Abstract
In this paper, we proposed a high accurate and stable Legendre transform algorithm, which can reduce the potential instability for a very high order at a very small increase in the computational time. The error analysis of interpolative decomposition for Legendre transform is presented. By employing block partitioning of the Legendre-Vandermonde matrix and butterfly algorithm, a new Legendre transform algorithm with computational complexity O(Nlog2N /loglogN) in theory and O(Nlog3N) in practical application is obtained. Numerical results are provided to demonstrate the efficiency and numerical stability of the new algorithm.
Highlights
Legendre transform (LT) plays an important part in many scientific applications, such as astrophysical [1], numerical weather prediction and climate models [2,3]
Motivated and inspired by the ongoing research in these areas, we present a theoretical method to analyze the error of LT using butterfly algorithm, and provide a numerically stability Legendre transform algorithm based on block partitioning and butterfly algorithm
Interpolative decomposition, CMAX is the number of columns in each sub-matrix on level 0)
Summary
Legendre transform (LT) plays an important part in many scientific applications, such as astrophysical [1], numerical weather prediction and climate models [2,3]. Jacobi transform by non-oscillatory phase functions shows an optimal computational complexity O(Nlog N/loglogN) in reference [24]. Legendre transform algorithm in ButterflyLab [25], which adopts interpolative butterfly factorization (IBF) [14,26] and non-oscillatory phase functions method to evaluate the Legendre polynomials [24], does not show high accuracy as Fourier transform using IBF. Fast Legendre transform (FLT) based on IBF and non-oscillatory phase functions and its extension to the associated Legendre functions need further study. Fast Legendre transform algorithm based on FFT deserved more attentions for its optimal computational complexity O(Nlog N/loglogN). Hale and Townsend [27] firstly presented a fast Chebyshev-Legendre transform, and developed a non-uniform discrete cosine transform which use a Taylor series expansion for Chebyshev polynomials about -spaced points in the frequency domain. The novel aspect is the mitigation of the potential instability of LT using butterfly algorithm at a very small increase of computational cost
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