implemented Tkhonov regularized least-squares using SVD

src/least_squares.jl

+"""
+Calculates the Tikhonov regularized estimate min. ||Ax-b|| + λ||x|| using SVD
+for a vector with different values of λ
+
+`x, normE, normX = dsvd(A,U,S,V,b,λ)`
+
+`x, normE, normX = dsvd(A,b,λ)`
+"""
+function dsvd(A,U,S,V,b,λ)
+    if minimum(λ) < 0
+        error("Illegal regularization parameter λ")
+    end
+    UTb     = U'b
+    N       = size(U,1)
+    n       = size(V,1)
+    nl      = length(λ)
+    x       = zeros(n,nl)
+    normE   = zeros(nl)
+    normX   = zeros(nl)
+    print("dsvd: ")
+    for (i,λi) in enumerate(λ)
+        x[:,i] = V*diagm(S./(S.^2+λi))*UTb
+        normX[i] = norm(x[:,i])
+        normE[i] = norm(A*x[:,i] - b)
+        print("[$(round(i/length(λ),2)),$(round(λi,2))]  ")
+    end
+    println("Done ")
+    return x, normE, normX
+end
+
+function dsvd(A,b,λ)
+    if minimum(λ) < 0
+        error("Illegal regularization parameter λ")
+    end
+    U,S,V = svd(A)
+    dsvd(A,U,S,V,b,λ)
+end
+
+function Lcurve(normE, normX, λ)
+    plot(normE,normX,xscale=:log10,yscale=:log10,m=:o)
+    annotations = [(normE[i],normX[i],"λ=$(round(λ[i],4))") for i in 1:length(λ)]
+    annotate!(annotations)
+    xlabel!("RMSE"); ylabel!("||k||"); title!("L-curve")
+end