# householder: Householder Reflections

### Description

Householder reflections and QR decomposition

### Usage

 `1` ``` householder(A) ```

### Arguments

 `A` numeric matrix with `nrow(A)>=ncol(A)`.

### Details

The Householder method applies a succession of elementary unitary matrices to the left of matrix `A`. These matrices are the so-called Householder reflections.

### Value

List with two matrices `Q` and `R`, `Q` orthonormal and `R` upper triangular, such that `A=Q%*%R`.

### References

Trefethen, L. N., and D. Bau III. (1997). Numerical Linear Algebra. SIAM, Society for Industrial and Applied Mathematics, Philadelphia.

`givens`

### Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ```## 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)); x ```

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