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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.

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 · 3/3 declared partitions (golden, brent, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
xmin and fmin of poly, sin, or exp on [a, b] by Brent’s 1-D minimizer, with an engine-linked figure of f and xmin.
Scope
The Result Card includes an engine-linked figure of f on [a, b] (marker is published xmin, fmin; poly coeffs from inputs, not a REST field). Same fnValue as REST/MCP.
Verification
Engine tested · Specification checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published methods· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-22.numerical-cvp
Verification revision
2026-09-22.numerical-cvp · 1/1 property · digest 0995f329cc5e
Legacy regression
11/11 tests · Production surface contract 4/4
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 1/1 (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
3/3 golden · 2/2 CVP boundary · 4/4 invalid · 1/1 property · 3/3 cross-interface · 1/1 cross-calculator · 4/4 CVP contract · Manifest
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
3 legacy golden · 2 legacy boundary · legacy regression suite · 3/3 oracle-backed golden · 4/4 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.0 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.

Parabolafit through (v,f(v)), (w,f(w)), (x,f(x))
Fallbackgolden section when the parabola is not acceptable
iMinimize only. This is not Dekker–Brent for f(x)=0. Polynomial coefficients are constant-first. Default tol = 10⁻⁸.

How to use

1

Enter integrand and [a, b]

Poly needs coeffs. a must be strictly less than b. Golden section lives on the sibling discovery URL.

2

Read xmin and fmin

An exact minimizer is shown when f is poly of degree ≤ 3, sin, or exp. The figure shows f on [a, b] and xmin.

Example calculations

Common configurations with formula and result.

ϟ

min (x−1)² on [0, 3]

coeffs 1,−2,1

xmin = 1
1
ϟ

min sin on [π, 2π]

interior min

3π/2
4.712…

Brent Method calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Brent’s 1-D minimizer combines golden-section search with inverse parabolic interpolation on a bracket [a, b]. This page is method=brent on math.numerical.optimize_1d. Not Brent root-finding. Not a second optimizer.
What it calculates
xmin and fmin of poly, sin, or exp on [a, b] by Brent’s 1-D minimizer, with an engine-linked figure of f and xmin.
Inputs
  • integrand
  • a
  • b
  • tol?
  • coeffs?
Outputs
  • xmin
  • fmin
  • exact_xmin
  • exact_fmin
  • n_eval
Formula
Brent (golden section + inverse parabolic interpolation)
Assumptions
  • The Result Card includes an engine-linked figure of f on [a, b] (marker is published xmin, fmin; poly coeffs from inputs, not a REST field). Same fnValue as REST/MCP.
  • Minimize only
  • Not root-finding Brent, not Nelder–Mead, not CAS
Units
  • dimensionless
Boundary conditions
  • a ≥ b → INVALID_INPUT
  • unknown method (including nelder) → INVALID_MODE
Example
brent poly 1,-2,1 on [0,3] → xmin=1
Validation cases

2 published on this page · 11/11 tests · Production surface contract 4/4 · View evidence

  • integrand=poly coeffs=1,-2,1 a=0 b=3 → exact_xmin=1
  • method=nelder integrand=sin a=0 b=1 → INVALID_MODE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.0.0

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

→ 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. › → 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 Brent’s method for roots?

No. Root-finding Brent is out of scope. This page minimizes a scalar on [a, b]. Numerical roots live on /calc/math/numerical-root.

Is this a second optimization engine?

No. It is method=brent on math.numerical.optimize_1d, the same engine as /calc/math/one-dimensional-optimization.