Gauss-Chebyshev (First Kind) Nodes and Weights

Enter the order n to list the Gauss-Chebyshev (first kind) quadrature nodes x_i = cos((2i−1)π/2n) and weights w_i = π/n.

Input

Enter the order n to display the first-kind Gauss-Chebyshev quadrature nodes x_i = cos((2i−1)π/(2n)) and weights w_i = π/n.

Order n

Number of nodes (1–512)

Weight w_i = π/n

0.6283185307

The weight is identical for every node.

Order n

Number of nodes

Nodes and weights

Nodes x_i and weights w_i on the standard interval [−1, 1].

First-kind Gauss-Chebyshev quadrature approximates integrals with the weight function 1/√(1−x²) on [−1, 1], i.e. ∫ f(x)/√(1−x²) dx.
For order n the nodes are x_i = cos((2i−1)π/(2n)) for i = 1, 2, …, n, which coincide with the zeros of the first-kind Chebyshev polynomial T_n(x).
All weights are equal: w_i = π/n. This uniform weight is a distinctive feature of the first-kind Gauss-Chebyshev rule.
The quadrature reads ∫_{−1}^{1} f(x)/√(1−x²) dx ≈ (π/n) Σ f(x_i) and is exact for polynomials of degree up to 2n−1.
The nodes lie strictly inside the interval and never include the endpoints ±1; higher orders cluster nodes near the endpoints.

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