## [Runge Phenomenon](/runge-phenomenon-0) Definition The tendency for high-degree polynomial interpolants at equispaced nodes to exhibit large oscillations near the interval endpoints, producing poor uniform approximation despite convergence at many interior points. ## Principle Principle Polynomial interpolation with equidistant nodes amplifies conditioning errors and the influence of high-degree basis polynomials near boundaries, causing nonuniform convergence. ## Demonstration Demonstration Interpolating the function f(x)=1/(1+25x^2) on [-1,1] with equally spaced nodes and increasing polynomial degree yields growing oscillations at the ends, whereas using Chebyshev nodes mitigates the effect. ## Misapplication Misapplication Assuming that increasing polynomial degree with uniformly spaced samples always improves uniform approximation leads to degraded results and numerical instability. ## Consequence Consequence Drives the adoption of nonuniform node distributions, piecewise interpolation (splines), or spectral methods with orthogonal polynomials to obtain stable uniform approximations. ## Reversal Reversal Replacing global high-degree polynomials on equispaced nodes by piecewise low-degree polynomials or by global expansions at Chebyshev-type nodes reverses the oscillatory behavior. ## Boundary Boundary Pertains specifically to polynomial interpolation on compact intervals with equispaced nodes; does not preclude good approximation by other bases or by polynomials with adapted node sets. ## Semantic Tension Semantic Tension Contrasts with spectral convergence claims for orthogonal polynomial expansions where appropriate node selection yields rapid uniform convergence for smooth functions. ## Synthesis Synthesis An interpolation instability phenomenon showing that naive high-degree polynomial fitting on equispaced points can worsen approximation near boundaries, motivating better node choices or local methods.