Why does a'*b yield a (slightly) different result than dot(a,b)?

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EQ 2017-11-28

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回答： Walter Roberson 2017-11-28

I was working an building a matrix entry-wise and checking two different methods against each other and eventually found that dot(a,b) is giving me a different answer than a'*b, where a and b are both real 9948x1 column vectors. I can see that the implementation of dot uses conj(a)*b, but my vectors are real. In one example, dot(a,b) = -1.0351e-11 and a'*b = -8.6402e-12.

I think it's likely a floating point precision issue, but I'm curious about the implementation of the "*" operator and what might be leading to it. Can anyone shed some light on this?

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EQ 2017-11-28

funny_inputs.mat

I'm attaching two vectors a and b that show the issue. I'm using R2015b.

Guillaume 2017-11-28

Indeed with your inputs I get the same discrepancy in R2017b. No idea why.

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Answer 1

Walter Roberson 2017-11-28

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For sufficiently large arrays, a'*b is going to invoke the high performance libraries, which are going to calculate the multiplications in multiple threads, doing partial summations, before eventually doing a final summation. Because the summations could be done in a different order than sum(conj(a).*b), the round-off can be different.