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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 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 · 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 or Brent, 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 are read from inputs, not a REST field). Same fnValue as REST/MCP. Iterate history is not a REST field.
Verification
Engine tested · Source checked · v1.0.3 · CVP VERIFIED · 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 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 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
Sources
Sources
Evidence
3 legacy golden · 2 legacy boundary · legacy regression suite · 15/15 oracle-backed golden · 14/14 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.3 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.

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. 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, golden or Brent

xmin = 0
0
ϟ

min sin on [π, 2π]

Brent

3π/2
4.712…

One-Dimensional Optimization calculator specification

Version 1.0.3 · 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, with an engine-linked figure of f and xmin.
Inputs
  • method
  • integrand
  • a
  • b
  • tol?
  • coeffs?
Outputs
  • xmin
  • fmin
  • exact_xmin
  • exact_fmin
  • n_eval
Formula
golden-section search / Brent
Assumptions
  • The Result Card includes an engine-linked figure of f on [a, b] (marker is published xmin, fmin; poly coeffs are read from inputs, not a REST field). Same fnValue as REST/MCP. Iterate history is not a REST field.
  • Minimize only; f treated as unimodal on [a, b]
  • Not multivariate and not CAS
  • Polynomial degree ≤ 8
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
  • 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 4/4 · 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
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
    Supports: Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.
  • Brent, Algorithms for Minimization without Derivatives — golden section and bounded Brent
    Supports: Golden-section search with φ̂=(√5−1)/2; Brent golden + inverse parabolic interpolation
  • SciPy — minimize_scalar
    Supports: Golden-section and Brent-family scalar minimization methods, bounds, and local-minimum assumptions.
Calculation version
1.0.3

Background

Interpretation and common distinctions.

What does this one-dimensional optimization calculator compute?

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.

The published formula or method is golden-section search / Brent. The following references support the stated mathematics or method:

  • SciPy — minimize_scalar: Golden-section and Brent-family scalar minimization methods, bounds, and local-minimum assumptions.

Which inputs and limits should you check?

These are scalar numerical minimizers. A local search or interval search does not establish a global minimum for an arbitrary multimodal function.

The worked example uses coeffs 0,0,1, golden or Brent with xmin = 0. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.numerical.optimize_1d v1.0.3 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

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.

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.