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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 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 · 6/6 declared partitions (golden, brent, small-scale-quadratic, scaled-cubic-reference, numerical-range, 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 golden-section search, 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 · v1.0.3 · CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Core assurance · all published methods· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.3 · CVP protocol 1.0.0-proposed · Evidence 2026-10-07.optimization-range-guards
Verification revision
2026-10-07.optimization-range-guards · 1/1 property · digest 422e879f92c3
Legacy regression
11/11 tests · Production surface contract 4/4
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 3/3 (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
15/15 golden · 2/2 CVP boundary · 14/14 invalid · 1/1 property · 7/7 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 · 15/15 oracle-backed golden · 14/14 invalid · Artifact integrity PASS
STALE · Build schema 1.0.3 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.

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. The figure shows f on [a, b] and xmin.

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 1.0.3 · 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, 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
golden-section search, φ̂=(√5−1)/2
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 Brent / Nelder–Mead / CAS
Units
  • dimensionless
Boundary conditions
  • Nonfinite interval width, overflowing evaluations/results, exp underflow to zero or unsafe sin phase index → NUMERICAL_RANGE_ERROR
  • 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 4/4 · 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
1.0.3

Other calculators in this family: Brent Method Calculator, Definite Integral Calculator, Euler Method Calculator, Gauss Quadrature Calculator, Nonlinear Equation 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. › → 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 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.