Is there half an iteration of the QR algorithm? Every real square matrix $M$ has a QR decomposition $M = QR$ where $Q^{-1}=Q^T$ and $R$ is an upper triangular matrix with non-negative reals on the diagonal. Call the function $f(QR)=RQ$ the Francis function. The Francis function is continuous over invertible matrices, and is well-behaved in a certain way over PSD matrices (see my video series; I'm not sure how to describe it formally). Is there a functional square root of the Francis function over PSD matrices? My visualisations suggest that there could be one.
 A: Look for the Toda flow; that should do exactly what you want.
A: Let $L_{n\times n}=L(t)$ being a real or complex matrix, with $t$ being a real number. You can define $$B(t)=L^*(t)_+-L(t)_-,$$ in which $L_+$ is the upper triangular part, and $L_-$ is the low triangular part of the matrix $L$, without the diagonal. It follows that $B^*=-B$.
Let us consider the function $U(t)$ as the solution to the initial value problem
$$\frac{d\,U}{dt}=BU,\qquad U(0)=I,$$ in wich
$I$ denotes the identity  $n\times n$ matrix.
This means that
$$\frac{d\,U^*U}{dt}=\left(\frac{d\,U}{dt}\right)^*U+U^*\frac{d\,U}{dt}=(BU)^*U+U^*BU=-U^*BU+U^*BU=0,$$ and follows that $U^*(t)U(t)=U^*(0)U(0)=I$, that is, $U(t)$ and $Q(t)=U^*(t)$ are bounded orthogonal matrix to each $t\in \mathbb{R}$.
It follows that the initial value problem
$$\frac{d\,L}{dt}=BL-LB,\qquad L(0)=L_0,$$ has a unique solution $$L(t)=Q^*(t)L_0Q(t).$$
Lemma: If $L_0=L_0^*$, then $L(t)$ converges to a diagonal matrix, containing each eigenvalues of $L_0$,  as $|t|\to \infty$.
Now, if you let $$B=B(t)=\left[G(L(t))^*\right]_+-G(L(t))_-,$$ to some  analytical function $G(z)$, to  $z$ in an open set of $\mathbb{C}$, containig the eigenvalues of $L_0$. It follows that $B^*=-B$, and that
$$M(t)=G(L(t))=U(t)G(L_0)U^*(t)$$ is the solution to the initial value problem $$ \frac{d\,M}{dt}=BM-MB,\qquad M(0)=G(L_0).$$
Lemma: If $L_0=L_0^*$ and $G(z)$ is real and one to one. Then  $e^{tG(L_0)}=Q(t)R(t)$, in which $R(t)$ is upper triangular with positive diagonal.
Theorem: $e^{G(L(m))}$ is the m-th iteration  to the $QR$  iteration to $e^{G(L_0)}$. In particular, if $L_0=\pm A^*A$, we can choose $G(z)=\ln(\pm z)$.
You can find details in the paper "Nanda, T., Differential Equations and the QR Algorithm, SIAM Journal on Numerical Analysis
Vol. 22, No. 2, 310--321, 1985".
You can find related results searching for "(
\frac{dX}{dt} ,=, B(X)X-XB(X), \quad X(0),=, A,
) " and for "Toda Flow" on SearchOnMath.
