John Boyd, Department of Atmospheric, Oceanic and Space Science, University of Michigan
Exponentially accurate Rung-free approximation from samples on an evenly-spaced grid for non-periodic functions
Approximating a function from its values at a f(xi) set of evenly spaced points xi through (N+1)-point polynomial interpolation often fails because of divergence near the endpoints, the ‘runge Phenomenon”. The present study shows how to achieve an error that decreases exponentially fast with N. Normalizing the span of the points to [-1, 1], the new strategy applies a filtered trigonometric interpolant on the subinterval [-1+D, 1-D] and ordinary polynomial interpolation in these two remaining subintervals. Convergence is guaranteed because the width D of the polynomial interpolation intervals decreases as N→∞, being proportional to 1/N. Applications to Gibbs Phenomenon and hydrodynamic shocks are discussed.
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Name: Ian Cunningham