Add a standard LU performance benchmark

Same as SciML/SciMLBenchmarks.jl#623

Author: ChrisRackauckas

benchmarks/lu.jl

@@ -0,0 +1,51 @@
+using BenchmarkTools, Random, VectorizationBase
+using LinearAlgebra, LinearSolve, MKL_jll
+nc = min(Int(VectorizationBase.num_cores()), Threads.nthreads())
+BLAS.set_num_threads(nc)
+BenchmarkTools.DEFAULT_PARAMETERS.seconds = 0.5
+
+function luflop(m, n = m; innerflop = 2)
+    sum(1:min(m, n)) do k
+        invflop = 1
+        scaleflop = isempty((k + 1):m) ? 0 : sum((k + 1):m)
+        updateflop = isempty((k + 1):n) ? 0 :
+                     sum((k + 1):n) do j
+            isempty((k + 1):m) ? 0 : sum((k + 1):m) do i
+                innerflop
+            end
+        end
+        invflop + scaleflop + updateflop
+    end
+end
+
+algs = [LUFactorization(), GenericLUFactorization(), RFLUFactorization(), MKLLUFactorization(), FastLUFactorization(), SimpleLUFactorization()]
+res = [Float64[] for i in 1:length(algs)]
+
+ns = 4:8:500
+for i in 1:length(ns)
+    n = ns[i]
+    @info "$n × $n"
+    rng = MersenneTwister(123)
+    global A = rand(rng, n, n)
+    global b = rand(rng, n)
+    global u0= rand(rng, n)
+    
+    for j in 1:length(algs)
+        bt = @belapsed solve(prob, $(algs[j])).u setup=(prob = LinearProblem(copy(A), copy(b); u0 = copy(u0), alias_A=true, alias_b=true))
+        push!(res[j], luflop(n) / bt / 1e9)
+    end
+end
+
+using Plots
+__parameterless_type(T) = Base.typename(T).wrapper
+parameterless_type(x) = __parameterless_type(typeof(x))
+parameterless_type(::Type{T}) where {T} = __parameterless_type(T)
+
+p = plot(ns, res[1]; ylabel = "GFLOPs", xlabel = "N", title = "GFLOPs for NxN LU Factorization", label = string(Symbol(parameterless_type(algs[1]))), legend=:outertopright)
+for i in 2:length(res)
+    plot!(p, ns, res[i]; label = string(Symbol(parameterless_type(algs[i]))))
+end
+p
+
+savefig("lubench.png")
+savefig("lubench.pdf")