Golden Section Search Calculator

What does this golden section search calculator compute?

Golden-section search shrinks [a, b] with ratio φ̂=(√5−1)/2 until the remaining interval is within tol.

The published formula or method is golden-section search, φ̂=(√5−1)/2. 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 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 v1.0.5 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.

What do convergence and precision mean?

Use finite decimal numbers or decimal text. Method/function/bound/coefficient aliases must agree. Explicit empty fields and decimal conversion underflow are rejected. Coefficients accept2–9 constant-first entries as text or an array; degree at most8, constants use c0,0. Default tol=1e-8 and max_iter=100 apply only when absent; tol must be positive and at most1, max_iter an integer1–100.

Golden-section and bounded Brent assume a unimodal interval. Multimodal inputs may return a local candidate; no general global-minimum guarantee is made. The original closed-interval endpoints are compared with the search candidate. converged means the internal search bracket width met the requested absolute tol; it is not a proof of global accuracy. termination distinguishes tolerance, max_iter, and resolution_limit. The reported bracket is the internal search state; a selected original endpoint may lie outside that smaller bracket.

Exponential ordering uses monotonicity and sine comparisons use a stable difference identity to reduce rounded-value plateaus. Polynomial ordering uses exact dyadic intermediates of binary64 coefficients and trial points, so adding a large constant cannot erase the ordering of small polynomial variations. Coordinates retain binary64 precision to stay inside [a,b]; objective values and signed reference differences are published to12 significant digits. Required endpoint evaluations and final values outside nonzero/finite binary64 range are rejected. Sine/exponential values and the optional analytic baseline use binary64 library arithmetic; the legacy exact_xmin/exact_fmin fields name that baseline, not an exact certificate. Tied minima can have different valid locations. Plot samples are illustrative binary64 values and may be unavailable at extreme scales.

Calculations run locally in this page. Shared input URLs may appear in server or third-party logs; do not put sensitive data into a shared link. REST/MCP are remote services.