MATLAB: Does a’*b yield a (slightly) different result than dot(a,b)

*dotfloating point precisionmtimes

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 9948×1 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?

Best Answer

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.

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MATLAB: How are vector – vector products calculated

It is different, if you type

type dot

You will see that the actual computation done in dot after all the dimension checks are:

D2=sum(conj(A).*B);

The sum .* and * functions are built in functions so it is hard to give a better answer then; yes the algorithms are different (see mtimes , sum and times)

But, it seems that * (or mtimes) is multithreaded. So it probably splits the vector into chunks, multiplies and sums them, then sum them all together in the end. You can confirm this by typing

test1=A'*B;
LastN=maxNumCompThreads(1);
test2=A'*B;
isequal(test1,test2)
ans =
       0

maxNumCompThreads does not persist through sessions, but to avoid having to restart matlab to enable multiple threads use

maxNumCompThreads(LastN);

MATLAB: Dot product for complex vector

help dot
 dot  Vector dot product.
    C = dot(A,B) returns the scalar product of the vectors A and B.
    A and B must be vectors of the same length.  When A and B are both
    column vectors, dot(A,B) is the same as A'*B.
    
 dot(A,B), for N-D arrays A and B, returns the scalar product
    along the first non-singleton dimension of A and B. A and B must
    have the same size.

So what are A and B?

A = [1+i 1-i -1+i -1-i];
B = [3-4i 6-2i 1+2i 4+3i];

They are row vectors, complex row vectors. So MATLAB forms the result as:

dot(A,B)
ans =
            1 -          5i
sum(conj(A).*B)
ans =
            1 -          5i

That is, when A and B are both vectors, MATLAB treats them the same as if A and B were column vectors. It effectively thinks of them as both column vectors. Then it forms the result by conjugating A, takes an element-wise product, then sums those terms.

Best Answer

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MATLAB: How are vector – vector products calculated

MATLAB: Dot product for complex vector

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