Why different result when sum matrix by different order

When I sum a matrix by different order, the summation is different, for example:

a=rand(10)/pi;

sum(sum(a,1),2)-sum(sum(a,2),1)

the code will return a non-zeros result sometimes, which cause some problem in my calculation.

the matlab edition Im using is R2016a

Answer 1

[Chunru]

This is due to the numerical accuracy of the floating point operations.

Let's say we have a CPU with 4 significant digits. and we want to perform 0.1234+ 0.2345 + 345.6.

If we add in the first two small numbers, the result is 0.3579; Then we add to 345.6, we will get 346.0 (all results, intermediate or final, having 4 significant digits).

Now if we perform the addition in a reversed order. 345.6 + 0.2345 ~= 345.8; Then 345.8 + 0.1234 ~= 345.9. Thus we have different results by summing in different order.

In numerical calculation, one obtains more accurate result by performing summation of small numbers first. For example, sum of the series 1/n^2, n=1,..N, in reversed order, usually is a better choice (especially when N is very big).

Hope this help.

[Li Fujian]

thanks for the helping. the difference is typically 10^-15 to the summation, so it doesn't make much problem. thanks again!