Gauss-Chebyshev (Second Kind) Nodes & Weights Calculator

Enter the order n to list the nodes x_i=cos(iπ/(n+1)) and weights w_i=π/(n+1)·sin²(iπ/(n+1)) of the second-kind Gauss-Chebyshev quadrature in a table.

Input

Compute the nodes and weights of second-kind Gauss-Chebyshev quadrature (weight function √(1−x²)) from the order n.

Order n

An integer from 1 to 256. This is the number of nodes.

Result

Order n

xᵢ = cos(iπ/(n+1)), wᵢ = π/(n+1)·sin²(iπ/(n+1))

Number of nodes

Sum of weights

1.5707963268

Exact value (π/2)

1.5707963268

Nodes and weights

Nodes x are sorted in ascending order. The angle θ = iπ/(n+1) is in radians.

#	Node xᵢ	Weight wᵢ	Angle θᵢ
1	-0.9396926208	0.0408329477	2.7925268032
2	-0.7660444431	0.1442256008	2.4434609528
3	-0.5	0.2617993878	2.0943951024
4	-0.1736481777	0.3385402271	1.745329252
5	0.1736481777	0.3385402271	1.3962634016
6	0.5	0.2617993878	1.0471975512
7	0.7660444431	0.1442256008	0.6981317008
8	0.9396926208	0.0408329477	0.3490658504

How it works

Second-kind Gauss-Chebyshev quadrature uses the weight function w(x)=√(1−x²) on [−1,1], giving ∫₋₁¹√(1−x²)f(x)dx ≈ Σ wᵢ f(xᵢ).
The nodes are the zeros of the second-kind Chebyshev polynomial Uₙ(x), given in closed form by xᵢ=cos(iπ/(n+1)) for i=1,…,n.
The weights are wᵢ=π/(n+1)·sin²(iπ/(n+1)), which are all positive.
The sum of the weights equals π/2 exactly.
Because the nodes and weights are available in closed form, this tool evaluates them directly without any iteration.

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