# How to sum elements raised to a power

Let’s assume that `X` is a matrix. I want to efficiently compute `sum(X.^2)` but without computing `X.^2` which will allocate some memory for nothing. So I’m trying to do something like `sum(^(2),X)` which throws a method error. I thought this could work because `sum(>(0.2),X)` works for instance.
I know that I could easily create an auxillary function `f(x)=x^2` and then compute `sum(f,X)`. But it’s not the best in terms of readability. So my question is, is there a way to properly use `^(x, y)` to achieve my purpose?

Use an anonymous function.

``````sum(x -> x^2, X)
``````
5 Likes

You could also do

``````sum(abs2, X)
``````

which will handle both real- and complex-valued matrices, and should have the same performance as `x->x^2`.

3 Likes

I use abs2 simply for ease in this type of situation

You can have a faster sum (not as safe as the default sum, which takes some care to not overflow things), using a parallel sum with `@tturbo`:

``````julia> function mysum(f,x)
s = zero(eltype(f(x[begin])))
@tturbo for i in eachindex(x)
s += f(x[i])
end
return s
end
mysum (generic function with 2 methods)

julia> x = rand(1000,1000);

julia> @btime mysum(abs2,\$x) # 4 threads (julia -t4)
83.500 μs (0 allocations: 0 bytes)
333115.0216837497

julia> #vs
@btime sum(abs2,\$x)
327.982 μs (0 allocations: 0 bytes)
333115.0216837495

``````

If that is within a peformance critical part of the code and the numbers are well behaved, it is a good alternative.

2 Likes

While the `abs2` solution is good enough for this case, I think generalizing the `^(n)` would be a nice addition to Julia, the same as we do `sqrt`.

Note that `abs2` is actually different from `x -> x^2` for complex numbers.

``````julia> X = rand(ComplexF64, 2, 2)
2×2 Matrix{ComplexF64}:
0.681033+0.323086im   0.756258+0.454746im
0.704345+0.603132im  0.0489073+0.595638im

julia> sum(X.^2)
0.5044940375620476 + 2.035762251252853im

julia> sum(abs2, X)
2.563956079161798

julia> sum(x -> x^2, X)
0.5044940375620476 + 2.035762251252853im
``````

Yeah, that’s sort of what I meant by saying that it also handles complex values. But if the OP actually wants to sum squared complex values, my suggestion won’t fly.

what about `LinearAlgerbra.norm`?

I don’t think you need `eltype` here, btw.

1 Like