我有以下代码(抱歉,它不是太小,我已经尝试从原来的代码中减少它)。
基本上,我在运行 eval_s()
方法/函数时遇到性能问题,我在其中:
1) 使用 eigvalsh()
2) 将特征值的倒数求和到一个变量result
3) 然后我对许多由 x,y,z
参数化的矩阵重复步骤 1 和 2,将累积和存储在 result
中。
我在第 3 步中重复计算(求特征值和求和)的次数取决于我代码中的变量 ksep
,我需要这个数字来增加我的实际代码(即, ksep
必须减少)。
但是 eval_s()
中的计算在 x,y,z
上有一个 for 循环,我猜这确实会减慢速度。
[试试 ksep=0.5
看看我的意思。]
有没有办法对我的示例代码中指示的方法进行矢量化(或者通常,涉及查找参数化矩阵的特征值的函数)?
代码:
import numpy as np
import sympy as sp
import itertools as it
from sympy.abc import x, y, z
class Solver:
def __init__(self, vmat):
self._vfunc = sp.lambdify((x, y, z),
expr=vmat,
modules='numpy')
self._q_count, self._qs = None, [] # these depend on ksep!
################################################################
# How to vectorize this?
def eval_s(self, stiff):
assert len(self._qs) == self._q_count, "Run 'populate_qs' first!"
result = 0
for k in self._qs:
evs = np.linalg.eigvalsh(self._vfunc(*k))
result += np.sum(np.divide(1., (stiff + evs)))
return result.real - 4 * self._q_count
################################################################
def populate_qs(self, ksep: float = 1.7):
self._qs = [(kx, ky, kz) for kx, ky, kz
in it.product(np.arange(-3*np.pi, 3.01*np.pi, ksep),
np.arange(-3*np.pi, 3.01*np.pi, ksep),
np.arange(-3*np.pi, 3.01*np.pi, ksep))]
self._q_count = len(self._qs)
def test():
vmat = sp.Matrix([[1, sp.cos(x/4+y/4), sp.cos(x/4+z/4), sp.cos(y/4+z/4)],
[sp.cos(x/4+y/4), 1, sp.cos(y/4-z/4), sp.cos(x/4 - z/4)],
[sp.cos(x/4+z/4), sp.cos(y/4-z/4), 1, sp.cos(x/4-y/4)],
[sp.cos(y/4+z/4), sp.cos(x/4-z/4), sp.cos(x/4-y/4), 1]]) * 2
solver = Solver(vmat)
solver.populate_qs(ksep=1.7) # <---- Performance starts to worsen (in eval_s) when ksep is reduced!
print(solver.eval_s(0.65))
if __name__ == "__main__":
import timeit
print(timeit.timeit("test()", setup="from __main__ import test", number=100))
附注代码的 sympy 部分可能看起来很奇怪,但它在我的原始代码中有用。
最佳答案
你可以,方法如下:
def eval_s_vectorized(self, stiff):
assert len(self._qs) == self._q_count, "Run 'populate_qs' first!"
mats = np.stack([self._vfunc(*k) for k in self._qs], axis=0)
evs = np.linalg.eigvalsh(mats)
result = np.sum(np.divide(1., (stiff + evs)))
return result.real - 4 * self._q_count
这仍然留下未向量化的 Sympy 表达式的计算。这部分向量化有点棘手,主要是因为输入矩阵中的 1
。您可以通过修改 Solver
来制作代码的完全矢量化版本,以便它用 vmat
中的数组常量替换标量常量:
import itertools as it
import numpy as np
import sympy as sp
from sympy.abc import x, y, z
from sympy.core.numbers import Number
from sympy.utilities.lambdify import implemented_function
xones = implemented_function('xones', lambda x: np.ones(len(x)))
lfuncs = {'xones': xones}
def vectorizemat(mat):
ret = mat.copy()
# get the first element of the set of symbols that mat uses
for x in mat.free_symbols: break
for i,j in it.product(*(range(s) for s in mat.shape)):
if isinstance(mat[i,j], Number):
ret[i,j] = xones(x) * mat[i,j]
return ret
class Solver:
def __init__(self, vmat):
self._vfunc = sp.lambdify((x, y, z),
expr=vectorizemat(vmat),
modules=[lfuncs, 'numpy'])
self._q_count, self._qs = None, [] # these depend on ksep!
def eval_s_vectorized_completely(self, stiff):
assert len(self._qs) == self._q_count, "Run 'populate_qs' first!"
evs = np.linalg.eigvalsh(self._vfunc(*self._qs.T).T)
result = np.sum(np.divide(1., (stiff + evs)))
return result.real - 4 * self._q_count
def populate_qs(self, ksep: float = 1.7):
self._qs = np.array([(kx, ky, kz) for kx, ky, kz
in it.product(np.arange(-3*np.pi, 3.01*np.pi, ksep),
np.arange(-3*np.pi, 3.01*np.pi, ksep),
np.arange(-3*np.pi, 3.01*np.pi, ksep))])
self._q_count = len(self._qs)
测试/计时
对于小型 ksep
,矢量化版本比原始版本快 2 倍,完全矢量化版本快 20 倍:
# old version for ksep=.3
import timeit
print(timeit.timeit("test()", setup="from __main__ import test", number=10))
-85240.46154500882
-85240.46154500882
-85240.46154500882
-85240.46154500882
-85240.46154500882
-85240.46154500882
-85240.46154500882
-85240.46154500882
-85240.46154500882
-85240.46154500882
118.42847006605007
# vectorized version for ksep=.3
import timeit
print(timeit.timeit("test()", setup="from __main__ import test", number=10))
-85240.46154498367
-85240.46154498367
-85240.46154498367
-85240.46154498367
-85240.46154498367
-85240.46154498367
-85240.46154498367
-85240.46154498367
-85240.46154498367
-85240.46154498367
64.95763925800566
# completely vectorized version for ksep=.3
import timeit
print(timeit.timeit("test()", setup="from __main__ import test", number=10))
-85240.46154498367
-85240.46154498367
-85240.46154498367
-85240.46154498367
-85240.46154498367
-85240.46154498367
-85240.46154498367
-85240.46154498367
-85240.46154498367
-85240.46154498367
5.648927717003971
矢量化版本结果中的舍入误差与原始版本略有不同。这大概是由于 result
中的总和的计算方式不同所致。
关于python - 如何在 python 中向量化包含 eigvalsh 的复杂代码,我们在Stack Overflow上找到一个类似的问题: https://stackoverflow.com/questions/53689075/