High dimension matrix summation: a + b != b + a

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Tian
Tian 2021-8-23
评论： Tian 2021-8-23
I have two matrices X1 and X2. Both are 10x15x10x10 dimensions. There are three scalars: ,and .
Let i={i1,i2,i3,i4} denote the index of an element. For elements with the same index in X1 and X2, say X1(i) and X2(i), I want to compute a simple weighted sum:
+ *X1(i) + *X2(i)
Since the γ's are all scalars, I think I can do this multiplication and summation at the matrix level and get the result as a 10x15x10x10 matrix. However, when I change the order of summation, the results are also different. That is:
A = + *X1 + *X2;
B = + *X2 + *X1;
The differences in all entries of A and B sum up to 2.3856e-12. However, since this is just element-wise summation, A and B should be exactly the same.
Can someone explain what is going wrong?
I do notice that if I take out , I get the same answer... Why does adding a constant affect the final result?
Please see attached for the sample data and codes.
Thank you for your help!!

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Wan Ji
Wan Ji 2021-8-23
编辑：Wan Ji 2021-8-23
It is a problem of machine error. If you have learned numerical analysis lessons, you will notice that big number will 'eat' small number when they are added.
a = 1;
b = eps;
for i = 1:10000
a = a+b;
end
c = a+10000*eps
c-a
Well the result is
ans =
2.220446049250313e-12
When you follow this code, error will not occur!
A = gamma0 + (gamma1*X1 + gamma2*X2);
B = gamma0 + (gamma2*X2 + gamma1*X1);
A = A(:);
B = B(:);
sum(sum(B-A))
ans =
0
This is beacuse the add operator order now is now the same for two expressions
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Wan Ji
Wan Ji 2021-8-23
So, when you do numerical analysis, avoid adding a large number with a small number or dividing a large number by small number. The error will explode if you do so. Also, you cannot minus a number far too close to the minuend.
An example will also show why:
calculate where x=1000001, y = 1000000; (use single precision)
We use
a = sqrt(single(1000001)) - sqrt(single(1000000))
The answer is
single
4.8828125e-04
While with the following transformation
a = single(1)/(sqrt(single(1000001)) + sqrt(single(1000000)))
The answer is
a = single(1)/(sqrt(single(1000001)) + sqrt(single(1000000)))
a =
single
4.9999985e-04
The answer with double precision is
a =
4.999998750000625e-04
You can see what I mean.
Tian
Tian 2021-8-23
Thank you for the comments. I am very new to numerical analysis and this is great to know! My problem is to simulate results for a set of given coefficients. Some coefficients are large. To prevent errors from exploding:
(1) Are there any rule of thumbs to follow?
(2) How do we know whether the numbers are right or "exploded"?
I think I can make sure small numbers come first when I sum them...

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