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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.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 11/11 tests · Production surface contract 2/2 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
xmin and fmin of poly, sin, or exp on [a, b] by golden-section search.
Scope
Minimize only
Verification
Engine tested · 11/11 tests · Production surface contract 2/2 · Specification checked · v1.0.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
3 golden · 2 boundary · 6 property · Production surface contract 2/2 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (1 capability; 163 remain CURRENT) @ 2026-09-19T00:00:17.039Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

Interior pointsc = b − φ̂(b−a), d = a + φ̂(b−a)
Conjugateφ̂ = (√5−1)/2
iMinimize only. Polynomial coefficients are constant-first. Brent lives on the sibling discovery URL. Default tol = 10⁻⁸.

How to use

1

Enter integrand and [a, b]

Poly needs coeffs. a must be strictly less than b.

2

Read xmin and fmin

An exact minimizer is shown when f is poly of degree ≤ 3, sin, or exp.

Example calculations

Common configurations with formula and result.

ϟ

min x² on [−1, 2]

coeffs 0,0,1

xmin = 0
0
ϟ

min eˣ on [0, 1]

monotonic, left endpoint

xmin = 0
0

Golden Section Search calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Golden-section search shrinks [a, b] with ratio φ̂=(√5−1)/2 until the remaining interval is within tol. This page is method=golden on math.numerical.optimize_1d. Not a second optimizer. Not Brent.
What it calculates
xmin and fmin of poly, sin, or exp on [a, b] by golden-section search.
Inputs
  • integrand
  • a
  • b
  • tol?
  • coeffs?
Outputs
  • xmin
  • fmin
  • exact_xmin
  • exact_fmin
  • n_eval
Formula
golden-section search, φ̂=(√5−1)/2
Assumptions
  • Minimize only
  • Not Brent / Nelder–Mead / CAS
Units
  • dimensionless
Boundary conditions
  • a ≥ b → INVALID_INPUT
  • unknown method (including nelder) → INVALID_MODE
Example
golden poly 0,0,1 on [-1,2] → xmin=0
Validation cases

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

  • integrand=poly coeffs=0,0,1 a=-1 b=2 → exact_xmin=0
  • 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: Brent Method Calculator, Definite Integral Calculator, Gauss Quadrature Calculator, Numerical Derivative Calculator, Numerical Interpolation Calculator, Numerical Root Calculator, One-Dimensional Optimization Calculator, Simpson Rule 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. › → 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. › → Numerical Derivative Calculator Finite-difference derivative of a polynomial, sin, or exp. Central, forward, or backward. 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 a real root of a polynomial with Newton, bisection, or secant. One capability, three methods. Not the nth-root page. Runs locally. › → One-Dimensional Optimization Calculator Minimize a polynomial, sin, or exp on [a, b] by golden-section search or Brent. Not multivariate, not max, not CAS. Runs locally. › → Simpson Rule Calculator Composite Simpson 1/3 of a polynomial, sin, or exp on [a, b]. Same engine as Definite Integral. Not indefinite CAS. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this a second optimization engine?

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

Is this Brent or Nelder–Mead?

No. Brent is the sibling discovery URL. Nelder–Mead, gradients, and maximization stay INVALID_MODE.