Givens Rotation

Givens Rotations and QR decomposition

givens(A)

Arguments

A

numeric square matrix.

Details

givens(A) returns a QR decomposition (or factorization) of the square matrix A by applying unitary 2-by-2 matrices U such that U * [xk;xl] = [x,0] where x=sqrt(xk^2+xl^2)

Value

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

References

Golub, G. H., and Ch. F. van Loan (1996). Matrix Computations. Third edition, John Hopkins University Press, Baltimore.

See also

householder

Examples

##  QR decomposition
A <- matrix(c(0,-4,2, 6,-3,-2, 8,1,-1), 3, 3, byrow=TRUE)
gv <- givens(A)
(Q <- gv$Q); (R <- gv$R)
#>      [,1]  [,2]  [,3]
#> [1,]  0.0 -0.80  0.60
#> [2,]  0.6 -0.48 -0.64
#> [3,]  0.8  0.36  0.48
#>      [,1] [,2] [,3]
#> [1,]   10   -1   -2
#> [2,]    0    5   -1
#> [3,]    0    0    2
zapsmall(Q %*% R)
#>      [,1] [,2] [,3]
#> [1,]    0   -4    2
#> [2,]    6   -3   -2
#> [3,]    8    1   -1

givens(magic(5))
#> $Q
#>           [,1]       [,2]        [,3]        [,4]        [,5]
#> [1,] 0.5233867  0.5057542 -0.67346976  0.12154153  0.04410384
#> [2,] 0.7081114 -0.6965734  0.01772741 -0.08154163  0.08000231
#> [3,] 0.1231498  0.1367418  0.35575062  0.63074410  0.66463461
#> [4,] 0.3078745  0.1910574  0.41223097  0.42467160 -0.72002083
#> [5,] 0.3386620  0.4514387  0.49963121 -0.63276736  0.17744103
#> 
#> $R
#>          [,1]     [,2]     [,3]      [,4]      [,5]
#> [1,] 32.48076 26.63115 21.39728 23.706339 25.861461
#> [2,]  0.00000 19.89427 12.32344  1.943926  4.085579
#> [3,]  0.00000  0.00000 24.39855 11.631551  3.741480
#> [4,]  0.00000  0.00000  0.00000 20.098200  9.973936
#> [5,]  0.00000  0.00000  0.00000  0.000000 16.000463
#>