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sum(w) and ones(1,size(w,2))*w' results totally different numbers

Asked by Abalfazl Zareei on 10 Apr 2013
hi, I have a vector w and it is a matrix with one row and 10 columns.
>> sum(w)
ans =
-0.1563
Then, I calculate the following, which is the product of a Ones vector and w, in which in matrix algebra, this is equivalent to summation of w's elements. >> ones(1,size(w,2))*w'
ans =
1
Both results are different. Would you please let me know why?
w =
1.0e+15 *
Columns 1 through 6
-0.0497 2.4484 -3.2273 0.1944 0.1592 0.4407
Columns 7 through 10
-0.4389 0.1548 0.1592 0.1592

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3 Answers

Answer by James Tursa
on 11 Apr 2013
Edited by James Tursa
on 11 Apr 2013

The answer is on the order of eps of the numbers you are dealing with. You have massive cancellation going on in the operation, and the order of the operation will make a difference (apparently sum and mtimes are doing the calculation in a slightly different order). The answer will have a lot of garbage bits. E.g., what do you get when you do this:
eps(max(abs(w)))
And then compare that result with your answer and you will see that they are about the same size.

  1 Comment

Exactly. Try this (insert all digits, if you have them):
w = [-0.0497e15, 2.4484e15, -3.2273e15, 0.1944e15, 0.1592e15, ...
0.4407e15, -0.4389e15, 0.1548e15, 0.1592e15, 0.1592e15];
format long g
sum(w)
sum(sort(w))
[dummy, index] = sort(abs(w));
sum(w(index))
Do you see a difference already? For a sum with error correction see FEX: XSum

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Answer by bym
on 10 Apr 2013

try
ones(size(w))*w.'

  2 Comments

It's not that; it is related to the way MATLAB computes the product between a vector whose elements are ~1e15 and an other whose elements are 1's.
The same answer:
>> ones(size(w))*w.'
ans =
1

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Answer by Ahmed A. Selman on 10 Apr 2013

The line
ones(1,size(w,2))*w'
means creating a ones matrix with dimensions (1,size(w,2)) multiplied with w', and
sum(w)
means summation of the vector w.
I really don't know if it is right to compare these two, entirely different things.
So use
sum(ones(1,size(w,2))*w')
and compare :)

  2 Comments

Multiplying an [1xN] with an [Nx1] vector means the dot-product. This is mathematically the same as summing the multiplied elements. Using a vector of 1's as one of the vectors results in a sum. Therefore the shown procedures are not different in theory.
Indeed it is the same. I thought w was N-by-N, my bad. Thank you for the correction.

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