AlgorithmAlgorithm: Literature-sourcedProof: Not applicablePipeline-generated

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Chebyshev–Gauss–Lobatto polynomial differentiation matrix

Definition

Input $N \geq 1$ and real nodal values $U_{0}, \dots, U_{N}$ at the descending grid $x_{j} = \cos (j π / N)$ from Chebyshev–Gauss–Lobatto collocation grid. Set $z_{k} = x_{N - k}$ and $Y_{k} = U_{N - k}$ , so $z_{0} < \dots < z_{N}$ . Form the cardinal polynomials $ℓ_{k}$ for the increasing nodes $z_{0}, \dots, z_{N}$ by Cardinal-polynomial interpolation. Define

(D_{N})_{i j} = ℓ_{N - j}^{'} (x_{i}), 0 \leq i, j \leq N,

and return the derivative vector $D_{N} U$ . Equivalently, form $p (x) = \sum_{k = 0}^{N} Y_{k} ℓ_{k} (x)$ and return $(p^{'} (x_{i}))_{i = 0}^{N}$ . The procedure is exact-real arithmetic and makes no stability claim for a floating-point implementation.

Depends on

Used by

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jie Shen and Tao Tang, Spectral and High-Order Methods with Applications, §2.1 (standard reference, not scraped)
Lloyd N. Trefethen, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, Ch. 8 (standard reference, not scraped)