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Nonlinear Equation Calculator

One real root of a polynomial, sin, or exp by Newton, bisection, or secant. Same engine as Numerical Root. Not a second root engine. Not CAS. Not a system. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 13/13 tests · Production surface contract 1/1 · v1.1.0
Input interpretation
Enter values to calculate.
Result
—
Model
One real root of poly, sin, or exp by Newton, bisection, or secant.
Scope
Not symbolic CAS solve
Verification
Engine tested · 13/13 tests · Production surface contract 1/1 · Specification checked · v1.1.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
3 golden · 3 boundary · 7 property · Production surface contract 1/1 · Artifact integrity PASS
Production
Embedded snapshot: STALE · Last attested schema matched 1.1.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status STALE (2 capabilities; 162 remain CURRENT) @ 2026-09-20T09:01:12.147Z
Semantic contract
PASS

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 and exp are f(x)=sin x and eˣ. Newton needs x0; bisection needs a sign-change bracket.

2

Read one real root

All n polynomial roots stay at /polynomial-roots. Residual and iteration count are in the result.

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 1.1.0 · 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 is math.numerical.root. Polynomial, sin, and exp. Newton, bisection, and secant are methods — not three product pages. Not a second engine. Not CAS. Not a linear system.
What it calculates
One real root of poly, sin, or exp by Newton, bisection, or secant.
Inputs
  • method
  • integrand
  • x0?
  • x1?
  • a?
  • b?
  • coeffs?
Outputs
  • root
  • residual
  • iterations
Formula
x ← x − f(x)/f′(x) or a sign-change bracket
Assumptions
  • Not symbolic CAS solve
  • Not a system of equations
  • Not all-roots of a polynomial
  • Not ⁿ√a
  • Polynomial degree ≤ 8
Units
  • dimensionless
Boundary conditions
  • no sign change → INVALID_INPUT
  • unknown method (including cas, halley, multivariable) → INVALID_MODE
  • no convergence → NO_CONVERGENCE
Example
newton integrand=sin x0=3 → π
Validation cases

2 published on this page · 13/13 tests · Production surface contract 1/1 · View evidence

  • method=newton integrand=sin x0=3 → root≈3.14159265359
  • method=cas integrand=sin x0=1 → INVALID_MODE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.1.0

Other calculators in this family: Bessel I Calculator, Bessel J Calculator, Bessel K Calculator, Bessel Y Calculator, Beta Function Calculator, Brent Method Calculator, Definite Integral Calculator, DFT Calculator . Explore all Numerical Calculus.

→ Bessel I Calculator I_n(x) for integer n=0…8. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Bessel J Calculator J_n(x) for integer n=0…8. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Bessel K Calculator K_n(x) for integer n=0…8 and x>0. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Bessel Y Calculator Y_n(x) for integer n=0…8 and x>0. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Beta Function Calculator Complete Euler B(x, y) = Γ(x)Γ(y)/Γ(x+y) for x>0, y>0. Same engine as Special Functions. Not incomplete beta. Not CAS. Runs locally. › → 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. › → DFT Calculator Forward DFT X[k]=Σ x[n] e^{−2πi kn/N}, N≤64, unnormalized. Same engine as Discrete Fourier Transform. Radix-2 FFT is on /fft. Not inverse, not Laplace. Runs locally. ›
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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 returns one real root of poly, sin, or exp.

Do you ship a CAS solver or a nonlinear system?

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