It’s obvious that [Givens rotation](https://en.wikipedia.org/wiki/Givens_rotation) works better with sparse matrices. But I don’t know why [Householder reflection](https://en.wikipedia.org/wiki/Householder_transformation) is better for dense matrices. Does it require less computations? Or it’s numerically more stable than Givens rotation?


([Original post](https://math.stackexchange.com/questions/1597360/why-householder-reflection-is-better-that-givens-rotation-in-dense-linear-algebr))



Why Householder reflection is better than Givens rotation in dense linear algebra?
Implementing the QR factorization with Householder rotations is cheaper ($2n^2m$ vs $3n^2m$ for a $m\times n$ matrix), and equally accurate in practice. See Section 19.6 of Higham's *Accuracy and Stability of Numerical Algorithms*, or Golub-Van Loan for more explicit algorithms.


Moreover, in a Householder-based implementation there is a higher fraction of [level-2 BLAS](https://en.wikipedia.org/wiki/Basic_Linear_Algebra_Subprograms#Level_2) operations vs level 1, which makes them easier to optimize on a real-world computer.


