HomeCalculatorsMathNumerical CalculusOne-Dimensional Optimization Calculator
Math calculator

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.

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 or Brent.
Scope
Minimize only; f treated as unimodal on [a, b]
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Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (1 capability; 163 remain CURRENT)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.

Goldenc = b − φ̂(b−a), d = a + φ̂(b−a)
Brentgolden section + inverse parabolic interpolation
iφ̂ = (√5−1)/2. Minimize only. Polynomial coefficients are constant-first. Degree ≤ 8. Default tol = 10⁻⁸. Discovery /golden-section and /brent-method lock the method.

How to use

1

Choose method and f

Poly needs coeffs. sin and exp are f(x)=sin x and eˣ. Brent usually needs fewer evaluations.

2

Enter [a, b]

a must be strictly less than b. The 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, golden or Brent

xmin = 0
0
ϟ

min sin on [π, 2π]

Brent

3π/2
4.712…

One-Dimensional Optimization calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Golden-section search and Brent’s method shrink a bracket [a, b] to a minimizer of a unimodal scalar f. Polynomial, sin, and exp have a closed-form check on this seed. Not multivariate, not maximization, not a CAS.
What it calculates
xmin and fmin of poly, sin, or exp on [a, b] by golden section or Brent.
Inputs
  • method
  • integrand
  • a
  • b
  • tol?
  • coeffs?
Outputs
  • xmin
  • fmin
  • exact_xmin
  • exact_fmin
  • n_eval
Formula
golden-section search / Brent
Assumptions
  • Minimize only; f treated as unimodal on [a, b]
  • Not multivariate and not CAS
  • Polynomial degree ≤ 8
Units
  • dimensionless
Boundary conditions
  • a ≥ b → INVALID_INPUT
  • tol ≤ 0 → VALUE_MUST_BE_POSITIVE
  • 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

  • method=golden 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, Golden Section Search Calculator, Numerical Derivative Calculator, Numerical Interpolation Calculator, Numerical Root 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. › → 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. › → 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. › → 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. ›
}

Frequently asked questions

Key distinctions behind the calculation.

Does this maximize, or optimize in several variables?

No. This seed minimizes a scalar poly, sin, or exp on a closed interval. Nelder–Mead, gradients, and maximization are INVALID_MODE.

Where does this run?

Locally in the browser by default. REST and MCP call the same 1-D optimization engine. /golden-section and /brent-method are discovery URLs, not a second engine.