An fpga implementation of the Smooth Particle Mesh Ewald Reciprocal Sum Compute Engine (rsce)
Figure 3 - 3-D View of the Charge Grid
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- *64)*64*2)+1] = θ 1 [
- +n*4] x θ 3 [ ith3 +n*4]*q
- Computation of F -1 (Q) using 3D-IFFT
- In do_pmesh_kspace() function in ffcall.c...
Figure 3 - 3-D View of the Charge Grid 
Figure 4 - Interpolating a Charge @ (15.369, 8.850, 39.127) in a 2-D Mesh Table 4 - Interpolating the Charge @ (15.369, 8.850, 39.127) in 3-D (O(P*P*P) Steps)
Program output for charge grid composition: Inside fill_charge_grid() function in charge_grid.c N=20739, order=4, nfft1=64, nfft2=64, nfft3=64 nfftdim1=65, nfftdim2=65, nfftdim3=65 theta1_dim1=4, theta2_dim1=4, theta3_dim1=4 q_dim2=65, q_dim3=65
n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[31][32][31] = -0.017630 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[32][32][31] = -0.004407 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[33][32][31] = 0.000000 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[30][33][31] = -0.046805 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[31][33][31] = -0.187218 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[32][33][31] = -0.046805 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[33][33][31] = 0.000000 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[30][34][31] = -0.028217 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[31][34][31] = -0.112868 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[32][34][31] = -0.028217 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[33][34][31] = 0.000000 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[30][35][31] = -0.000389 : n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[33][34][32] = 0.000000 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[30][35][32] = -0.006465 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[31][35][32] = -0.025858 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[32][35][32] = -0.006465 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[33][35][32] = 0.000000 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[30][32][33] = -0.060405 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[31][32][33] = -0.241621 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[32][32][33] = -0.060405 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[33][32][33] = 0.000000 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[30][33][33] = -0.641467 : n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[33][35][33] = 0.000000 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[30][32][34] = -0.001803 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[31][32][34] = -0.007214 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[32][32][34] = -0.001803 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[33][32][34] = 0.000000 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[30][33][34] = -0.019152 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[31][33][34] = -0.076608 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[32][33][34] = -0.019152 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[33][33][34] = 0.000000 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[30][34][34] = -0.011546 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[31][34][34] = -0.046185 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[32][34][34] = -0.011546 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[33][34][34] = 0.000000 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[30][35][34] = -0.000159 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[31][35][34] = -0.000636 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[32][35][34] = -0.000159 n=1, fr1=32.000000, fr2=34.308041, fr3=33.426081, q[33][35][34] = 0.000000 :
n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[63][32][30] = 0.000022 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[64][32][30] = 0.000005 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[1][32][30] = 0.000000 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[62][33][30] = 0.001768 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[63][33][30] = 0.007071 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[64][33][30] = 0.001768 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[1][33][30] = 0.000000 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[62][34][30] = 0.004306 : n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[63][35][31] = 0.174262 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[64][35][31] = 0.043565 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[1][35][31] = 0.000000 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[62][32][32] = 0.000592 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[63][32][32] = 0.002368 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[64][32][32] = 0.000592 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[1][32][32] = 0.000000 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[62][33][32] = 0.193902 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[63][33][32] = 0.775608 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[64][33][32] = 0.193902 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[1][33][32] = 0.000000 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[62][34][32] = 0.472327 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[63][34][32] = 1.889307 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[64][34][32] = 0.472327 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[1][34][32] = 0.000000 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[62][35][32] = 0.070553 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[63][35][32] = 0.282213 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[64][35][32] = 0.070553 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[1][35][32] = 0.000000 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[62][32][33] = 0.000054 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[63][32][33] = 0.000216 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[64][32][33] = 0.000054 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[1][32][33] = 0.000000 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[62][33][33] = 0.017691 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[63][33][33] = 0.070766 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[64][33][33] = 0.017691 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[1][33][33] = 0.000000 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[62][34][33] = 0.043095 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[63][34][33] = 0.172378 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[64][34][33] = 0.043095 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[1][34][33] = 0.000000 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[62][35][33] = 0.006437 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[63][35][33] = 0.025749 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[64][35][33] = 0.006437 n=3, fr1=0.000000, fr2=34.831113, fr3=32.683046, q[1][35][33] = 0.000000 3.2.8 Computation of F-1(Q) using 3D-IFFT & Q Array Update The 3D IFFT is done in fft_back() function in the fftcall.c. It invokes the dynamic library libpubfft.a to perform the inverse 3D-FFT operation. The transformed elements are stored in the original charge array; hence, this is called in-place FFT operation. 
There may be a confusion on why the program is calculating the inverse FFT when the equation 4.7 shows that the forward FFT F(Q)(m1, m2, m3) is needed. The reason is that the F(Q)(-m1, -m2, -m3) is mathematically equivalent to F-1(Q) except for the scaling factor. When one thinks about this, the multiplication of F(Q)(m1, m2, m3) and F(Q)(-m1, -m2, -m3) should result in something like: (X*eiθ) * (X*e-iθ) => (Xcosθ + iXsinθ) * (Xcosθ - iXsinθ) => X2cos2θ + X2sin2θ
(X*e-iθ) => Real = Xcosθ Imaginary = Xsinθ => Real2 + Imaginary2 => X2cos2θ + X2sin2θ
fft_back(&q[1], &fftable[1], &ffwork[1], nfft1, nfft2, nfft3, &nfftdim1, &nfftdim2, &nfftdim3, &nfftable, &nffwork);
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