Julia "strange" behaviour using fill() and .+= - sum

Question

I observe an unexpected behaviour for ".+=" in my code (it's probably just me, I'm rather new to Julia). Consider the following example:
julia> b = fill(zeros(2,2),1,3)
1×3 Array{Array{Float64,2},2}:
[0.0 0.0; 0.0 0.0] [0.0 0.0; 0.0 0.0] [0.0 0.0; 0.0 0.0]
julia> b[1] += ones(2,2)
2×2 Array{Float64,2}:
1.0 1.0
1.0 1.0
julia> b
1×3 Array{Array{Float64,2},2}:
[1.0 1.0; 1.0 1.0] [0.0 0.0; 0.0 0.0] [0.0 0.0; 0.0 0.0]
julia> b[2] .+= ones(2,2)
2×2 Array{Float64,2}:
1.0 1.0
1.0 1.0
julia> b
1×3 Array{Array{Float64,2},2}:
[1.0 1.0; 1.0 1.0] [1.0 1.0; 1.0 1.0] [1.0 1.0; 1.0 1.0]
As it can be seen, the last command changed not only the value of b[2] but also of b[3], while b[1] remains the same as before (*), as we can confirm running:
julia> b[2] .+= ones(2,2)
2×2 Array{Float64,2}:
2.0 2.0
2.0 2.0
julia> b
1×3 Array{Array{Float64,2},2}:
[1.0 1.0; 1.0 1.0] [2.0 2.0; 2.0 2.0] [2.0 2.0; 2.0 2.0]
Now, using simply "+=" instead I can obtain the behaviour I would have expected also for ".+=", that is:
julia> b = fill(zeros(2,2),1,3); b[2]+=ones(2,2); b
1×3 Array{Array{Float64,2},2}:
[0.0 0.0; 0.0 0.0] [1.0 1.0; 1.0 1.0] [0.0 0.0; 0.0 0.0]
Can anyone explain me why does it happen? I can use of course just +=, or maybe something different from an Array of Arrays, but since I'm striving for speed (I have a code that needs to perform these operations millions of times, and on much larger matrices) and .+= is considerably faster I would like to understad if I can still exploit this feature.
Thank you all in advance!
EDIT: (*) apparently only because b[1] was not zero. If I run:
julia> b = fill(zeros(2,2),1,3); b[2]+=ones(2,2);
julia> b[1] .+= 10 .*ones(2,2); b
[10.0 10.0; 10.0 10.0] [1.0 1.0; 1.0 1.0] [10.0 10.0; 10.0 10.0]
you can see that only the zero-values are changed. This beats me.

Answer 1

This happens because of the combination of several factors. Let's try and make things clearer.
First, b = fill(zeros(2,2),1,3) does not create a new zeros(2,2) for each element of b; instead it creates one 2x2 array of zeros, and sets all elements of b to that unique array. In short, this line behaves equivalently to
z = zeros(2,2)
b = Array{Array{Float64,2},2}(undef, 1, 3)
for i in eachindex(b)
b[i] = z
end
therefore, modifying z[1,1] or any of the b[i,j][1,1] would modify the other values as well. To illustrate this:
julia> b = fill(zeros(2,2),1,3)
1×3 Array{Array{Float64,2},2}:
[0.0 0.0; 0.0 0.0] [0.0 0.0; 0.0 0.0] [0.0 0.0; 0.0 0.0]
# All three elements are THE SAME underlying array
julia> b[1] === b[2] === b[3]
true
# Mutating one of them mutates the others as well
julia> b[1,1][1,1] = 42
42
julia> b
1×3 Array{Array{Float64,2},2}:
[42.0 0.0; 0.0 0.0] [42.0 0.0; 0.0 0.0] [42.0 0.0; 0.0 0.0]
Second, b[1] += ones(2,2) is equivalent to b[1] = b[1] + ones(2,2). This implies a succession of operations:
a new array (let's call it tmp) is created to hold the sum of b[1] and ones(2,2)
b[1] is rebound to that new array, thereby losing its connection to z (or all other elements of b.
This is a variation on the classical theme that although both involve = signs in their notations, mutation and assignment are not the same thing. Again, to illustrate:
julia> b = fill(zeros(2,2),1,3)
1×3 Array{Array{Float64,2},2}:
[0.0 0.0; 0.0 0.0] [0.0 0.0; 0.0 0.0] [0.0 0.0; 0.0 0.0]
# All elements are THE SAME underlying array
julia> b[1] === b[2] === b[3]
true
# But that connection is lost when `b[1]` is re-bound (not mutated) to a new array
julia> b[1] = ones(2,2)
2×2 Array{Float64,2}:
1.0 1.0
1.0 1.0
# Now b[1] is no more the same underlying array as b[2]
julia> b[1] === b[2]
false
# But b[2] and b[3] still share the same array (they haven't be re-bound to anything else)
julia> b[2] === b[3]
true
Third, b[2] .+= ones(2,2) is a different beast altogether. It does not imply re-binding anything to a newly created array; instead, it mutates the array b[2] in place. It effectively behaves like:
for i in eachindex(b[2])
b[2][i] += 1 # or b[2][i] = b[2][i] + 1
end
Neither b itself nor even b[2] is re-bound to anything, only elements of it are modified in place. And in your example this affects b[3] as well, since both b[2] and b[3] are bound to the same underlying array.

Answer 2

Becasue b is filled with the same matrix, not 3 identical matrices. .+= change the content of the matrix, thus all content in b are changed. += on the other hand, create a new matrix and assign it back to b[1]. To see this, you can use the === operator:
b = fill(zeros(2,2),1,3)
b[1] === b[2] # true
b[1] += zeros(2, 2) # a new matrix is created and assigned back to b[1]
b[1] == b[2] # true, they are all zeros
b[1] === b[2] # false, they are not the same matrix
There is actually an example in the help message of fill function pointing out exactly this problem. You can find it by running ?fill in the REPL.
...
If x is an object reference, all elements will refer to the same object:
julia> A = fill(zeros(2), 2);
julia> A[1][1] = 42; # modifies both A[1][1] and A[2][1]
julia> A
2-element Array{Array{Float64,1},1}:
[42.0, 0.0]
[42.0, 0.0]
There are various ways to create an array of independent matrices. One is using list comprehension:
c = [zeros(2,2) for _ in 1:1, _ in 1:3]
c[1] === c[2] # false