QR decomposition methods for dense Hessenberg matrices

The QR decomposition of dense Hessenberg matrices arises as a part of one step of the QR algorithm. However, in modern variants of the QR algorithm, this problem is solved implicitly using the fact that the product RQ must be calculated later at the same step, again in an implicit manner. Depending on the chosen shift strategy, one uses the implicit scheme with either a single shift or a double shift. The former is based on the Givens method, while the latter is based on the Householder method.