Gauss Quadrature Calculator

What does this gauss quadrature calculator compute?

n-point Gauss–Legendre quadrature approximates ∫_a^b f(x) dx from nodes on [-1, 1] mapped to [a, b]. Exact for polynomials of degree ≤ 2n−1.

The published formula or method is Gauss–Legendre n-point. The following references support the stated mathematics or method:

• NIST DLMF §3.5 — Quadrature: Trapezoidal, Simpson, and Gaussian quadrature formulas and their assumptions.

Which inputs and limits should you check?

Choose a method suitable for the integrand and interval. Refinement diagnostics are not a universal error bound for discontinuous or otherwise unsupported functions.

The worked example uses 2-point is exact with 1/2. 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.definite_integral v1.1.5 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.

Input precision, complete results, and privacy

Inputs are finite decimal numbers or decimal text; a nonzero decimal below binary64 range is rejected, never silently changed to zero. Polynomial input is 2–9 constant-first coefficients; a constant uses c0,0. Method/function aliases must agree. n is an integer: 2–10000 panels for Simpson/trapezoid or 2–8 Gauss nodes. An odd Simpson count advances to the next even count, which is shown in the result. Missing n uses 100 panels or 5 Gauss nodes; explicitly empty inputs remain errors.

The output includes the chosen method, function, full polynomial coefficients, bounds, effective count, approximation, binary64 analytic baseline, signed raw difference, and formula. The historical field abs_err means approximation minus baseline, before 12 significant digit publication; it is not an absolute error or certified bound. This is a fixed rule, not adaptive integration. Sample range failures, unresolved nonzero underflow, and severe cancellation are rejected; split the interval or rescale the function. Figures may be omitted at extreme scales while numeric results remain available.

Normal form calculations run locally without an account. Copy JSON and CSV retain complete inputs, results, and calculation version. A shared query URL sends its parameters to the server when opened and may appear in server or third-party logs. Function and Numerical Workspace links use a fragment and recompute the inputs; stored result claims are ignored. Version notices identify current computation, and machine version pins reject unavailable versions.

Sine bounds are limited to magnitude 1e6 (NUMERICAL_RANGE_ERROR beyond it) because a binary64 fixed grid loses phase resolution for very large arguments. Binary64 node rounding and cancellation near zeros remain precision limitations; the 12 digit publication format is not a uniform accuracy guarantee. A true zero from the selected quadrature rule is allowed even when the analytic baseline is nonzero.