  A common algorithm for the computation of eigenvalues of real symmetric
tridiagonal matrices is the iteration of certain special maps $F_\sigma$ called
shifted $QR$ steps. Such maps preserve spectrum and a natural common domain is
${\cal T}_\Lambda$, the manifold of real symmetric tridiagonal matrices
conjugate to the diagonal matrix $\Lambda$. More precisely, a (generic) shift
$s \in \RR$ defines a map $F_s: {\cal T}_\Lambda \to {\cal T}_\Lambda$. A
strategy $\sigma: {\cal T}_\Lambda \to \RR$ specifies the shift to be applied
at $T$ so that $F_\sigma(T) = F_{\sigma(T)}(T)$. Good shift strategies should
lead to fast deflation: some off-diagonal coordinate tends to zero, allowing
for reducing of the problem to submatrices. For topological reasons, continuous
shift strategies do not obtain fast deflation; many standard strategies are
indeed discontinuous. Practical implementation only gives rise systematically
to bottom deflation, convergence to zero of the lowest off-diagonal entry
$b(T)$. For most shift strategies, convergence to zero of $b(T)$ is cubic,
$|b(F_\sigma(T))| = \Theta(|b(T)|^k)$ for $k = 3$. The existence of arithmetic
progressions in the spectrum of $T$ sometimes implies instead quadratic
convergence, $k = 2$. The complete integrability of the Toda lattice and the
dynamics at non-smooth points are central to our discussion. The text does not
assume knowledge of numerical linear algebra.
