Final answer:
The Classic Gram Schmidt (CGS) algorithm for QR factorization is implemented in R by creating a function that projects columns of an input matrix onto an orthogonal space and then normalizes them, sequentially building up the Q and R matrices of the QR factorization.
Step-by-step explanation:
Classic Gram Schmidt (CGS) in R
To implement the Classic Gram Schmidt (CGS) algorithm for QR factorization in R, you can follow this step-by-step process. Start by creating a function that takes a matrix as input. Then, initialize the matrices Q and R, which will store the orthogonal vectors and the upper triangular matrix, respectively. Loop through the columns of the input matrix, at each step projecting the current column onto the space spanned by the vectors in Q and subtracting this projection from the column to make it orthogonal to Q. Normalize the resulting vector and add it to Q. Finally, calculate the corresponding coefficients for the R matrix.
Here's a sample R code implementing CGS:
cgs <- function(A) {
n <- ncol(A)
Q <- matrix(0, nrow = nrow(A), ncol = n)
R <- matrix(0, nrow = n, ncol = n)
for (j in 1:n) {
v <- A[,j]
for (i in 1:(j-1)) {
R[i,j] <- sum(Q[,i] * A[,j])
v <- v - R[i,j] * Q[,i]
}
R[j,j] <- sqrt(sum(v^2))
Q[,j] <- v / R[j,j]
}
return(list(Q = Q, R = R))
}
After defining the function, you can apply it to a matrix to obtain the QR factorization.