QR decomposition
In this crude form the iterations are relatively expensive. This can be mitigated by first bringing the matrix "A" to upper Hessenberg form (which costs formula_2 arithmetic operations using a technique based on Householder reduction), with a finite sequence of orthogonal similarity transforms, somewhat like a two-sided QR decomposition. (For QR decomposition, the Householder reflectors are multiplied only on the left, but for the Hessenberg case they are multiplied on both left and right.) Determining the QR decomposition of an upper Hessenberg matrix costs formula_3 arithmetic operations. Moreover, because the Hessenberg form is already nearly upper-triangular (it has just one nonzero entry below each diagonal), using it as a starting point reduces the number of steps required for convergence of the QR algorithm.