# Memory allocation in matrix multiplication

**URL:** <https://discourse.julialang.org/t/memory-allocation-in-matrix-multiplication/42707>\
**Category:** Performance\
**Created:** [July 8, 2020, 3:59am UTC](https://discourse.julialang.org/t/memory-allocation-in-matrix-multiplication/42707 "2020-07-08T03:59:59Z")\
**Posts on this page:** 3\
**Page:** 1

<div class="post-metadata">

**Author:** ![aaraujo71](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/aaraujo71/32/10635_2.png) [@aaraujo71](https://discourse.julialang.org/u/aaraujo71)\
**Post date:** [July 8, 2020, 3:59am UTC](https://discourse.julialang.org/t/memory-allocation-in-matrix-multiplication/42707/1 "2020-07-08T03:59:59Z")

</div>

I am using matrix multiplication inside large loops and I wonder how I can reduce memory allocation. For example, the following (useless) code

```julia
function run_cycle!(R, A, B)
    for i in 1:1000
        R = A*B
    end
end

function test_timing()
    A = rand(5, 5)
    B = rand(5, 5)
    R = Array{Float64,2}(undef, 5, 5)
    @time run_cycle!(R, A, B)
end

test_timing()

```

returns

```julia
0.000682 seconds (1000 allocations: 281.250 KiB)

```

Is there any way to eliminate this allocation? Before, in element-wise operations with matrices, I used to use macro `@.` to eliminate memory allocation. However, although allocation is eliminated, in this case `@. R = A*B` does something totally different.

---

<div class="post-metadata">

**Author:** ![Elrod](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/elrod/32/22461_2.png) [@Elrod](https://discourse.julialang.org/u/Elrod)\
**Post date:** [July 8, 2020, 4:13am UTC](https://discourse.julialang.org/t/memory-allocation-in-matrix-multiplication/42707/2 "2020-07-08T04:13:25Z")

</div>

This works:

```julia
using LinearAlgebra
mul!(R, A, B)

```

If your arrays are small (I’ll use 5x5 as an example), then odds are a different library will give you better performance.  
Two options include `StaticArrays.jl`, a popular choice that uses stack allocated arrays:

```julia
using StaticArrays, LinearAlgebra, BenchmarkTools
As = @SMatrix rand(5,5);
Bs = @SMatrix rand(5,5);
Aa = Array(As); Ba = Array(Bs); Ca = similar(Aa, size(Aa,1), size(Ba,2));

```

This yields:

```julia
julia> @btime As2 * $Bs setup = (As2 = $(Ref(As))[])
  21.886 ns (0 allocations: 0 bytes)
5×5 SArray{Tuple{5,5},Float64,2,25} with indices SOneTo(5)×SOneTo(5):
 1.15682 0.961188 1.08718 0.848822 1.10338
 0.584637 1.26248 0.896451 1.05197 0.892125
 1.22157 1.71552 1.4453 1.5288 1.28231
 1.40697 2.4277 1.80974 1.78083 2.07344
 0.963769 1.29204 1.19697 1.30232 1.07738

julia> @btime mul!($Ca, $Aa, $Ba)
  136.604 ns (0 allocations: 0 bytes)
5×5 Array{Float64,2}:
 1.15682 0.961188 1.08718 0.848822 1.10338
 0.584637 1.26248 0.896451 1.05197 0.892125
 1.22157 1.71552 1.4453 1.5288 1.28231
 1.40697 2.4277 1.80974 1.78083 2.07344
 0.963769 1.29204 1.19697 1.30232 1.07738

```

Because static arrays are stack allocated, the Julia compiler cheats the benchmark if you just do `@btime $As * $Bs`.

My library `PaddedMatrices.jl` isn’t well documented and is still missing a lot of features, but aims to be much faster than `StaticArrays.jl`:

```julia
using PaddedMatrices
Ap = FixedSizeMatrix{5,5,Float64}(undef) .= Aa;
Bp = FixedSizeMatrix{5,5,Float64}(undef) .= Ba;
Cp = FixedSizeMatrix{5,5,Float64}(undef);

```

yields

```julia
julia> @btime mul!($Cp, $Ap, $Bp)
  5.734 ns (0 allocations: 0 bytes)
5×5 FixedSizeArray{Tuple{5,5},Float64,2,Tuple{1,5},32,0,0,false} with indices 1:5×1:5:
 1.15682 0.961188 1.08718 0.848822 1.10338
 0.584637 1.26248 0.896451 1.05197 0.892125
 1.22157 1.71552 1.4453 1.5288 1.28231
 1.40697 2.4277 1.80974 1.78083 2.07344
 0.963769 1.29204 1.19697 1.30232 1.07738

```

---

<div class="post-metadata">

**Author:** ![aaraujo71](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/aaraujo71/32/10635_2.png) [@aaraujo71](https://discourse.julialang.org/u/aaraujo71)\
**Post date:** [July 8, 2020, 11:39am UTC](https://discourse.julialang.org/t/memory-allocation-in-matrix-multiplication/42707/3 "2020-07-08T11:39:53Z")

</div>

thank you very much 🙂