1 * X' * X is not symmetric

I find the following snippet unexpected.

srand(1);
X = rand(1000, 1000) - 0.5;
(X' *X) == (X' *X)' #true
(1 * X' *X) == (1 * X' *X)' #false

Multiplying by 1 changes the symmetry, probably due to floating-point arithmetic issues. But I fail to understand where the failure happens.

Let A = X' * X. Naively I expect that:

A[i,j] == sum(X[k,i] * X[k,j] for k = 1 : N)
A[j,i] == sum(X[k,j] * X[k,i] for k = 1 : N)

Since multiplication is commutative (this holds in floating-point arithmetic as well), it follows that A[i,j] == A[j,i], so A is symmetric. I don’t see how multiplying by 1 changes this.

After some discussion in the slack chat (see https://julialang.slack.com/archives/C67910KEH/p1528205759000199), it seems that the culprit code is here:

1 * A creates a copy of A, which the fails the identity test A === B, which results in a call to gemm_wrapper!

But I don’t understand why that affects the result and breaks the symmetry.

I think that 1 * X' * X is evaluated as (1 * X') * X, and 1 * X' cannot be recognized as coming from the same matrix X.

You should use 1 * (X' * X), or even better, Symmetric(X' * X) * c for some generic real c (I am assuming the 1 is just for the MWE).

Different algorithms computing the same thing will give different results in floating point arithmetic. Don’t use == to compare floating point values.

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Thanks. However I am not trying to fix the issue. I want to understand where it comes from to understand the kind of floating point errors I might be committing here.

What exactly is the difference between syrk_wrapper! and gemm_wrapper!?

Those are different BLAS routines. If not mistaken syrk (symmetric rank-k update) takes advantage of crossproducts while gemm (General Matrix Multiply) is general.

Is there a reason for === instead of == in the snippet of code of mul!?

A good starting point would be

TL;DR: order matters, if you know (from theory) that two things are equal and want to use it like that, don’t calculate it twice, or discard one of them. Hence Symmetric.

Yes, A === B checks if A and B are the same matrix rather than checking if they are equal.

Wouldn’t the same thing work with == instead of ===? Why do you need them to be the same matrix, instead of just having the same numerical values?

Because checking === is very fast: it only requires comparing two pointers. Checking == is expensive: in general, it requires looking at all elements of both matrices (which may be very large).

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I see. Thanks!