Tabulation methods aim at reducing the time spent in performing the Ewald sum by tabulating the energy
and the three components of forces as a function of the distance vector 
. A table is
constructed as a function of the distance
between a pair of particles, one of which is at the center of the original unit cell while the other
spans a grid. Interpolation is used to approximate untabulated force/potential components.
To reduce the run time, the table can be computed off-line.
For this approach to be feasible, the interpolation method used must be simple and
the table has to be ``moderately'' fine for it to be sufficiently accurate.
Therefore, a trade-off exists between the interpolation scheme and grid discretization used, and
the resultant efficiency and accuracy.
Sangester et al. [43] used simple 3-D linear interpolation.
In their method,
the vectors 
were tabulated with components satisfying:
and thus making use of the symmetry of the cubic cell to realize savings in storage.
Sangester's tabulation method [43] initially calculates and stores the energy and force
components at a particle i, centered at the basic cell, and particle j at a mesh point defined
by the segment above.
All nearest-image particles however are excluded from the tabulation and are calculated on-line
for accuracy.
To obtain the tabulated force between particle i and all j's images (except the nearest image),
a transformation must be first applied to find the corresponding
nearest-image vector 
which is followed by interpolation and backward transformation
to yield the interpolated force.
The total inter-particle force felt by particle i due to particles 
is therefore the sum
of the contribution from all image particles (tabulated) and the nearest particle (on-line).
The above process is repeated for all particles j.
Compared to the standard Ewald sum, this approach accelerates Ewald's computation; however,
due to the direct force evaluation, the method is still 
. In addition,
interpolation and forward/backward transformation may end up having a large runtime overhead.
Alternatively, one can tabulate only the direct-space sum, provided that the reciprocal-space sum
can be computed efficiently.
As discussed earlier, the direct-space sum, an 
operation, is reducible
to 
with the appropriate choice of Ewald parameters.
For example, the choice of 
can bias the intensity of calculations in either space at
the expense of the other. Therefore, depending on the Ewald parameters used,
tabulation may benefit the overall runtime.
Belhadj et al. [6] reported a speedup of four when look-up
tables were incorporated in their water simulations.
Another approach can tabulate only the exponent, 
, and/or the complimentary error function,
, if they are more expensive to evaluate explicitly on a particular computer.
Our own tests show a speedup of about 
when the table look-up is used for both
and 
.
Moreover, interpolation schemes that compute and store an estimate of the first derivative
such as spline interpolation [24], offer a good approximation for the tabulated function and its derivative.
For example, 
, in Eq.(8),
is obtained for free when 
is tabulated. This is based on the observation that:
.
In summary, tabulation methods attempt to strike a balance between accuracy versus speed, a trade-off
that is dependent on the size of the system and the objective of the simulation. These methods
can produce a major impact in speeding up MD/MC applications where medium
accuracy is acceptable. Higher accuracy simulations with tabulation may suffer in performance due
to low resolution in coarse tabulation and/or expensive interpolation,
or run out of storage for high-resolution tables.