Besides its simple structure and only O(N log N + M) arithmetic operations, this algorithm utilises m FFTs of size n compared to only one in the NFFT approach, uses a medium amount of extra memory, and is not suited for highly accurate computations; see Example 4.
Thus, for fixed [sigma] > 1, the approximation error introduced by the NFFT decays exponentially with the number m of summands in (2.
In the following, we suggest different methods for the compressed storage and application of the matrix B which are all available within our NFFT library by choosing particular flags in a simple way during the initialisation phase.
d] extra multiplications per node and is used within the NFFT by the flag PRE_LIN_PSI.
j] and these 2m + 1 values are computed only once within the NFFT and their amount is negligible.
Furthermore, it is faster than the default NFFT until an break even of N = 128.
a, p, m [member of] N, [sigma] [member of] Q (parameters for the NFFT summation algorithm)
The complexity estimates assume that the parameter [eta] for the NFFT summation algorithm is set to N.
In this step, the NFFT summation algorithm (Algorithm 3) is applied O([N.
k], the number of near-field evaluations performed in step 4 of the NFFT summation algorithm is bounded by O(aN log N), resulting in an overall complexity for the spherical filter of O([N.
1 tells us that step 4 of the NFFT summation algorithm (Algorithm 3) has a complexity of O(aN log N).
Since step 2 of the fast spherical filter (Algorithm 4) calls the NFFT summation O([N.