private void ComputeKernelTilde()
{
//Evaluate the kernel at the interpolation nodes and form the embedded generating kernel vector for a circulant matrix
int n_fft_coeffs = 2 * n_interpolation_points_1d;
for (int i = 0; i < n_interpolation_points_1d; i++)
{
for (int j = 0; j < n_interpolation_points_1d; j++)
{
for (int k = 0; k < n_interpolation_points_1d; k++)
{
fft[((n_interpolation_points_1d + i) * n_fft_coeffs + n_interpolation_points_1d + j) * n_fft_coeffs + n_interpolation_points_1d + k]
= fft[((n_interpolation_points_1d - i) * n_fft_coeffs + n_interpolation_points_1d + j) * n_fft_coeffs + n_interpolation_points_1d + k]
= fft[((n_interpolation_points_1d + i) * n_fft_coeffs + n_interpolation_points_1d - j) * n_fft_coeffs + n_interpolation_points_1d + k]
= fft[((n_interpolation_points_1d - i) * n_fft_coeffs + n_interpolation_points_1d - j) * n_fft_coeffs + n_interpolation_points_1d + k]
= fft[((n_interpolation_points_1d + i) * n_fft_coeffs + n_interpolation_points_1d + j) * n_fft_coeffs + n_interpolation_points_1d - k]
= fft[((n_interpolation_points_1d - i) * n_fft_coeffs + n_interpolation_points_1d + j) * n_fft_coeffs + n_interpolation_points_1d - k]
= fft[((n_interpolation_points_1d + i) * n_fft_coeffs + n_interpolation_points_1d - j) * n_fft_coeffs + n_interpolation_points_1d - k]
= fft[((n_interpolation_points_1d - i) * n_fft_coeffs + n_interpolation_points_1d - j) * n_fft_coeffs + n_interpolation_points_1d - k]
= SquaredCauchy(tilde[0], tilde[i], tilde[j], tilde[k]);
}
}
}
// Precompute the FFT of the kernel generating matrix
Fourier.ForwardMultiDim(fft, new int[] { n_fft_coeffs, n_fft_coeffs, n_fft_coeffs }, FourierOptions.NoScaling);
for (int i = fft.Length - 1; i >= 0; i--)
{
kernel_tilde[i] = fft[i].Real;
}
}
private void NBodyFFT()
{
//Compute the values v_{m, n} at the equispaced nodes, multiply the kernel matrix with the coefficients w
int n_fft_coeffs = 2 * n_interpolation_points_1d;
for (int d = 0; d < n_terms; d += 2)
{
Array.Clear(fft, 0, fft.Length);
for (int i = 0; i < n_interpolation_points_1d; i++)
{
for (int j = 0; j < n_interpolation_points_1d; j++)
{
for (int k = 0; k < n_interpolation_points_1d; k++)
{
fft[(i * n_fft_coeffs + j) * n_fft_coeffs + k] = new Complex(tilde_values[((i * n_interpolation_points_1d + j) * n_interpolation_points_1d + k) * n_terms + d],
(d == n_terms - 1) ? 0 : tilde_values[((i * n_interpolation_points_1d + j) * n_interpolation_points_1d + k) * n_terms + d + 1]);
}
}
}
Fourier.ForwardMultiDim(fft, new int[] { n_fft_coeffs, n_fft_coeffs, n_fft_coeffs });
// Take the Hadamard product of two complex vectors
for (int i = kernel_tilde.Length - 1; i >= 0; i--)
{
fft[i] *= kernel_tilde[i];
}
// Invert the computed values at the interpolated nodes
Fourier.InverseMultiDim(fft, new int[] { n_fft_coeffs, n_fft_coeffs, n_fft_coeffs });
for (int i = 0; i < n_interpolation_points_1d; i++)
{
for (int j = 0; j < n_interpolation_points_1d; j++)
{
for (int k = 0; k < n_interpolation_points_1d; k++)
{
tilde_values[((i * n_interpolation_points_1d + j) * n_interpolation_points_1d + k) * n_terms + d] = fft[((n_interpolation_points_1d + i) * n_fft_coeffs + n_interpolation_points_1d + j) * n_fft_coeffs + n_interpolation_points_1d + k].Real;
if (d != n_terms - 1)
{
tilde_values[((i * n_interpolation_points_1d + j) * n_interpolation_points_1d + k) * n_terms + d + 1] = fft[((n_interpolation_points_1d + i) * n_fft_coeffs + n_interpolation_points_1d + j) * n_fft_coeffs + n_interpolation_points_1d + k].Imaginary;
}
}
}
}
}
}