Gauss-Lobatto Quadrature Calculator

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

Input

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

Integrand f(x)

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

Lower limit a

Upper limit b

Order n

Number of nodes (2 to 256)

Integral ∫ f(x) dx

0.7468241328

Interval [0, 1]

Order n

Nodes

Gauss-Lobatto quadrature includes both endpoints a and b as nodes.

Nodes, weights, and function values

Shows the nodes x_i, weights w_i, and function values f(x_i) mapped onto [a, b]. The first and last rows are endpoints.

#	Type	Node x_i	Weight w_i	f(x_i)
1	Endpoint	0	0.01785714	1
2	Interior	0.06412993	0.10535211	0.9958958
3	Interior	0.20414991	0.17056135	0.95917937
4	Interior	0.39535039	0.2062294	0.85530091
5	Interior	0.60464961	0.2062294	0.69377946
6	Interior	0.79585009	0.17056135	0.53079608
7	Interior	0.93587007	0.10535211	0.41650667
8	Endpoint	1	0.01785714	0.36787944

Gauss-Lobatto quadrature approximates the definite integral of f(x) while always including both endpoints a and b as nodes. Because it samples the function at the interval ends, it captures boundary behavior well.
The rule works on the standard interval from −1 to 1. The two ends ±1 are fixed nodes, and the interior nodes are the zeros of the derivative of the Legendre polynomial P(n-1).
Weights use the formula w ＝ 2 ÷ (n × (n-1) × P(n-1)(x)²). At the endpoints P(n-1)(±1) ＝ ±1, so w ＝ 2 ÷ (n × (n-1)).
An order-n rule uses n nodes (2 endpoints plus n-2 interior) and integrates polynomials up to degree 2n-3 exactly.
The standard nodes and weights are linearly mapped onto the interval a to b, and the integral is the weighted sum of the function values. The table lists every node, weight, and function value.
Enter the order n as an integer from 2 to 256. Provide finite and distinct values for a and b.

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