# Matrix multiplication

**URL:** <https://discourse.julialang.org/t/matrix-multiplication/50293>\
**Category:** Performance\
**Created:** [November 17, 2020, 11:52am UTC](https://discourse.julialang.org/t/matrix-multiplication/50293 "2020-11-17T11:52:32Z")\
**Posts on this page:** 7\
**Page:** 1

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**Author:** ![nhavt](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/nhavt/32/16356_2.png) [@nhavt](https://discourse.julialang.org/u/nhavt)\
**Post date:** [November 17, 2020, 11:52am UTC](https://discourse.julialang.org/t/matrix-multiplication/50293/1 "2020-11-17T11:52:32Z")

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Hi there,

I need to perform many matrix multiplications as follows. Here I proposed two functions (f1 and f2). I am wondering whether we could improve the speed of the function f2 further? Thanks in advance!

```julia
using LinearAlgebra
f1(C,X,Y,Z,T,x) = C .= .-x[1].*X*X' .- x[2].*Y*Y' .+ x[3].*Z*T;
function f2(C,X,Y,Z,T,x)
    x1 = -x[1]
    x2 = -x[2]
    x3 = x[3]
    fill!(C,0.0)    
    BLAS.syrk!('L', 'N', x1, X, 1.0, C)
    BLAS.syrk!('L', 'N', x2, Y, 1.0, C)
    conSym(C)
    mul!(C, Z, T, x3, 1.0)
end
function conSym(A)
   @inbounds for ii=1:size(A,1)-1
           for jj=ii+1:size(A,1)
             A[ii,jj] = A[jj,ii]   
           end
        end
end
v = 600;
r = 3;
b1 = 60;
b2 = 90;
x = rand(3);
C1 = zeros(Float64,v,v);
C2 = zeros(Float64,v,v);
C3 = zeros(Float64,v,v);

X = rand(v,b1);
Y = rand(v,b2);
Z = rand(v,v*r);
T = rand(v*r,v) ;         
@time f1(C1,X,Y,Z,T,x);
@time f2(C2,X,Y,Z,T,x);

@test all(C1 .≈ C2);

```

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<div class="post-metadata">

**Author:** ![lmiq](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/lmiq/32/18314_2.png) [@lmiq](https://discourse.julialang.org/u/lmiq)\
**Post date:** [November 17, 2020, 12:12pm UTC](https://discourse.julialang.org/t/matrix-multiplication/50293/2 "2020-11-17T12:12:16Z")

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> [@nhavt](#):
>
> `@time f2(C2,X,Y,Z,T,x);`

Are you aware that you are measuring the compilation time there as well? Add a call to `f2` before that and you will get a better time estimate:

```julia
f2(C2,X,Y,Z,T,x);
@time f2(C2,X,Y,Z,T,x);
  0.012986 seconds

```

(instead of ` 0.063080 seconds (36.77 k allocations: 1.934 MiB)`, that you get in the first call)

Or even better, use `BenchmarkTools`.

---

<div class="post-metadata">

**Author:** ![tamasgal](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/tamasgal/32/27946_2.png) [@tamasgal](https://discourse.julialang.org/u/tamasgal)\
**Post date:** [November 17, 2020, 12:13pm UTC](https://discourse.julialang.org/t/matrix-multiplication/50293/3 "2020-11-17T12:13:05Z")

</div>

You should read the [Performance Tips](https://docs.julialang.org/en/v1/manual/performance-tips/) section in the docs. For example, you are using global variables which affect type stability and the `@time` macro called once will also include the time needed for the first compilation. Use `@btime` from `BenchmarkTools` instead.

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<div class="post-metadata">

**Author:** ![nhavt](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/nhavt/32/16356_2.png) [@nhavt](https://discourse.julialang.org/u/nhavt)\
**Post date:** [November 17, 2020, 1:43pm UTC](https://discourse.julialang.org/t/matrix-multiplication/50293/4 "2020-11-17T13:43:02Z")

</div>

> Are you aware that you are measuring the compilation time there as well? Add a call to `f2` before that and you will get a better time estimate:

Yes, I am aware of that. I actually don’t compare the computing times of these functions for a first call. Also, the function f1 is just a way that I make sure that the calculation of function f2 is correct.

> (instead of `0.063080 seconds (36.77 k allocations: 1.934 MiB)`, that you get in the first call)  
> Or even better, use `BenchmarkTools`.

Indeed, I actually used the command `@btime`. But somehow I got trouble when using this command `@test all(C1 .≈ C2)`. This is because I thought that these commands `@btime f1($C1,$X,$Y,$Z,$T,$x); @btime f2($C2,$X,$Y,$Z,$T,$x);` will not change variables `C1` and `C2`. Therefore, I used these commands `@time f1(C1,X,Y,Z,T,x); @time f2(C2,X,Y,Z,T,x);` in the original post so that my codes will run smoothly. But probably, I am wrong here. I am sorry about this. Nevertheless, this is computing times from my computer:

```julia
@btime f1($C1,$X,$Y,$Z,$T,$x);
  23.513 ms (12 allocations: 17.17 MiB)
@btime f2($C2,$X,$Y,$Z,$T,$x);
  17.317 ms (0 allocations: 0 bytes)

```

As you can see, the function f2 is not much faster than function f1 as I expected. I had an idea to write a loop. Since the dimensions of matrices X, Y, Z, and T are different, it is not straightforward to do that. So, do you have any suggestions? Thanks in advance!

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<div class="post-metadata">

**Author:** ![lmiq](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/lmiq/32/18314_2.png) [@lmiq](https://discourse.julialang.org/u/lmiq)\
**Post date:** [November 17, 2020, 2:12pm UTC](https://discourse.julialang.org/t/matrix-multiplication/50293/5 "2020-11-17T14:12:43Z")

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To benchmark that with `BenchmarkTools` you should use:

```julia
setup = (
C3 = zeros(Float64,v,v);
Xin = X;
Yin = Y;
Zin = Z;
Tin = T;
)
@btime f2(C3,Xin,Yin,Zin,Tin,$x) setup=setup evals=1

```

On what remains, no, I do not see any obvious change to accelerate that code.

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<div class="post-metadata">

**Author:** ![nhavt](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/nhavt/32/16356_2.png) [@nhavt](https://discourse.julialang.org/u/nhavt)\
**Post date:** [November 17, 2020, 2:39pm UTC](https://discourse.julialang.org/t/matrix-multiplication/50293/6 "2020-11-17T14:39:52Z")

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> To benchmark that with `BenchmarkTools` you should use:

Yes, thanks for your tip!

---

<div class="post-metadata">

**Author:** ![moeddel](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/moeddel/32/18641_2.png) [@moeddel](https://discourse.julialang.org/u/moeddel)\
**Post date:** [November 17, 2020, 7:45pm UTC](https://discourse.julialang.org/t/matrix-multiplication/50293/7 "2020-11-17T19:45:22Z")

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> [@lmiq](#):
>
> On what remains, no, I do not see any obvious change to accelerate that code.

Maybe less obvious. Depending on the application, one could switch to `Float32`, which should speed things up.