问题描述
我了解数学上等效的算术运算如何由于数值错误(例如,以不同顺序求和浮点数)而导致不同的结果.
I understand how mathematically-equivalent arithmentic operations can result in different results due to numerical errors (e.g. summing floats in different orders).
然而,令我惊讶的是,将零添加到 sum
会改变结果.我认为这始终适用于浮点数,无论如何:x + 0. == x
.
However, it surprises me that adding zeros to sum
can change the result. I thought that this always holds for floats, no matter what: x + 0. == x
.
这是一个例子.我希望所有的行都完全为零.谁能解释一下为什么会这样?
Here's an example. I expected all the lines to be exactly zero. Can anybody please explain why this happens?
M = 4 # number of random values
Z = 4 # number of additional zeros
for i in range(20):
a = np.random.rand(M)
b = np.zeros(M+Z)
b[:M] = a
print a.sum() - b.sum()
-4.4408920985e-16
0.0
0.0
0.0
4.4408920985e-16
0.0
-4.4408920985e-16
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
2.22044604925e-16
0.0
4.4408920985e-16
4.4408920985e-16
0.0
M
和 Z
的较小值似乎不会发生这种情况.
It seems not to happen for smaller values of M
and Z
.
我还确定了 a.dtype==b.dtype
.
这里还有一个例子,它也展示了 python 的内置 sum
的行为符合预期:
Here is one more example, which also demonstrates python's builtin sum
behaves as expected:
a = np.array([0.1, 1.0/3, 1.0/7, 1.0/13, 1.0/23])
b = np.array([0.1, 0.0, 1.0/3, 0.0, 1.0/7, 0.0, 1.0/13, 1.0/23])
print a.sum() - b.sum()
=> -1.11022302463e-16
print sum(a) - sum(b)
=> 0.0
我正在使用 numpy V1.9.2.
I'm using numpy V1.9.2.
推荐答案
简答:你看到了两者的区别
a + b + c + d
和
(a + b) + (c + d)
因为浮点数不准确所以不一样.
which because of floating point inaccuracies is not the same.
长答案: Numpy 将成对求和作为速度(它允许更容易矢量化)和舍入误差的优化.
Long answer: Numpy implements pair-wise summation as an optimization of both speed (it allows for easier vectorization) and rounding error.
numpy sum-implementation 可以在 here(函数pairwise_sum_@TYPE@
).它基本上做了以下事情:
The numpy sum-implementation can be found here (function pairwise_sum_@TYPE@
). It essentially does the following:
- 如果数组的长度小于 8,则执行常规的 for 循环求和.这就是为什么如果
W < 没有观察到奇怪的结果.4
在您的情况下 - 在两种情况下都将使用相同的 for 循环求和. - 如果长度在 8 到 128 之间,则在 8 个 bin
r[0]-r[7]
中累加总和,然后通过((r[0] + r[1]) + (r[2] + r[3])) + ((r[4] + r[5]) + (r[6] + r[7]))
.李> - 否则,它将递归地对数组的两半求和.
- If the length of the array is less than 8, a regular for-loop summation is performed. This is why the strange result is not observed if
W < 4
in your case - the same for-loop summation will be used in both cases. - If the length is between 8 and 128, it accumulates the sums in 8 bins
r[0]-r[7]
then sums them by((r[0] + r[1]) + (r[2] + r[3])) + ((r[4] + r[5]) + (r[6] + r[7]))
. - Otherwise, it recursively sums two halves of the array.
因此,在第一种情况下,您会得到 a.sum() = a[0] + a[1] + a[2] + a[3]
而在第二种情况下 b.sum() = (a[0] + a[1]) + (a[2] + a[3])
这导致 a.sum() - b.sum() != 0
.
Therefore, in the first case you get a.sum() = a[0] + a[1] + a[2] + a[3]
and in the second case b.sum() = (a[0] + a[1]) + (a[2] + a[3])
which leads to a.sum() - b.sum() != 0
.
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