← x − f(x)/f′(x) or a sign-change bracket. 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 Newton from 3 with x ← x − sin x / cos x. 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.

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, Numerical Derivative Calculator, Numerical Interpolation Calculator, Numerical Root 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. › → 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. › → 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. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a second root engine?

No. It is math.numerical.root, the same engine as /calc/math/numerical-root. Newton, bisection, and secant stay methods of that capability.

Is this all roots of a polynomial?

No. /calc/math/polynomial-roots returns every root, including complex conjugates. This page approximates one real zero of a polynomial or sin x.

Do you ship a CAS solver or a nonlinear system?

No. method=cas, symbolic, and multivariable are INVALID_MODE. Scalar f(x)=0 only.