Exponentially accurate Runge-free approximation of non-periodic functions from samples on an evenly spaced grid
✍ Scribed by John P. Boyd
- Book ID
- 104000309
- Publisher
- Elsevier Science
- Year
- 2007
- Tongue
- English
- Weight
- 477 KB
- Volume
- 20
- Category
- Article
- ISSN
- 0893-9659
No coin nor oath required. For personal study only.
✦ Synopsis
Approximating a function from its values f (x i ) at a set of evenly spaced points x i through (N + 1)-point polynomial interpolation often fails because of divergence near the endpoints, the "Runge Phenomenon". This report 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 x ∈ [-1 + D, 1 -D] and ordinary polynomial interpolation in the two remaining subintervals. Convergence is guaranteed because the width D of the polynomial interpolation subintervals decreases as N → ∞, being proportional to 1/ √ N . Applications to the Gibbs Phenomenon and hydrodynamic shocks are discussed.