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## Algorithm for the new FMM

The algorithm for the new FMM is given as follows:

Steps 1-3. Same as the steps 1-3 in 3.2.
Step 4. Computation of the coefficients of the exponential expansion:

Compute the coefficients of the exponential expansion at each cell using (34)-(47), taking the origin (O) at the centroid of the cell.

Step 5. Computation of the coefficients of the local expansion:

We compute the coefficients of the local expansion of cells of level l, starting from l=2 and increasing l. Now we consider cell C and cell C' which is contained in the interaction list of C. Considering the position of C' relative to C, we translate the coefficients of the exponential expansion via (48) and (49) as the centre of the exponential expansion is shifted from the centroid of C (O) to that of C' ( ) and then use appropriate formulae in (50)-(63) to convert the coefficients of the exponential expansion to the coefficients of the local expansion. After carrying out these conversions about all cells in the interaction list of C, we add them together via (64) and (65) to obtain the contribution from the interaction list of C to the coefficients of the local expansion. To these contributions we add the coefficients of the local expansion of the parent of C with the origin shifted from the centroid of the parent to that of C via (21) and (22) to obtain the coefficients of the local expansion associated with C.

Step 6. Same as the 5th step in 3.2.

Next: Numerical Examples Up: New FMM Previous: Rotation of coefficients
Ken-ichi Yoshida
2001-03-26