# givens: Givens Rotation In pracma: Practical Numerical Math Functions

## Description

Givens Rotations and QR decomposition

## Usage

 `1` ```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.

`householder`

## Examples

 ```1 2 3 4 5 6 7``` ```## 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) zapsmall(Q %*% R) givens(magic(5)) ```

### Example output

```     [,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
[,1] [,2] [,3]
[1,]    0   -4    2
[2,]    6   -3   -2
[3,]    8    1   -1
\$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
```

pracma documentation built on Dec. 11, 2021, 9:57 a.m.