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Memory Cards — Chapter 3
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Lagrange basis
Lagrange polynomials give a transparent formula, while barycentric form gives stable practical evaluation.
Vandermonde warning
The Vandermonde formulation is mathematically direct but can be severely ill-conditioned.
Splines
Splines trade one high-degree global polynomial for piecewise low-degree polynomials with smooth joining.
Runge phenomenon
Equally spaced high-degree interpolation can oscillate near endpoints. Chebyshev nodes reduce the danger.
Interpolation problem
Interpolation constructs \(p\) so that \(p(x_i)=y_i\). Existence is not enough; stability and error matter.
Newton form
Divided differences allow incremental construction and efficient updates when new nodes are added.