奇怪的乘法结果 [英] Strange multiplication result

问题描述

在我的代码中，我在C ++代码中具有这种乘法，所有变量类型均为double []

f1[0] = (f1_rot[0] * xu[0]) + (f1_rot[1] * yu[0]); 
f1[1] = (f1_rot[0] * xu[1]) + (f1_rot[1] * yu[1]); 
f1[2] = (f1_rot[0] * xu[2]) + (f1_rot[1] * yu[2]); 

f2[0] = (f2_rot[0] * xu[0]) + (f2_rot[1] * yu[0]); 
f2[1] = (f2_rot[0] * xu[1]) + (f2_rot[1] * yu[1]);
f2[2] = (f2_rot[0] * xu[2]) + (f2_rot[1] * yu[2]);

对应于这些值

Force Rot1 : -5.39155e-07, -3.66312e-07
Force Rot2 : 4.04383e-07, -1.51852e-08

xu: 0.786857, 0.561981, 0.255018
yu: 0.534605, -0.82715, 0.173264

F1: -6.2007e-07, -4.61782e-16, -2.00963e-07
F2: 3.10073e-07, 2.39816e-07, 1.00494e-07

这种乘法特别会产生-4.61782e-16而不是1.04745e-13的错误值

f1[1] = (f1_rot[0] * xu[1]) + (f1_rot[1] * yu[1]);  

我亲自在计算器上验证了其他乘法，它们似乎都产生了正确的值.

这是一个开放的mpi编译代码，上面的结果用于运行单个处理器，运行多个处理器时会有不同的值，例如40个处理器由于F1 [1]乘法而产生1.66967e-13.

这是某种MPI错误吗?还是类型精度问题?为什么对其他乘法工作还可以呢?

解决方案

您的问题是所谓的灾难性总结的明显结果: 我们知道，双精度浮点数可以处理约16个有效小数的数字.

f1[1] = (f1_rot[0] * xu[1]) + (f1_rot[1] * yu[1])
      = -3.0299486605499998e-07 + 3.0299497080000003e-07
      = 1.0474500005332475e-13

这是我们使用您在示例中提供的数字获得的结果. 请注意，(-7) - (-13) = 6对应于示例中给出的浮点数中的小数位数:(例如:-5.39155e-07 -3.66312e-07，每个尾数的精度为6个小数).这意味着您在这里使用了单精度浮点数.

我确信在您的计算中，数字的精度更高，这就是为什么您找到更精确的结果的原因.

无论如何，如果您使用单精度浮点数，则不能指望有更高的精度.使用双精度，您可以找到高达16的精度.您不应该相信两个数字之间的差，除非它大于尾数:

• 简单精度浮点数:(a-b)/b> =〜1e-7
• 双精度浮点数:(a-b)/b> =〜4e-16

有关更多信息，请参见这些例子... 或本文中的表格...

In my code I have this multiplications in a C++ code with all variable types as double[]

f1[0] = (f1_rot[0] * xu[0]) + (f1_rot[1] * yu[0]); 
f1[1] = (f1_rot[0] * xu[1]) + (f1_rot[1] * yu[1]); 
f1[2] = (f1_rot[0] * xu[2]) + (f1_rot[1] * yu[2]); 

f2[0] = (f2_rot[0] * xu[0]) + (f2_rot[1] * yu[0]); 
f2[1] = (f2_rot[0] * xu[1]) + (f2_rot[1] * yu[1]);
f2[2] = (f2_rot[0] * xu[2]) + (f2_rot[1] * yu[2]);

corresponding to these values

Force Rot1 : -5.39155e-07, -3.66312e-07
Force Rot2 : 4.04383e-07, -1.51852e-08

xu: 0.786857, 0.561981, 0.255018
yu: 0.534605, -0.82715, 0.173264

F1: -6.2007e-07, -4.61782e-16, -2.00963e-07
F2: 3.10073e-07, 2.39816e-07, 1.00494e-07

this multiplication in particular produces a wrong value -4.61782e-16 instead of 1.04745e-13

f1[1] = (f1_rot[0] * xu[1]) + (f1_rot[1] * yu[1]);  

I hand verified the other multiplications on a calculator and they all seem to produce the correct values.

this is an open mpi compiled code and the above result is for running a single processor, there are different values when running multiple processors for example 40 processors produces 1.66967e-13 as result of F1[1] multiplication.

Is this some kind of mpi bug ? or a type precision problem ? and why does it work okay for the other multiplications ?

解决方案

Your problem is an obvious result of what is called catastrophic summations: As we know, a double precision float can handle numbers of around 16 significant decimals.

f1[1] = (f1_rot[0] * xu[1]) + (f1_rot[1] * yu[1])
      = -3.0299486605499998e-07 + 3.0299497080000003e-07
      = 1.0474500005332475e-13

This is what we obtain with the numbers you have given in your example. Notice that (-7) - (-13) = 6, which corresponds to the number of decimals in the float you give in your example: (ex: -5.39155e-07 -3.66312e-07, each mantissa is of a precision of 6 decimals). It means that you used here single precision floats.

I am sure that in your calculations, the precision of your numbers is bigger, that's why you find a more precise result.

Anyway, if you use single precision floats, you can't expect a better precision. With a double precision, you can find a precision up to 16. You shouldn't trust a difference between two numbers, unless it is bigger than the mantissa:

• Simple precision floats: (a - b) / b >= ~1e-7
• Double precision floats: (a - b) / b >= ~4e-16

For further information, see these examples ... or the table in this article ...

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