A note on spectrum distribution of constraint preconditioned generalized saddle point matrices

In this note, results concerning the eigenvalue distribution and form of the eigenvectors of the constraint preconditioned generalized saddle point matrix and its minimal polynomial are given. These results extend previous ones that appeared in the literature. Copyright © 2009 John Wiley & Sons, Ltd.     

Symmetric nonnegative realizability via partitioned majorization

A sufficient condition for symmetric nonnegative realizability of a spectrum is given in terms of (weak) majorization of a partition of the negative eigenvalues by a selection of the positive eigenvalues. If there are more than two positive eigenvalues, an additional condition, besides majorization, is needed on the partition. This generalizes observations of Suleǐmanova and Loewy about the cases of one and two positive eigenvalues, respectively. It may be used to provide insight into realizability of 5-element spectra and beyond.     

Implicit QR algorithms for palindromic and even eigenvalue problems

In the spirit of the Hamiltonian QR algorithm and other bidirectional chasing algorithms, a structure-preserving variant of the implicit QR algorithm for palindromic eigenvalue problems is proposed. This new palindromic QR algorithm is strongly backward stable and requires less operations than the standard QZ algorithm, but is restricted to matrix classes where a preliminary reduction to structured Hessenberg form can be performed. By an extension of the implicit Q theorem, the palindromic QR algorithm is shown to be equivalent to a previously developed explicit version. Also, the classical convergence theory for the QR algorithm can be extended to prove local quadratic convergence. We briefly demonstrate how even eigenvalue problems can be addressed by similar techniques. D. S. Watkins partly supported by Deutsche Forschungsgemeinschaft through Matheon, the DFG Research Center Mathematics for key technologies in Berlin.     

关于反中心对称矩阵的某些性质探讨

利用反中心对称矩阵的定义以及翻转矩阵等技巧,给出了反中心对称矩阵的伴随矩阵、特征值及特征向量的一些新结论.     

Connected signed graphs of fixed order,size, and number of negative edges with maximal index

In this paper we focus on connected signed graphs of fixed number of vertices, positive edges and negative edges that maximize the largest eigenvalue (also called the index) of their adjacency matrix. In the first step we determine these signed graphs in the set of signed generalized theta graphs. Concerning the general case, we use the eigenvector techniques for getting some structural properties of resulting signed graphs. In particular, we prove that positive edges induce nested split subgraphs, while negative edges induce double nested signed subgraphs. We observe that our concept can be applied when considering balancedness of signed graphs (the property that is extensively studied in both mathematical and non-mathematical context).     

Systematic Study of Selected Diagonalization Methods for Configuration Interaction Matrices

Several modifications to the Davidson algorithm are systematically explored to establish their performance for an assortment of configuration interaction (CI) computations. The combination of a generalized Davidson method, a periodic two‐vector subspace collapse, and a blocked Davidson approach for multiple roots is determined to retain the convergence characteristics of the full subspace method. This approach permits the efficient computation of wave functions for large‐scale CI matrices by eliminating the need to ever store more than three expansion vectors ( b _i) and associated matrix‐vector products ( σ _i), thereby dramatically reducing the I/O requirements relative to the full subspace scheme. The minimal‐storage, single‐vector method of Olsen is found to be a reasonable alternative for obtaining energies of well‐behaved systems to within μE_h accuracy, although it typically requires around 50% more iterations and at times is too inefficient to yield high accuracy (ca. 10^?10 E_h) for very large CI problems. Several approximations to the diagonal elements of the CI Hamiltonian matrix are found to allow simple on‐the‐fly computation of the preconditioning matrix, to maintain the spin symmetry of the determinant‐based wave function, and to preserve the convergence characteristics of the diagonalization procedure. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1574–1589, 2001     

An improved Hardy-Sobolev inequality and its application

For , a bounded domain, and for , we improve the Hardy-Sobolev inequality by adding a term with a singular weight of the type . We show that this weight function is optimal in the sense that the inequality fails for any other weight function more singular than this one. Moreover, we show that a series of finite terms can be added to improve the Hardy-Sobolev inequality, which answers a question of Brezis-Vazquez. Finally, we use this result to analyze the behaviour of the first eigenvalue of the operator as increases to for .

     

Accurate computation of the smallest eigenvalue of a diagonally dominant -matrix

If each off-diagonal entry and the sum of each row of a diagonally dominant -matrix are known to certain relative accuracy, then its smallest eigenvalue and the entries of its inverse are known to the same order relative accuracy independent of any condition numbers. In this paper, we devise algorithms that compute these quantities with relative errors in the magnitude of the machine precision. Rounding error analysis and numerical examples are presented to demonstrate the numerical behaviour of the algorithms.

     

Perturbation Analysis for the Eigenvalue Problem of a Formal Product of Matrices

We study the perturbation theory for the eigenvalue problem of a formal matrix product A ₁ ^{s 1} ··· A _p ^{s p}, where all A _k are square and s _k {–1, 1}. We generalize the classical perturbation results for matrices and matrix pencils to perturbation results for generalized deflating subspaces and eigenvalues of such formal matrix products. As an application we then extend the structured perturbation theory for the eigenvalue problem of Hamiltonian matrices to Hamiltonian/skew-Hamiltonian pencils.