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