Figure 5.1(b),(c) illustrates the error of Algorithm 2 using the butterfly fast Fourier transform (BSFFT) and the nonequispaced fast Fourier transform (NFFT), respectively.
Figure 5.2(b) displays the same for Algorithm 2 using the BSFFT (symbol +) and the NFFT (*).
POTTS, Using NFFT 3--a software library for various nonequispaced fast Fourier transforms, ACM Trans.
In case of a Cartesian trajectory, this step is implemented by the inverse fast Fourier transform; for nonequidistant sampling patterns, the NFFT
's adjoint operation can be used.
In contrast, the approximate NFFT takes only O([N.sub.[pi]] log [N.sub.[pi]]+M) floating point operations (flops), where the constant of proportionality depends in theory solely on the prescribed target accuracy and on the space dimension d.
In each of these applications the actual computation of the NDFT is the computationally dominant task and one has to deal with different requirements on the NFFT with respect to the target accuracy, the usage of memory, and the actual computation time.
In each iteration step, the product between this Fourier-type matrix and an arbitrary vector of length [??] can be computed with the NFFT
by O(N log N + L [absolute value of log [epsilon]]) flops, where [epsilon] > 0 is the wanted accuracy; see .
Expansions in these polynomials can be efficiently handled with FFT and NFFT
techniques publicly available in software libraries; see [12, 25].
Instead, we will use a fast summation algorithm  that is based on the Nonequispaced Fast Fourier Transform (NFFT) .
The algorithm is based on the Nonequispaced Fast Fourier Transform (NFFT), an approximate algorithm for computing the Discrete Fourier Transform at nonequispaced nodes (see, for example, [6, 1, 18, 17]).