这是使用blas 的优化版本。 blas 具有用于对称或 Hermitian 矩阵上的线性代数的特殊例程。这些使用专门的打包存储，只保留上三角或下三角，而忽略（冗余）镜像条目。这样 blas 不仅节省了 ~ 一半的存储空间，而且还节省了 ~ 一半的触发器。
我已经放置了很多 cmets 以使其可读。
import numpy as np
import itertools
from scipy.linalg.blas import zhpr, dspr2, zhpmv
def acc_vect(pos, mas):
n = mas.size
d2 = pos@(-2*pos.T)
diag = -0.5 * np.einsum('ii->i', d2)
d2 += diag + diag[:, None]
np.einsum('ii->i', d2)[...] = 1
return np.nansum((pos[:, None, :] - pos) * (mas[:, None] * d2**-1.5)[..., None], axis=0)
def acc_blas(pos, mas):
n = mas.size
# trick: use complex Hermitian to get the packed anti-symmetric
# outer difference in the imaginary part of the zhpr answer
# don't want to sum over dimensions yet, therefore must do them one-by-one
trck = np.zeros((3, n * (n + 1) // 2), complex)
for a, p in zip(trck, pos.T - 1j):
zhpr(n, -2, p, a, 1, 0, 0, 1)
# does a -> a + alpha x x^H
# parameters: n -- matrix dimension
# alpha -- real scalar
# x -- complex vector
# ap -- packed Hermitian n x n matrix a
# i.e. an n(n+1)/2 vector
# incx -- x stride
# offx -- x offset
# lower -- is storage of ap lower or upper
# overwrite_ap -- whether to change a inplace
# as a by-product we get pos pos^T:
ppT = trck.real.sum(0) + 6
# now compute matrix of squared distances ...
# ... using (A-B)^2 = A^2 + B^2 - 2AB
# ... that and the outer sum X (+) X.T equals X ones^T + ones X^T
dspr2(n, -0.5, ppT[np.r_[0, 2:n+1].cumsum()], np.ones((n,)), ppT,
1, 0, 1, 0, 0, 1)
# does a -> a + alpha x y^T + alpha y x^T in packed symmetric storage
# scale anti-symmetric differences by distance^-3
np.divide(trck.imag, ppT*np.sqrt(ppT), where=ppT.astype(bool),
out=trck.imag)
# it remains to scale by mass and sum
# this can be done by matrix multiplication with the vector of masses ...
# ... unfortunately because we need anti-symmetry we need to work
# with Hermitian storage, i.e. complex numbers, even though the actual
# computation is only real:
out = np.zeros((3, n), complex)
for a, o in zip(trck, out):
zhpmv(n, 0.5, a, mas*-1j, 1, 0, 0, o, 1, 0, 0, 1)
# multiplies packed Hermitian matrix by vector
return out.real.T
def accelerations(positions, masses, epsilon=1e-6, gravitational_constant=1.0):
'''Params:
- positions: numpy array of size (n,3)
- masses: numpy array of size (n,)
'''
n_bodies = len(masses)
accelerations = np.zeros([n_bodies,3]) # n_bodies * (x,y,z)
# vectors from mass(i) to mass(j)
D = np.zeros([n_bodies,n_bodies,3]) # n_bodies * n_bodies * (x,y,z)
for i, j in itertools.product(range(n_bodies), range(n_bodies)):
D[i][j] = positions[j]-positions[i]
# Acceleration due to gravitational force between each pair of bodies
A = np.zeros((n_bodies, n_bodies,3))
for i, j in itertools.product(range(n_bodies), range(n_bodies)):
if np.linalg.norm(D[i][j]) > epsilon:
A[i][j] = gravitational_constant * masses[j] * D[i][j] \
/ np.linalg.norm(D[i][j])**3
# Calculate net accleration of each body
accelerations = np.sum(A, axis=1) # sum of accel vectors for each body
return accelerations
from numpy.linalg import norm
def acc_pm(positions, masses, G=1):
'''Params:
- positions: numpy array of size (n,3)
- masses: numpy array of size (n,)
'''
mass_matrix = masses.reshape((1, -1, 1))*masses.reshape((-1, 1, 1))
disps = positions.reshape((1, -1, 3)) - positions.reshape((-1, 1, 3)) # displacements
dists = norm(disps, axis=2)
dists[dists == 0] = 1 # Avoid divide by zero warnings
forces = G*disps*mass_matrix/np.expand_dims(dists, 2)**3
return forces.sum(axis=1)/masses.reshape(-1, 1)
n = 500
pos = np.random.random((n, 3))
mas = np.random.random((n,))
from timeit import timeit
print(f"loops: {timeit('accelerations(pos, mas)', globals=globals(), number=1)*1000:10.3f} ms")
print(f"pmende: {timeit('acc_pm(pos, mas)', globals=globals(), number=10)*100:10.3f} ms")
print(f"vectorized: {timeit('acc_vect(pos, mas)', globals=globals(), number=10)*100:10.3f} ms")
print(f"blas: {timeit('acc_blas(pos, mas)', globals=globals(), number=10)*100:10.3f} ms")
A = accelerations(pos, mas)
AV = acc_vect(pos, mas)
AB = acc_blas(pos, mas)
AP = acc_pm(pos, mas)
assert np.allclose(A, AV) and np.allclose(AB, AV) and np.allclose(AP, AV)
示例运行；与 OP 相比，我的纯 numpy 矢量化和@P Mende 的。
loops: 3213.130 ms
pmende: 41.480 ms
vectorized: 43.860 ms
blas: 7.726 ms
我们可以看到
1) P Mende 在矢量化方面比 I 稍好
2) blas 的速度是 ~5 倍；请注意，我的 blas 不是很好；我怀疑使用优化的 blas 可能会变得更好（不过，预计 numpy 在更好的 blas 上也会运行得更快）
3) 任何答案都比循环快得多