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The discussion in the previous section has been restricted to the case
where C_{t} is in +x_{3} direction of C_{s}. In this section we
shall remove this assumption to generalise the discussion. We divide the
interaction list of C_{s} into 6 lists: uplist, downlist, northlist,
southlist, eastlist and westlist. The uplist and downlist contain target
cells located in +x_{3} and x_{3} directions of C_{s},
respectively. The northlist and southlist contain target cells located
in +x_{2} and x_{2} directions of C_{s} except those in the uplist or
downlist, respectively. The eastlist and westlist contain target cells
located in +x_{1} and x_{1} directions of C_{s} except those in the
uplist, downlist, northlist or southlist, respectively. The situation in
the previous section can be described as the case where C_{t} is
contained in the uplist of C_{s}. If the target cell is included in
lists except the uplist of C_{s} we rotate the coordinate system so that
the target cell is in the positive
direction viewed from
the source cell, where
denotes the new axis. In general the
multipole moments in the new coordinate system are obtained as follows:
where
is the coefficient of
rotation,
is a unit vector parallel to the rotation axis,
is a rotation angle and
is a rotation
matrix. The explicit form of
is
given by (See Biedenharn and Louck[25])
where
and
.
The summation in (33) is carried out over such k that the powers
in the numerator are all nonnegative.
We next describe the generalised M2L translation process in the new
FMM.
 1.
 Rotation:
First we rotate the multipole moments via (31) and
(32) so as to make the procedure presented in
4.1 applicable. The specific forms of
(31) and (32) depend on the location of C_{t} and are described as follows:
where
is the base vector for the Cartesian coordinates
and superposed indices {U, D, N, S, E, W}
correspond to the initial letters of
{uplist, downlist, northlist, southlist, eastlist, westlist},
respectively.
 2.
 Compute the coefficients of the exponential expansion:
Compute the coefficients of the exponential expansion via
(25) and (26) as follows:
where
is an element of
.
 3.
 Translation of the coefficients of the exponential expansion:
As the centre of the exponential expansion is shifted from
the centroid of C_{s} (O) to the centroid of C_{t} (
),
the coefficients of the exponential expansion is translated
according to (27) and (28) as follows:
where
is an element of
.
 4.
 Compute the coefficients of the local expansion:
Compute the coefficients of the local expansion from the
exponential expansion according to (29) and (30)
as follows:
where
is an element of
.
Then rotate
as follows:

uplist

downlist

northlist

southlist

eastlist

westlist
Finally add
L^{U},L^{D},L^{N},L^{S},L^{E} and L^{W} together
to obtain the coefficients of the local expansion.
Next: Algorithm for the new
Up: New FMM
Previous: Formulation for the new
Kenichi Yoshida
20010326