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Dynamics of the symmetric eigenvalue problem with shift strategies

A common algorithm for the computation of eigenvalues of real symmetric tridiagonal matrices is the iteration of certain special maps $F_σ$ called shifted $QR$ steps. Such maps preserve spectrum and a natural common domain is ${\cal T}_Λ$, the manifold of real symmetric tridiagonal matrices conjugate to the diagonal matrix $Λ$. More precisely, a (generic) shift $s \in \RR$ defines a map $F_s: {\cal T}_Λ\to {\cal T}_Λ$. A strategy $σ: {\cal T}_Λ\to \RR$ specifies the shift to be applied at $T$ so that $F_σ(T) = F_{σ(T)}(T)$. 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, $|b(F_σ(T))| = Θ(|b(T)|^k)$ 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.