ChebyshevGaussLobatto

ChebyshevGaussLobatto refers to the Chebyshev-Gauss-Lobatto nodes, a specific set of points in the interval [-1, 1] given by x_k = cos(kπ/n) for k = 0, 1, ..., n. These endpoints x_0 = cos(0) = 1 and x_n = cos(π) = -1 are included, making the grid suitable for boundary value problems and spectral methods.

The nodes arise from the theory of Chebyshev polynomials and Gauss-Lobatto quadrature. They are the extremal points of the Chebyshev polynomial T_n, satisfy T_n(x_k) = cos(kπ) = (-1)^k, and are used in a quadrature rule for the weight function w(x) = 1/√(1 - x^2). The corresponding Gauss-Lobatto quadrature approximates integrals of the form ∫_{-1}^{1} f(x)/√(1 - x^2) dx by a weighted sum at the nodes: ∑_{k=0}^{n} w_k f(x_k), with weights w_0 = w_n = π/(2n) and w_k = π/n for k = 1, ..., n-1.

In numerical analysis, these nodes are widely used for Chebyshev interpolation and spectral collocation methods. They enable efficient, highly accurate polynomial approximations of smooth functions via Chebyshev polynomials, often implemented with barycentric interpolation formulas and collocation differentiation matrices. The clustering of points near the endpoints yields excellent resolution for boundary layers but can affect conditioning for high-degree systems.

Applications include solving ordinary and partial differential equations with spectral methods, fluid dynamics simulations, and other problems where high-order accuracy is advantageous. The approach emphasizes global polynomial representations and spectral convergence, while practitioners consider endpoint effects and numerical stability in implementation.