A Comparison of Numerical Algorithms for Fourier Extension of the First, Second, and Third Kinds
✍ Scribed by John P. Boyd
- Publisher
- Elsevier Science
- Year
- 2002
- Tongue
- English
- Weight
- 488 KB
- Volume
- 178
- Category
- Article
- ISSN
- 0021-9991
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✦ Synopsis
The range of Fourier methods can be significantly increased by extending a nonperiodic function f (x) to a periodic function f on a larger interval. When f (x) is analytically known on the extended interval, the extension is straightforward. When f (x) is unknown outside the physical interval, there is no standard recipe. Worse still, like a radarless aircraft groping through fog, the algorithm may wreck on the "mountain-in-fog" problem: a function f (x) which is perfectly well behaved on the physical interval may very well have singularities in the extended domain. In this article, we compare several algorithms for successfully extending a function f (x) into the "fog" even when the analytic extension is singular. The best third-kind extension requires singular value decomposition with iterative refinement but achieves accuracy close to machine precision.