Choosing Commercial Real Estate for Business According to the Technology of the Hierarchy Analysis Method

The problem of choosing commercial real estate for business depending on the market situation is considered on the example of a city (region). A method for analyzing hierarchies is proposed, which serves to justify decision-making in conditions of cer-tainty and multi-criteria. The stages of the hierarchy analysis method are considered. criteria for evaluating commercial real estate have been developed. The technology of constructing pairwise comparison matrices is shown. Conclusions from the matrix anal-ysis are presented.

Convergence and Preconditioning of Inexact Inverse Subspace Iteration for Generalized Eigenvalue Problems

This paper focuses on the inner iteration that arises in inexact inverse subspace iteration for computing a small deflating subspace of a large matrix pencil. First, it is shown that the method achieves linear rate of convergence if the inner iteration is performed with increasing accuracy. Then, as inner iteration, block-GMRES is used with preconditioners generalizing the one by Robbé, Sadkane and Spence [Inexact inverse subspace iteration with preconditioning applied to non-Hermitian eigenvalue problems, SIAM J. Matrix Anal. Appl. 31 2009, 1, 92–113]. It is shown that the preconditioners help to maintain the number of iterations needed by block-GMRES to approximately a small constant. The efficiency of the preconditioners is illustrated by numerical examples.

Rayleigh-Ritz Majorization Error Bounds for the Linear Response Eigenvalue Problem

In the linear response eigenvalue problem arising from computational quantum chemistry and physics, one needs to compute a few of smallest positive eigenvalues together with the corresponding eigenvectors. For such a task, most of efficient algorithms are based on an important notion that is the so-called pair of deflating subspaces. If a pair of deflating subspaces is at hand, the computed approximated eigenvalues are partial eigenvalues of the linear response eigenvalue problem. In the case the pair of deflating subspaces is not available, only approximate one, in a recent paper [SIAM J. Matrix Anal. Appl., 35(2), pp.765-782, 2014], Zhang, Xue and Li obtained the relationships between the accuracy in eigenvalue approximations and the distances from the exact deflating subspaces to their approximate ones. In this paper, we establish majorization type results for these relationships. From our majorization results, various bounds are readily available to estimate how accurate the approximate eigenvalues based on information on the approximate accuracy of a pair of approximate deflating subspaces. These results will provide theoretical foundations for assessing the relative performance of certain iterative methods in the linear response eigenvalue problem.

A concise proof to the spectral and nuclear norm bounds through tensor partitions

On estimations of the lower and upper bounds for the spectral and nuclear norm of a tensor, Li established neat bounds for the two norms based on regular tensor partitions, and proposed a conjecture for the same bounds to be hold based on general tensor partitions [Z. Li, Bounds on the spectral norm and the nuclear norm of a tensor based on tensor partition, SIAM J. Matrix Anal. Appl., 37 (2016), pp. 1440-1452]. Later, Chen and Li provided a solution to the conjecture [Chen B., Li Z., On the tensor spectral p-norm and its dual norm via partitions]. In this short paper, we present a concise and different proof for the validity of the conjecture, which also offers a new and simpler proof to the bounds of the spectral and nuclear norms established by Li for regular tensor partitions. Two numerical examples are provided to illustrate tightness of these bounds.

Cone-constrained rational eigenvalue problems

This work deals with the eigenvalue analysis of a rational matrix-valued function subject to complementarity constraints induced by a polyhedral cone $K$. The eigenvalue problem under consideration has the general structure \[ \left(\sum_{k=0}^d \lambda^k A_k + \sum_{k =1}^m \frac{p_k(\lambda)}{q_k(\lambda)} \,B_k\right) x = y , \quad K\ni x \perp y\in K^\ast, \] where $K^\ast$ denotes the dual cone of $K$. The unconstrained version of this problem has been discussed in [Y.F. Su and Z.J. Bai. Solving rational eigenvalue problems via linearization. \emph{SIAM J. Matrix Anal. Appl.}, 32:201--216, 2011.] with special emphasis on the implementation of linearization-based methods. The cone-constrained case can be handled by combining Su and Bai's linearization approach and the so-called facial reduction technique. In essence, this technique consists in solving one unconstrained rational eigenvalue problem for each face of the polyhedral cone $K$.

Convergences of the squareroot approximation scheme to the Fokker–Planck operator

We study the qualitative convergence behavior of a novel FV-discretization scheme of the Fokker–Planck equation, the squareroot approximation scheme (SQRA), that recently was proposed by Lie, Fackeldey and Weber [A square root approximation of transition rates for a markov state model, SIAM J. Matrix Anal. Appl. 34 (2013) 738–756] in the context of conformation dynamics. We show that SQRA has a natural gradient structure and that solutions to the SQRA equation converge to solutions of the Fokker–Planck equation using a discrete notion of G-convergence for the underlying discrete elliptic operator. The SQRA does not need to account for the volumes of cells and interfaces and is tailored for high-dimensional spaces. However, based on FV-discretizations of the Laplacian it can also be used in lower dimensions taking into account the volumes of the cells. As an example, in the special case of stationary Voronoi tessellations, we use stochastic two-scale convergence to prove that this setting satisfies the G-convergence property.

Joint spectral radius, Sturmian measures and the finiteness conjecture

The joint spectral radius of a pair of $2\times 2$ real matrices $(A_{0},A_{1})\in M_{2}(\mathbb{R})^{2}$ is defined to be $r(A_{0},A_{1})=\limsup _{n\rightarrow \infty }\max \{\Vert A_{i_{1}}\cdots A_{i_{n}}\Vert ^{1/n}:i_{j}\in \{0,1\}\}$, the optimal growth rate of the norm of products of these matrices. The Lagarias–Wang finiteness conjecture [Lagarias and Wang. The finiteness conjecture for the generalized spectral radius of a set of matrices. Linear Algebra Appl.214 (1995), 17–42], asserting that $r(A_{0},A_{1})$ is always the $n$th root of the spectral radius of some length-$n$ product $A_{i_{1}}\cdots A_{i_{n}}$, has been refuted by Bousch and Mairesse [Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture. J. Amer. Math. Soc.15 (2002), 77–111], with subsequent counterexamples presented by Blondel et al [An elementary counterexample to the finiteness conjecture. SIAM J. Matrix Anal.24 (2003), 963–970], Kozyakin [A dynamical systems construction of a counterexample to the finiteness conjecture. Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference (Seville, Spain, December 2005). IEEE, Piscataway, NJ, pp. 2338–2343] and Hare et al [An explicit counterexample to the Lagarias–Wang finiteness conjecture. Adv. Math.226 (2011), 4667–4701]. In this article, we introduce a new approach to generating finiteness counterexamples, and use this to exhibit an open subset of $M_{2}(\mathbb{R})^{2}$ with the property that each member $(A_{0},A_{1})$ of the subset generates uncountably many counterexamples of the form $(A_{0},tA_{1})$. Our methods employ ergodic theory; in particular, the analysis of Sturmian invariant measures. This approach allows a short proof that the relationship between the parameter $t$ and the Sturmian parameter ${\mathcal{P}}(t)$ is a devil’s staircase.

Principal Pivot Transforms of Quasidefinite Matrices and Semidefinite Lagrangian Subspaces

Lagrangian subspaces are linear subspaces that appear naturally in control theory applications, and especially in the context of algebraic Riccati equations. We introduce a class of semidefinite Lagrangian subspaces and show that these subspaces can be represented by a subset I ⊆ {1, 2, . . . , n} and a Hermitian matrix X ∈ C n×n with the property that the submatrix X II is negative semidefinite and the submatrix X I c I c is positive semidefinite. A matrix X with these definiteness properties is called I-semidefinite and it is a generalization of a quasidefinite matrix. Under mild hypotheses which hold true in most applications, the Lagrangian subspace associated to the stabilizing solution of an algebraic Riccati equation is semidefinite, and in addition we show that there is a bijection between Hamiltonian and symplectic pencils and semidefinite Lagrangian subspaces; hence this structure is ubiquitous in control theory. The (symmetric) principal pivot transform (PPT) is a map used by Mehrmann and Poloni [SIAM J. Matrix Anal. Appl., 33(2012), pp. 780–805] to convert between two different pairs (I, X) and (J , X 0 ) representing the same Lagrangian subspace. For a semidefinite Lagrangian subspace, we prove that the symmetric PPT of an I-semidefinite matrix X is a J -semidefinite matrix X 0 , and we derive an implementation of the transformation X 7→ X 0 that both makes use of the definiteness properties of X and guarantees the definiteness of the submatrices of X 0 in finite arithmetic. We use the resulting formulas to obtain a semidefiniteness-preserving version of an optimization algorithm introduced by Mehrmann and Poloni to compute a pair (I opt , X opt ) with M = max i,j |(X opt ) ij | as small as possible. Using semidefiniteness allows one to obtain a stronger inequality on M with respect to the general case.

On some previous results for the Drazin inverse of block matrices

This short paper is motivated by the paper of Bu et al. [C. Bu, C. Feng, P. Dong, A note on computational formulas for the Drazin inverse of certain block matrices, J. Appl. Math. Comput.(38) (2012) 631-640], where the authors gave additive formula for Drazin inverse for matrices under new conditions, and two representations under some specific conditions. Here is shown that the additive formula is not valid for all matrices which satisfy given conditions. Also, here is proved that the representations which were given in mentioned paper do not extend the results given by Hartwig et al. [R. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of a 2 _ 2 block matrix, SIAM J. Matrix. Anal. Appl. (27)(2006) 757-771 ], in fact they are equivalent.