diff --git a/project/testing.ipynb b/project/testing.ipynb
index ac57874..c6e6b41 100644
--- a/project/testing.ipynb
+++ b/project/testing.ipynb
@@ -14,16 +14,23 @@
   {
    "cell_type": "code",
    "execution_count": 2,
-   "id": "426735f0-859a-4f7d-83cf-f52f464a2737",
+   "id": "afd34800-7b9a-4006-9849-071fa197d962",
    "metadata": {},
    "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "x = [5.810770900369532e-16, 0.4999999999999994]\n"
+     ]
+    },
     {
      "data": {
       "text/plain": [
        "3-element Vector{Float64}:\n",
-       "  2.8498844747360086\n",
-       " -2.404529848146403\n",
-       " -0.3104749426792832"
+       " 0.9999999999999993\n",
+       " 1.9999999999999993\n",
+       " 2.999999999999999"
       ]
      },
      "execution_count": 2,
@@ -32,24 +39,26 @@
     }
    ],
    "source": [
-    "A = [1. 2; 3 4; 5 6]\n",
-    "A = qrhous!(A)\n",
-    "A * [1., 2, 3]"
+    "using LinearAlgebra: \\, qr\n",
+    "A = [1. 2.; 3. 4.; 5. 6.]\n",
+    "(Q, R) = qr(A)\n",
+    "x = A \\ [1., 2, 3]\n",
+    "@show x\n",
+    "A * x"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 3,
-   "id": "afd34800-7b9a-4006-9849-071fa197d962",
+   "id": "8b3d71f8-2f4f-44f0-8c97-dd9d78706163",
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "3-element Vector{Float64}:\n",
-       "  2.8498844747360064\n",
-       " -2.404529848146404\n",
-       " -0.31047494267928366"
+       "2-element Vector{Float64}:\n",
+       " -1.5012955407352218e-16\n",
+       "  0.5000000000000001"
       ]
      },
      "execution_count": 3,
@@ -58,10 +67,34 @@
     }
    ],
    "source": [
-    "using LinearAlgebra: qr!\n",
-    "A = [1. 2.; 3. 4.; 5. 6.]\n",
-    "(Q, R) = qr!(A)\n",
-    "Q * [1., 2, 3]"
+    "A = [1. 2; 3. 4.; 5. 6.]\n",
+    "A = qrhous!(A)\n",
+    "b = [1., 2, 3]\n",
+    "x = A \\ b"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "1883485c-b026-4a8a-8c34-b7c74ad79868",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "3-element Vector{Float64}:\n",
+       " 0.9999999999999991\n",
+       " 1.9999999999999996\n",
+       " 2.9999999999999996"
+      ]
+     },
+     "execution_count": 4,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "A * x"
    ]
   }
  ],
diff --git a/project/thinQR.jl b/project/thinQR.jl
index 3df71ec..31be4dd 100644
--- a/project/thinQR.jl
+++ b/project/thinQR.jl
@@ -1,9 +1,9 @@
 module thinQR
 
-import Base: size, show, getproperty, getfield, propertynames, *
-using LinearAlgebra: norm, I, triu, diagm
+import Base: size, show, getproperty, getfield, propertynames, \, *
+using LinearAlgebra: norm, I, triu, diagm, ldiv!
 
-export QRthin, qrhous!, qyhous
+export QRthin, qrhous!, qyhous, qyhoust
 
 mutable struct QRthin{T <: Real}
     A::AbstractVecOrMat{T}
@@ -43,6 +43,15 @@ function qyhous(A::QRthin{T}, y::AbstractArray{T}) where T
     return z
 end
 
+function qyhoust(A::QRthin{T}, y::AbstractArray{T}) where T
+    m, n = size(A.A)
+    z = deepcopy(y)
+    for j ∈ 1:n
+        z[j:m] = z[j:m] - A.A[j:m, j] * (A.A[j:m, j]' * z[j:m])
+    end
+    return z
+end
+
 function calculateQ(A::QRthin{T}) where T
     if A.AQ != nothing
         return A.AQ
@@ -61,7 +70,7 @@ function calculateR(A::QRthin{T}) where T
         return A.AR
     end
     m, n = size(A.A)
-    A.AR = triu(A.A[1:n, :], 1) + diagm(A.d)
+    A.AR = vcat(triu(A.A[1:n, :], 1) + diagm(A.d), zeros(m-n, n))
     return A.AR
 end
 
@@ -87,8 +96,18 @@ Base.propertynames(A::QRthin, private::Bool=false) = (:R, :Q, (private ? fieldna
 
 Base.size(A::QRthin) = size(getfield(A, :A))
 
-function *(A::QRthin{T}, b::AbstractVector{T}) where {T}
-    return qyhous(A, b)
+function (\)(A::QRthin{T}, b::AbstractVector{T}) where T
+    n, m = size(A)
+    v = qyhoust(A, b)
+    x = zeros(m)
+    for j ∈ m:-1:1
+        x[j] = (v[j] - x[j+1:m]' * A.A[j, j+1:m]) * A.d[j]^-1
+    end
+    return x
+end
+
+function (*)(A::QRthin{T}, x::AbstractVecOrMat{T}) where T
+    return qyhous(A, (A.R * x))
 end
 
 end # module
\ No newline at end of file