added LU decomposition. (TheAlgorithms#11)

* added project-rosalind folder and some starter

* added lu decomposition

matrix/lu_decompose.jl

@@ -0,0 +1,60 @@
+using LinearAlgebra # to test against the inbuilt function
+using Test
+
+"""
+Decomposes a `n x n` non singular matrix into a lower triangular matrix (L) and an upper triangular matrix (U)
+"""
+function lu_decompose(mat)
+	n = mat |> size |> first
+	L = zeros(n,n)
+	U = zeros(n,n)
+
+	for i in 1:n
+		for j in i:n
+			s = 0
+			for k in 1:i
+				s += L[i,k] * U[k,j]
+			end
+			U[i,j] = mat[i,j] - s
+		end
+
+		for k in i:n
+			if i == k
+				L[i,i] = 1
+			else
+				s = 0
+				for j in 1:i
+					s += L[k,j] * U[j,i]
+				end
+				L[k,i] = (mat[k,i] - s) / U[i,i]
+			end
+		end
+	end
+
+	return L,U
+end
+
+mat = [
+	2  -1 -2; 
+	-4   6  3; 
+	-4  -2  8
+]
+
+L = [
+	 1  0 0;    
+	-2  1 0;   
+	-2 -1 1
+]
+U = [
+	2 -1 -2;   
+	0  4 -1;    
+	0  0  3
+]
+
+@testset "LU decomposition" begin
+	@test lu_decompose(mat) == (L,U)
+end
+
+
+
+