Various Ways for Least Square Estimation

> library(RColorBrewer)
> cols <- brewer.pal(4, "Set1")
> A <- matrix(data = 0, nrow = 50, ncol = 12)
> x.t <- seq(0, 1, length.out = 50)
> for (i in 1:12) {
+     A[, i] <- x.t^(i - 1)
+ }
> b <- cos(4 * x.t)

> Q <- qr.Q(qr(A))
> b_prime <- t(Q) %*% b
> R <- qr.R(qr(A))
> x <- solve(R, b_prime)
> x1 <- x
> plot(10^8 * (A %*% x1 - cos(4 * x.t)))

> U <- svd(A)$u
> d <- diag(svd(A)$d)
> V <- svd(A)$v
> b_prime <- t(U) %*% b
> x <- V %*% solve(d, b_prime)
> x2 <- x
> plot(10^8 * (A %*% x2 - cos(4 * x.t)))

> cbind(x1, x2)
                        [,1]                   [,2]
 [1,]  1.000000000996605e+00  1.000000000996604e+00
 [2,] -4.227428691202565e-07 -4.227430102352229e-07
 [3,] -7.999981235689559e+00 -7.999981235685077e+00
 [4,] -3.187631876893836e-04 -3.187632430210450e-04
 [5,]  1.066943079560576e+01  1.066943079597327e+01
 [6,] -1.382028678074381e-02 -1.382028825931538e-02
 [7,] -5.647075630590243e+00 -5.647075626774281e+00
 [8,] -7.531601862032665e-02 -7.531602508485076e-02
 [9,]  1.693606956958508e+00  1.693606964113136e+00
[10,]  6.032113428498480e-03  6.032108446974022e-03
[11,] -3.742417053440816e-01 -3.742417033639870e-01
[12,]  8.804057640437810e-02  8.804057606180213e-02