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 $F_\sigma$ 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.Comment: 22 pages, 4 figures. This preprint borrows heavily from the unpublished preprint arXiv:0912.3376 but is adapted for a different audienc

An atlas for tridiagonal isospectral manifolds

Let ${\cal T}_\Lambda$ be the compact manifold of real symmetric tridiagonal matrices conjugate to a given diagonal matrix $\Lambda$ with simple spectrum. We introduce {\it bidiagonal coordinates}, charts defined on open dense domains forming an explicit atlas for ${\cal T}_\Lambda$. In contrast to the standard inverse variables, consisting of eigenvalues and norming constants, every matrix in ${\cal T}_\Lambda$ now lies in the interior of some chart domain. We provide examples of the convenience of these new coordinates for the study of asymptotics of isospectral dynamics, both for continuous and discrete time.Comment: Fixed typos; 16 pages, 3 figure

The Asymptotics of Wilkinson's Iteration: Loss of Cubic Convergence

One of the most widely used methods for eigenvalue computation is the $QR$ iteration with Wilkinson's shift: here the shift $s$ is the eigenvalue of the bottom $2\times 2$ principal minor closest to the corner entry. It has been a long-standing conjecture that the rate of convergence of the algorithm is cubic. In contrast, we show that there exist matrices for which the rate of convergence is strictly quadratic. More precisely, let $T_X$ be the $3 \times 3$ matrix having only two nonzero entries $(T_X)_{12} = (T_X)_{21} = 1$ and let $T_L$ be the set of real, symmetric tridiagonal matrices with the same spectrum as $T_X$. There exists a neighborhood $U \subset T_L$ of $T_X$ which is invariant under Wilkinson's shift strategy with the following properties. For $T_0 \in U$, the sequence of iterates $(T_k)$ exhibits either strictly quadratic or strictly cubic convergence to zero of the entry $(T_k)_{23}$. In fact, quadratic convergence occurs exactly when $\lim T_k = T_X$. Let $X$ be the union of such quadratically convergent sequences $(T_k)$: the set $X$ has Hausdorff dimension 1 and is a union of disjoint arcs $X^\sigma$ meeting at $T_X$, where $\sigma$ ranges over a Cantor set.Comment: 20 pages, 8 figures. Some passages rewritten for clarit

Linearizing Toda and SVD flows on large phase spaces of matrices with real spectrum

We consider different phase spaces for the Toda flows and the less familiar SVD flows. For the Toda flow, we handle symmetric and non-symmetric matrices with real simple eigenvalues, possibly with a given profile. Profiles encode, for example, band matrices and Hessenberg matrices. For the SVD flow, we assume simplicity of the singular values. In all cases, an open cover is constructed, as are corresponding charts to Euclidean space. The charts linearize the flows, converting it into a linear differential system with constant coefficients and diagonal matrix. A variant construction transform the flows into uniform straight line motion. Since limit points belong to the phase space, asymptotic behavior becomes a local issue. The constructions rely only on basic facts of linear algebra, making no use of symplectic geometry.Comment: 25 pages, 2 figure

Metal reduction in wine using PVI-PVP copolymer and its effects on chemical and sensory characters

We studied the influence of an adsorbent PVI-PVP resin (a copolymer of vinylimidazole and vinylpyrrolidone), on the removal of heavy metals in wines, mainly copper (Cu), iron (Fe), lead (Pb), cadmium (Cd) and aluminium (Al). The study also investigated the influence of PVI-PVP on the physical-chemical and sensory characteristics of white and red wines, comparing its effect when applied in the must and in the wine. The removal of metals was more effective when PVI-PVP was applied to the wine than to the must. The removal of Fe and Pb was more effective in white wines than in red wines, while the removal of Cu and Al was higher in red wines. In general, the higher the PVI-PVP dose, the greater the quantity of metallic elements (copper, iron, lead and aluminium) that are removed. PVI-PVP had a minor effect on phenolic composition. The wines showed some decrease in total acidity and an increase in pH with PVI-PVP. The application of PVI-PVP at the dose rates employed here did not affect the wine’s sensory characteristics significantly