Convolve two sequences in Stan with the FFT. Matches `convolve(x, rev(y), type = "open")` in R.

convolve.stan

functions {
  vector convolve(vector a, vector b) {
    int na = num_elements(a);       // Vector lengths
    int nb = num_elements(b);
    int n_zero_a = nb - 1;          // Zero padding lengths
    int n_zero_b = na - 1;

    vector[nb] b_rev = reverse(b); // The reversed b vector

    vector[na + n_zero_a] a_pad;    // Instantiate zero padded vectors
    vector[nb + n_zero_b] b_rev_pad;

    // Perform zero padding
    a_pad[1 : n_zero_a] = rep_vector(0, n_zero_a);
    b_rev_pad[(nb + 1) : (nb + n_zero_b)] = rep_vector(0, n_zero_b);
    
    // Fill in padded vector
    a_pad[(n_zero_a + 1) : (na + n_zero_a)] = a;
    b_rev_pad[1 : nb] = b_rev;

     // Stan automatically scales the inverse FFT, so no rescaling needed
    return get_real(inv_fft(fft(a_pad) .* conj(fft(b_rev_pad))));
  }
}