Chebyshev Nodes

Numerical Analysis · Axiom Academy

Optimal node placement for polynomial interpolation We saw that equally spaced nodes can cause the Runge phenomenon. What's the best way to place nodes? Recall the interpolation error: We can't control f⁽ⁿ⁺¹⁾(ξ), but we can control where we place the nodes x₀, x₁, ..., xₙ! Find nodes x₀, x₁, ..., xₙ in [-1, 1] that minimize: The solution involves the remarkable Chebyshev polynomials Tₙ(x): Key property: |Tₙ(x)| ≤ 1 for x ∈ [-1, 1], and it achieves ±1 exactly n+1 times (oscillates!). The roots of Tₙ₊₁(x) give the optimal interpolation nodes: Chebyshev nodes are roots of Tₙ₊₁(x) The node polynomial ω(x) = 2⁻ⁿTₙ₊₁(x) max|ω(x)| = 2⁻ⁿ — the smallest possible for monic degree-(n+1) polynomials! Among all monic polynomials of degree n+1, the polynomial Tₙ₊₁(x)/2ⁿ has the smallest maximum absolute value on [-1, 1]. This means Chebyshev nodes minimize the worst-case error bound! Nodes cluster near endpoints (denser near ±1) For interval [a, b], transform: xₖ → (a+b)/2 + (b-a)/2 · xₖ Eliminates Runge phenomenon for smooth functions Used in spectral methods and numerical integration Interpolation of 1/(1+25x²): Equal spacing (red) vs Chebyshev nodes (green)

This is the written version of the interactive lesson above. See the full Numerical Analysis course.