Gauss-Legendre Quadrature (Numerical Integration) Calculator

Enter f(x), the interval [a, b] and the order n to evaluate the definite integral ∫f(x)dx with high accuracy using Gauss-Legendre quadrature. Shows the nodes, weights and function values.

Input

Enter the integrand f(x), the interval [a, b] and the order n to evaluate the definite integral with Gauss-Legendre quadrature.

Integrand f(x)

e.g. exp(-x^2), sin(x)/x, x^2 + 1. See the method section for supported functions, constants and operators.

Lower limit a

Upper limit b

Order n

Number of nodes (1-256)

Result

Integral ∫ f(x) dx

0.7468241328

Interval [0, 1]

Order n

Nodes

Nodes, weights and values

Shows the nodes x_i (mapped to [a, b]), weights w_i and function values f(x_i).

#	Node x_i	Weight w_i	f(x_i)
1	0.40828268	0.18134189	0.84645796
2	0.2372338	0.15685332	0.94527454
3	0.10166676	0.11119052	0.9897171
4	0.01985507	0.05061427	0.99960585
5	0.98014493	0.05061427	0.38263105
6	0.89833324	0.11119052	0.44619348
7	0.7627662	0.15685332	0.55888459
8	0.59171732	0.18134189	0.70459692

How it works

Gauss-Legendre quadrature approximates an integral on the reference interval [-1, 1] by sampling the integrand at the roots x_i of the degree-n Legendre polynomial P_n(x) and combining them with weights w_i = 2 / ((1 - x_i^2)(P_n'(x_i))^2), giving ∫_{-1}^{1} g(t)dt ≈ Σ w_i g(x_i).
An arbitrary interval [a, b] is handled with the linear change of variable x = (b-a)/2·t + (a+b)/2 and dx = (b-a)/2·dt, so ∫_a^b f(x)dx ≈ (b-a)/2·Σ w_i f((b-a)/2·x_i + (a+b)/2).
With n nodes the rule is exact (in exact arithmetic) for any polynomial integrand up to degree 2n-1, so smooth functions are integrated to very high accuracy even with relatively few nodes.
The nodes (roots of the Legendre polynomial) are found by evaluating the polynomial through the recurrence (k+1)P_{k+1}(x) = (2k+1)xP_k(x) - kP_{k-1}(x) and refining initial approximations with Newton's method, exploiting the symmetry of the roots.
The function expression is parsed by an independent expression parser (no eval). Supported operators are + - * / ^ (power) with parentheses, unary minus and implicit multiplication; constants pi and e; and functions sin cos tan asin acos atan sinh cosh tanh exp log ln log10 sqrt cbrt abs.
Computations use double-precision floating point and are therefore subject to rounding error. Accuracy can degrade when the integrand has singularities or rapid oscillations inside the interval.

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