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- W140333869 abstract "The computation of discrete polynomial transforms is a fundamental operation in the applied mathematical sciences. Much effort has been placed on the development of fast polynomial transforms, particularly those based on the fast Fourier transform. Two exact fast transform models of particular importance are the polynomial division tree model and the three-term recurrence rule model.Recent work has introduced some new fast polynomial transforms based on the two models. These algorithms have complexity at most $O(N logsp2 N)$, as compared to the $O(Nsp2$) direct methods; however, in some cases they display numerical instability on reasonable inputs. This thesis presents some heuristics which greatly improve the numerical reliability without sacrificing efficiency in either the asymptotic or execution time sense.For the fast transforms based on polynomial division, a heuristic technique called symmetry stabilization greatly improves the numerical reliability for the case where the sample points are on the unit circle. This technique improves the numerical reliability for three sample point distributions which arise in important applications with little or no increase in the complexity. New fast transforms based on the polynomial division model are described for generalized Chebyshev polynomials, which may also benefit from the symmetry stabilization technique. This new transform unifies the fast Fourier transform and the fast cosine transform as a single algorithm parameterized by a variable $rho in$ (0, 1).For the fast transforms based on three-term recurrence relations, a modified algorithm is described which uses stability bypass operations to improve the numerical reliability. This method has been tested extensively for transforms onto sets of associated Legendre functions, or fast Legendre transforms. The fast Legendre transform is used to effect a fast spherical harmonic transform on the 2-sphere, thereby giving an $O(N logsp2 N$) algorithm for convolving two functions on the sphere. Some efficient variations of the algorithm are described which are based on a semi-naive approach. Finally, a leveled hypercube parallel algorithm for the fast Legendre transform is described which is work optimal, and when used in conjunction with well-known parallel FFT algorithms, effects a work-optimal parallel algorithm for fast spherical harmonic transforms." @default.
- W140333869 created "2016-06-24" @default.
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- W140333869 date "1995-11-20" @default.
- W140333869 modified "2023-09-26" @default.
- W140333869 title "Efficient stabilization methods for fast polynomial tranforms" @default.
- W140333869 hasPublicationYear "1995" @default.
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