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Math calculator

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.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · production STALE · 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 · 4/4 declared partitions (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; 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 · v1.1.3 · CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Core assurance · all published methods· View Manifest · CVP overview · Specification
Versions
Calculation 1.1.3 · CVP protocol 1.0.0-proposed · Evidence 2026-10-06.polynomial-value-underflow
Verification revision
2026-10-06.polynomial-value-underflow · 1/1 property · digest 9e6c673d4478
Legacy regression
19/19 tests · Production surface contract 5/5
Trust layers
Verification VERIFIED · Production STALE · Overall CVP state: VERIFIED_STALE under CVP 1.0.0-proposed
CVP status
STALE · Capability production binding: STALE · Site report: STALE · Evidence changed after the last successful production attestation. Re-attestation required.
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 4/4 (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
7/7 golden · 6/6 CVP boundary · 13/13 invalid · 1/1 property · 5/5 cross-interface · 1/1 cross-calculator · 5/5 CVP contract · Manifest
Sources
Sources
Evidence
5 legacy golden · 7 legacy boundary · legacy regression suite · 7/7 oracle-backed golden · 13/13 invalid · Artifact integrity PASS
STALE · Build schema 1.1.3 matches page · production binding STALE · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · CVP STALE · attestation STALE — Production CURRENT withheld · 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 1.1.3 · 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?
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 is pack.root on y=0; 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 solving normalizes finite coefficients before evaluating signs, steps, and scaled residuals. The published signed residual remains the raw binary64 evaluation. A zero scale at x=0 with c₀=0 is an exact root; an underflowed zero elsewhere is not accepted as an exact root.
Units
  • dimensionless
Boundary conditions
  • 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 · 19/19 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
1.1.3

Background

Interpretation and common distinctions.

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 raw binary64 evaluation, while the scaled residual is used for the polynomial stopping test. 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 1.1.3 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.

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 Find one approximate real zero of a polynomial or sin x with Newton, bisection, or secant. Uses the Numerical Root engine. 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. ›

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.