On implementations of the complex QR decomposition

Question

I have an mathematical theorem on the QR decomposition, which relies on the QR decomposition of an invertible square complex matrix always constructing a triangular matrix with real diagonal entries.

While at least in Octave this seems to be true, I wonder this can be relied upon in practice. I am looking for an implementation that does not produce such a triangular factor.

PS: Actually the theorem in question is the correctness of the double shifted QR iteration, as described in section 3.5 of http://people.inf.ethz.ch/arbenz/ewp/Lnotes/chapter3.pdf . The presentation seems to rely on a particular implementation of the QR algorithm.

Answer 1

A slight generalization of a Householder reflector, as seen in LAPACK's zlarfg, can be used to define a sequence of unitary transformations. In particular, the $j$'th transformation zeros the portion of the matrix below the $j$'th diagonal entry, and also ensures that the $j$'th diagonal entry becomes real. The unblocked algorithm zgeqr2 and the blocked algorithm zgeqrf make use of this approach.

On the other hand, suppose that you have an algorithm that provides $A=QR$, where $R$ is triangular but with complex diagonal entries. Then define the diagonal matrix $\Omega$ to have diagonal entries drawn from the unit circle such that $\Omega R$ has real diagonal entries, which would imply that the decomposition $A=(Q \bar \Omega)(\Omega R)$ is a QR decomposition with your desired properties.