sparse_matvec.jl

n = 2_500_000
p = 1.9e-5
X = sprand(n,n,p)
v = sprand(n,8/n)

function fast_A_mul_B(X::SparseMatrixCSC{TX,IX},v::SparseVector{TV,IV}) where {TX,IX,TV,IV}
    X.n == v.n || throw(DimensionMismatch())
    
    ind_out = promote_type(IX,IV)[]
    val_out = promote_type(TX,TV)[]
    
    for (vi,vv) in zip(v.nzind, v.nzval)
        lo_ptr = X.colptr[vi]
        hi_ptr = X.colptr[vi+1] - 1
        append!(ind_out, X.rowval[lo_ptr:hi_ptr])
        append!(val_out, X.nzval[lo_ptr:hi_ptr] * vv)
    end
    sparsevec(ind_out, val_out, X.m)
end

@time X*v # 0.018 seconds
@time fast_A_mul_B(X,v) # 0.000_051 seconds