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 This preprint borrows heavily from the unpublished preprint arXiv:0912.3376 but is adapted for a different audienc

Increased cell efficiency in InGaAs thin film solar cells with dielectric and metal back reflectors

Compound single junction and multijunction solar cells enable very high photovoltaic efficiencies by virtue of employing different band gap materials in seriesconnected tandem cells to access the full solar spectrum. Researchers focused on improving the electrical properties of solar cells by optimizing the material growth conditions, however relatively little work to date has been devoted to light trapping and enhanced absorption in III-V compound solar cells using back reflectors. We studied absorption enhancement in InGaAs and InGaAsP thin film solar cells by means of numerical modeling. Flat dielectric and metal back reflectors that might be introduced into the solar cell via wafer-bonding, epitaxial lift-off or deposition techniques have been shown to increase the short circuit current and the photovoltaic efficiency of solar cells