Pseudozeros of polynomials and pseudospectra of companion matrices

Summary. It is well known that the zeros of a polynomial are equal to the eigenvalues of the associated companion matrix . In this paper we take a geometric view of the conditioning of these two problems and of the stability of algorithms for polynomial zerofinding. The is the set of zeros of all polynomials obtained by coefficientwise perturbations of of size ; this is a subset of the complex plane considered earlier by Mosier, and is bounded by a certain generalized lemniscate. The is another subset of defined as the set of eigenvalues of matrices with ; it is bounded by a level curve of the resolvent of $A$. We find that if $A$ is first balanced in the usual EISPACK sense, then and are usually quite close to one another. It follows that the Matlab ROOTS algorithm of balancing the companion matrix, then computing its eigenvalues, is a stable algorithm for polynomial zerofinding. Experimental comparisons with the Jenkins-Traub (IMSL) and Madsen-Reid (Harwell) Fortran codes confirm that these three algorithms have roughly similar stability properties. Received June 15, 1993     

Behavioral measures and their correlation with IPM iteration counts on semi-definite programming problems

We study four measures of problem instance behavior that might account for the observed differences in interior-point method (IPM) iterations when these methods are used to solve semidefinite programming (SDP) problem instances: (i) an aggregate geometry measure related to the primal and dual feasible regions (aspect ratios) and norms of the optimal solutions, (ii) the (Renegar-) condition measure C(d) of the data instance, (iii) a measure of the near-absence of strict complementarity of the optimal solution, and (iv) the level of degeneracy of the optimal solution. We compute these measures for the SDPLIB suite problem instances and measure the sample correlation (CORR) between these measures and IPM iteration counts (solved using the software SDPT3) when these measures have finite values. Our conclusions are roughly as follows: the aggregate geometry measure is highly correlated with IPM iterations (CORR = 0.901), and provides a very good explanation of IPM iterations, particularly for problem instances with solutions of small norm and aspect ratio. The condition measure C(d) is also correlated with IPM iterations, but less so than the aggregate geometry measure (CORR = 0.630). The near-absence of strict complementarity is weakly correlated with IPM iterations (CORR = 0.423). The level of degeneracy of the optimal solution is essentially uncorrelated with IPM iterations. This research has been partially supported through the MIT-Singapore Alliance.     

Iridium‐ and Osmium‐decorated Reduced Graphenes as Promising Catalysts for Hydrogen Evolution

Renewable energy sources are highly sought after as a result of numerous worldwide problems concerning the environment and the shortage of energy. Currently, the focus in the field is on the development of catalysts that are able to provide water splitting catalysis and energy storage for the hydrogen evolution reaction (HER). While platinum is an excellent material for HER catalysis, it is costly and rare. In this work, we investigated the electrocatalytic abilities of various graphene–metal hybrids to replace platinum for the HER. The graphene materials were doped with 4f metals, namely, iridium, osmium, platinum and rhenium, as well as 3d metals, namely, cobalt, iron and manganese. We discovered that a few hybrids, in particular iridium‐ and osmium‐doped graphenes, have the potential to become competent electrocatalysts owing to their low costs and—more importantly—to their promising electrochemical performances towards the HER. One of the more noteworthy observations of this work is the superiority of these two hybrids over MoS₂, a well‐known electrocatalyst for the HER.     

A block coordinate gradient descent method for regularized convex separable optimization and covariance selection

We consider a class of unconstrained nonsmooth convex optimization problems, in which the objective function is the sum of a convex smooth function on an open subset of matrices and a separable convex function on a set of matrices. This problem includes the covariance selection problem that can be expressed as an ℓ ₁-penalized maximum likelihood estimation problem. In this paper, we propose a block coordinate gradient descent method (abbreviated as BCGD) for solving this class of nonsmooth separable problems with the coordinate block chosen by a Gauss-Seidel rule. The method is simple, highly parallelizable, and suited for large-scale problems. We establish global convergence and, under a local Lipschizian error bound assumption, linear rate of convergence for this method. For the covariance selection problem, the method can terminate in O(n³/e){O(n^3/\epsilon)} iterations with an e{\epsilon}-optimal solution. We compare the performance of the BCGD method with the first-order methods proposed by Lu (SIAM J Optim 19:1807–1827, 2009; SIAM J Matrix Anal Appl 31:2000–2016, 2010) for solving the covariance selection problem on randomly generated instances. Our numerical experience suggests that the BCGD method can be efficient for large-scale covariance selection problems with constraints.     

Gold-catalyzed cycloisomerizations of 1-(2-(tosylamino)phenyl)prop-2-yn-1-ols to 1H-indole-2-carbaldehydes and (E)-2-(iodomethylene)indolin-3-ols

A method to prepare 1H-indole-2-carbaldehydes and (E)-2-(iodomethylene)indolin-3-ols by gold(I)-catalyzed cycloisomerization of 1-(2-(tosylamino)phenyl)prop-2-yn-1-ols with N-iodosuccinimide (NIS) is reported. The reactions were shown to be operationally simplistic and proceed efficiently for a wide variety of substrates, affording the corresponding products in good to excellent yields (70-99%). The mechanism is suggested to involve activation of the alkyne moiety of the substrate by the gold(I) catalyst. This triggers intramolecular addition of the tethered aniline moiety to give a vinyl gold intermediate, which undergoes iododeauration with NIS to give the (E)-2-(iodomethylene)indolin-3-ol adduct. Subsequent 1,3-allylic alcohol isomerization (1,3-AAI) followed by formylation of this vinyl iodide intermediate then gives the 1H-indole-2-carbaldehyde.     

169Re的转动带结构研究

利用在束γ谱学方法,通过反应¹⁴⁴Sm(²⁸Si,1p3n)¹⁶⁹Re研究了¹⁶⁹Re的激发态能级结构.实验进行了X-γ符合、γ-γ符合、DCO系数和带内B(M1)/B(E2)比率测量.基于这些测量,建立了组态为π9/2^－[514]的强耦合带和组态为π1/2^－[541]的退耦合带.通过比较¹⁶⁹Re的转动带与邻近奇质子核已知转动带的结构和B(M1)/B(E2)比率,指定了¹⁶⁹Re转动带的组态.实验观测到π9/2^－[514]和π1/2^－[541]转动带的中子AB带交叉的转动频率分别为0.23和0.27MeV.着重讨论了¹⁶⁹Re转动带的中子AB带交叉频率、转动角动量顺排和旋称劈列等,并讨论了奇ARe核转动带结构的系统性     

Globally and Quadratically Convergent Algorithm for Minimizing the Sum of Euclidean Norms

For the problem of minimizing the sum of Euclidean norms (MSN), most existing quadratically convergent algorithms require a strict complementarity assumption. However, this assumption is not satisfied for a number of MSN problems. In this paper, we present a globally and quadratically convergent algorithm for the MSN problem. In particular, the quadratic convergence result is obtained without assuming strict complementarity. Examples without strictly complementary solutions are given to show that our algorithm can indeed achieve quadratic convergence. Preliminary numerical results are reported.