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Number of precision digits( sum(sum(A)) not equal to sum(A:)

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Hey guys. I am new to Matlab. And I need to compute a two-dimension matrix(65 x 65). called matrix A. sth like:
-0.043209482621010 -0.043209460507728 -0.043209425775628 -0.043209372126255
-0.043209488597448 -0.043209470284528 -0.043209441521445 -0.043209397092189
-0.043209517396291 -0.043209517396291 -0.043209517396291 -0.043209517396291
-0.043209590936714 -0.043209637700394 -0.043209711149500 -0.043209824603580
But when I do the operation
sum(sum(A)) it computes:1.847411112976260e-13
and the operation
sum(A(:)) it computes:5.652422974122828e-14
shouldn't they be the same consequence? I am confusing with that. Maybe it is about the precision digits. Can anyone help me?Thanks so much.


Stephen Cobeldick
Stephen Cobeldick on 25 Jun 2018
This is just accumulated floating point error, and is easy to demonstrate:
>> M = [+1e7,-1e7;1e-7,-1e-7]
M =
1.0000e+007 -1.0000e+007
1.0000e-007 -1.0000e-007
>> sum(sum(M))
ans = 0
>> sum(M(:))
ans = 0.00000000058284
This is exactly the same as every other question asked on this forum regarding floating point numbers: different operations on floating point numbers can (and usually do) produce different output values. Summing some values, then summing those sums is clearly a different set of operations than rearranging and then summing.

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Accepted Answer

Ameer Hamza
Ameer Hamza on 25 Jun 2018
Edited: Ameer Hamza on 25 Jun 2018
It might be happening because of finite precision of floating point numbers, Also, calling the sum() twice seems to be amplifying the errors caused by the finite precision. Since you are dealing with very small numbers, you might want to use arbitrary precision arithmetic to get accurate answers.


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