ISSN 0249-6399 ISRN INRIA/RR--5752--FR+ENG
a p p o r t
d e r e c h e r c h e
Thème NUM
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Multipole-to-local operator in the Fast Multipole Method: comparison of FFT, rotations and BLAS
improvements
Pierre Fortin
N° 5752
Novembre 2005
Unité de recherche INRIA Futurs Parc Club Orsay Université, ZAC des Vignes, 4, rue Jacques Monod, 91893 ORSAY Cedex (France)
Téléphone : +33 1 72 92 59 00 — Télécopie : +33 1 72 92 59 ??
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Φ(Z) = 1
k Z − Q k = 1
k (X 2 − X 1 ) − ((Q − X 1 ) − (Z − X 2 )) k .
adm hx>Z5egWjZ-x>shjm¬enm|sJh
R > r 1 + r 2
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k (Q − X 1 ) − (Z − X 2 ) k < k X 2 − X 1 k
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Φ(Z) =
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n=0
X n l= − n
( − 1) n I n − l ((Q − X 1 ) − (Z − X 2 )) O l n (X 2 − X 1 ) .
adm hx>Z
I n l ( − X) = ( − 1) n I n l (X)
¡jªZWSZJ®Φ(Z) =
+ ∞
X
n=0
X n l=−n
I n − l ((Z − X 2 ) − (Q − X 1 ))O l n (X 2 − X 1 ).
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¡uh$sjenum h³®Φ(Z) =
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n=0
X n l=−n
X n j=0
X j k=−j
( − 1) j I j k (Q − X 1 )I n − − l − j k (Z − X 2 )O n l (X 2 − X 1 ).
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Φ P (Z) = X P n=0
X n l= − n
X n j=0
X j k= − j
( − 1) j I j k (Q − X 1 )I n−j − l − k (Z − X 2 )O n l (X 2 − X 1 ).
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n
hj
z m|´JZ®Φ P (Z) = X P j=0
X j k= − j
X P n=j
X n l= − n
( − 1) j I j k (Q − X 1 )I n − − l − j k (Z − X 2 )O l n (X 2 − X 1 ).
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n 0 = n − j
®Φ P ( Z ) = X P j=0
X j k= − j
P−j X
n 0 =0 n X 0 +j l=−(n 0 +j)
( − 1) j I j k ( Q − X 1 )I n − 0 l − k ( Z − X 2 )O l n 0 +j ( X 2 − X 1 ),
h
l 0 = l + k
egs[sjenum h³®Φ P (Z) = X P j=0
X j k=−j
P − j
X
n 0 =0
n 0 +j+k
X
l 0 = − (n 0 +j)+k
( − 1) j I j k (Q − X 1 )I n − 0 l 0 (Z − X 2 )O l n 0 0 − +j k (X 2 − X 1 ).
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I n − 0 l 0 (x)
m Y[{scfZKc>®− n 0 ≤ l 0 ≤ n 0
© henWjZcgY[ZwªSb¡I j k (x)
m Y²{sJcgZLcL®| k | ≤ j
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¡juh³®k − j ≤ 0
©P_slnZ>sJZ>lK®j + k ≥ 0 ⇒ n 0 + j + k ≥ n 0 ,
k − j ≤ 0 ⇒ − (n 0 + j) + k = − n 0 + (k − j) ≤ − n 0 .
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n 0 +j+k
X
l 0 = − (n 0 +j)+k
=
n 0
X
l 0 =−n 0
.
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l = − l 0
uhn = n 0
¡jªZw¤huz z|b$WSZJ®Φ P (Z) = X P j=0
X j k= − j
P−j X
n=0
X n l= − n
( − 1) j I j k (Q − X 1 )I n l (Z − X 2 )O n+j − l − k (X 2 − X 1 ).
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X P j=0
P − j
X
n=0
≡ X
| j |≤ P
| n |≤ P, 0≤n+j≤P
≡ X P n=0
P X − n j=0
.
·Zseum h³®
Φ P (Z) = X P n=0
X n l= − n
P−n X
j=0
X j k= − j
( − 1) j I j k (Q − X 1 )O −l−k n+j (X 2 − X 1 )
I n l (Z − X 2 ).
sl±uz z
Q ∈ B 1
¡^enWjZNYDijz|egm {sJz|Z+Z}{hcgm|sJhcenZ>lnYcM j k
¡∀ (j, k) ∈ N 2
ªCm|egW0 ≤ | k | ≤ j
¡ZLm|hj3Z>¤hjZL db®M j k = ( − 1) j I j k (Q − X 1 )
¡jªZWSZwegWdicC{jlnsZKenWeK®Φ P ( Z ) = X P n=0
X n l= − n
L l n I n l ( Z − X 2 ),
ªCWjZLlgZenWjZ
L l n
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cL®L l n =
P X − n j=0
X j k=−j
M j k O − n+j l − k (X 2 − X 1 )
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ws = 1
slws = 2
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m h$sJlnZLlegs[Z>hcgijlnZ-¢£JcveCZLhjsijJWx>shdZLlgJZ>hx>Z©· m¬enW
ws = 1
¢sJlCZ}juY[{jz Z¡ªZ5WSZJ¡J¢sJlAx>Z>z z³su¢?cfmpZ
l
®r 1 = r 2 = √ 2 3
uhR ≥ 2
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√ 3 2
! P+1
= 1
R − (r 1 + r 2 ) (0.866) P+1 .
· m|egW
ws = 2
¡ §R ≥ 3
¨ ¡ªZsjenum h³®k Φ(Z) − Φ P (Z) k ≤ 1 R − (r 1 + r 2 )
√ 3 3
! P+1
= 1
R − (r 1 + r 2 ) (0.577) P+1 .
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r 2
R − r 1
P+1
§ws = 1 ⇒ (0.764) P+1
h
ws = 2 ⇒ (0.406) P+1
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ws = 1 ⇒ (0.577) P+1
hws = 2 ⇒ (0.346) P+1
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