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Nonlinear Equation Calculator

Find one approximate real zero of a polynomial or sin x with Newton, bisection, or secant. Uses the Numerical Root engine. Runs 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 is pack.root on y=0; seeds/bracket from inputs, not REST fields). Same fnValue as REST/MCP. Iterate history is not a REST field.
Verification
Engine tested · Source checked · v1.1.4 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published methods· View Manifest · CVP overview · Specification
Versions
Calculation 1.1.4 · CVP protocol 1.0.0-proposed · Evidence 2026-10-08.mixed-clustered-root
Verification revision
2026-10-08.mixed-clustered-root · 2/2 property · digest 9da1bdecdbb4
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 7/7 (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.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
107/107 golden · 6/6 CVP boundary · 13/13 invalid · 2/2 property · 12/12 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 · 13/13 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.1.4 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 1.1.4 · 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?
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 is pack.root on y=0; 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
  • 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
1.1.4

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 v1.1.4 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.

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 an approximate real zero of a polynomial or sin x with Newton, bisection, or secant. Enter a starting point or bracket and inspect the stopping criterion. Runs 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.