Optimizing function to evaluate elementary symmetric polynomials

[abraemer]

Two quick observations:

@Gattu_Mytraya

    for i in axes(roots, 1)
        for j in min(i, b):-1:1
            dest[j+1, :] .+= roots[i, :] .* dest[j, :]
        end
    end

This allocates for root[i,:] and dest[j,:] because slices copy in Julia. The former is also constant for the inner loop. So try instead

    @views for i in axes(roots, 1)
        tmp = roots[i, :]
        for j in min(i, b):-1:1
            dest[j+1, :] .+= temp .* dest[j, :]
        end
    end

You order of indices should be suboptimal since Julia Arrays are column-major, which means you should iterate the left-most index in the inner most loop. In your code you have written a function that takes Matrix inputs but conceptually it is a operation on vectors. You kinda baked in a form of vectorization. I would consider this an antipattern in Julia. And note that this antipattern is responsible for the issue with the index order in this case.
So I would recommend you to either switch the indices on your matrices, i.e. transpose them effectively, or rewrite this function to take in Vectors instead and perform a broadcast over the different instances at a higher level in the code.
If you come from Matlab or similar: Note that in Julia you don’t need to vectorize computations for performances. A for loop has the same speed as broadcasting.

Also another comment on style: You don’t need to type annotate the function arguments. You can write this function completely generically, which makes it a lot easier to test difference number types you are also almost there already since you use zero and one. Just drop the type annotations from the signature and use eltype(dest) where you hardcoded the types in the body.