Householder Reflections
Householder reflections and QR decomposition
householder(A)
A |
numeric matrix with |
The Householder method applies a succession of elementary unitary
matrices to the left of matrix A. These matrices are the so-called
Householder reflections.
List with two matrices Q and R, Q orthonormal and
R upper triangular, such that A=Q%*%R.
Trefethen, L. N., and D. Bau III. (1997). Numerical Linear Algebra. SIAM, Society for Industrial and Applied Mathematics, Philadelphia.
## QR decomposition
A <- matrix(c(0,-4,2, 6,-3,-2, 8,1,-1), 3, 3, byrow=TRUE)
S <- householder(A)
(Q <- S$Q); (R <- S$R)
Q %*% R # = A
## Solve an overdetermined linear system of equations
A <- matrix(c(1:8,7,4,2,3,4,2,2), ncol=3, byrow=TRUE)
S <- householder(A); Q <- S$Q; R <- S$R
m <- nrow(A); n <- ncol(A)
b <- rep(6, 5)
x <- numeric(n)
b <- t(Q) %*% b
x[n] <- b[n] / R[n, n]
for (k in (n-1):1)
x[k] <- (b[k] - R[k, (k+1):n] %*% x[(k+1):n]) / R[k, k]
qr.solve(A, rep(6, 5)); xPlease choose more modern alternatives, such as Google Chrome or Mozilla Firefox.