Rational Matrix Research Articles

Robust determination of the chemical potential in the pole expansion and selected inversion method for solving Kohn-Sham density functional theory.

Fermi operator expansion (FOE) methods are powerful alternatives to diagonalization type methods for solving Kohn-Sham density functional theory (KSDFT). One example is the pole expansion and selected inversion (PEXSI) method, which approximates the Fermi operator by rational matrix functions and reduces the computational complexity to at most quadratic scaling for solving KSDFT. Unlike diagonalization type methods, the chemical potential often cannot be directly read off from the result of a single step of evaluation of the Fermi operator. Hence multiple evaluations are needed to be sequentially performed to compute the chemical potential to ensure the correct number of electrons within a given tolerance. This hinders the performance of FOE methods in practice. In this paper, we develop an efficient and robust strategy to determine the chemical potential in the context of the PEXSI method. The main idea of the new method is not to find the exact chemical potential at each self-consistent-field (SCF) iteration but to dynamically and rigorously update the upper and lower bounds for the true chemical potential, so that the chemical potential reaches its convergence along the SCF iteration. Instead of evaluating the Fermi operator for multiple times sequentially, our method uses a two-level strategy that evaluates the Fermi operators in parallel. In the regime of full parallelization, the wall clock time of each SCF iteration is always close to the time for one single evaluation of the Fermi operator, even when the initial guess is far away from the converged solution. We demonstrate the effectiveness of the new method using examples with metallic and insulating characters, as well as results from ab initio molecular dynamics.

New Condition for FD, All Filters, and New KYP Lemma

AbstractThis paper presents a necessary and sufficient condition for the fault detection (FD) problem, under a quite general and natural assumption. The condition is actually a constant inertia on the extended imaginary axis (EIA) of a para‐hermitian rational matrix (RM). An observer solution is given. Another two main results are the presentation of all problem solutions, and a necessary and sufficient condition for constant inertia on the EIA or in a given frequency region, for a class of para‐hermitian RMs, in terms of linear matrix inequalities. The latter result can be regarded as a generalization of the Kalman‐Yakubovich‐Popov (KYP) lemma.

Stability of the procedure of explicit recovery of skew-selfadjoint Dirac systems from rational Weyl matrix functions

Procedures to recover explicitly discrete and continuous skew-selfadjoint Dirac systems on semi-axis from rational Weyl matrix functions are considered. Their stability is shown. Some new facts on asymptotics of pseudo-exponential potentials (i.e., of explicit solutions of inverse problems) are proved as well. The GBDT version of Bäcklund–Darboux transformation, methods from system theory and results on algebraic Riccati equations are used for this purpose.

Representation and Factorization of Discrete-Time Rational All-Pass Functions

In this paper, we obtain a general characterization of discrete-time all-pass rational matrix functions from state-space representations. We establish a general characterization of the solutions of LMI's and Riccati equations. Finally, we derive a complete factorization theory of all-pass functions. Our results are obtained in the most general setting, without introducing any ad hoc assumption and can be applied to a variety of problems such as the discrete-time counterpart of the $H^{\infty}$ model reduction problems solved by Glover in continuous-time.

Alternative Aerodynamic Uncertainty Modeling Approaches for Flutter Reliability Analysis

A newly developed Monte Carlo simulation-based computational capability for uncertain aeroelastic and aeroservoelastic systems is used to study flutter prediction statistics, accounting for aerodynamic and structural uncertainties. In the aerodynamics area an element-by-element aerodynamic influence coefficients based uncertainty scheme is introduced and is used to study the uncertainty of flutter predictions in the case of the historical AGARD 445.6 wing-test configuration at a sequence of transonic Mach numbers. Exploring the possibility that in an aerodynamic influence coefficient-based flutter predictions uncertainty is strongly linked to the Mach number zones in which corresponding aerodynamic boxes reside, the relative importance to the flutter velocity and frequency uncertainties of contribution of uncertainties associated with defined Mach number zones on the wing is ranked by using the method of global sensitivity analysis. An alternative approach to aerodynamic uncertainty modeling, focusing on rational function approximation matrix uncertainties, allows examination of the relative impact of uncertainty in aerodynamic stiffness, damping, inertia, or lag terms. The two approaches complement each other regarding the insights they provide into the nature of unsteady aerodynamic uncertainty. With the inclusion of structural uncertainties, the contributions of structural and aerodynamic variation to the uncertainties in flutter predictions for the AGARD 445.6 wing are compared and quantitatively ranked.

Numeric Algorithms for Corank Two Edge-bipartite Graphs and their Mesh Geometries of Roots

Following a Coxeter spectral analysis problems for positive edge-bipartite graphs (signed multigraphs with a separation property) introduced in [SIAM J. Discr. Math. 27(2013), 827-854] and [Fund. Inform. 123(2013), 447-490], we study analogous problems for loop-free corank two edge-bipartite graphs Δ = (Δ0,Δ1). i.e. for edge-bipartite graphs Δ, with at least n = 3 vertices such that their rational symmetric Gram matrix GΔ ∈ 𝕄n(ℚ) is positive semi-definite of rank n – 2. We study such connected edge-bipartite graphs by means of the non-symmetric Gram matrix ĞΔ ∈ 𝕄n(ℤ), the Coxeter matrix CoxΔ := –ĞΔ · ĞΔ–tr, its complex spectrum speccΔ ⊆ ℂ, and an associated simply laced Dynkin diagram DynΔ, with n – 2 vertices. Here ℤ means the ring of integers. It is well-known that if Δ ≈ℤ Δ′ (i.e., there exists B ∈ 𝕄n(ℤ) such that det B = ±1 and ĞΔ′ = Btr · ĞΔ · B) then speccΔ = speccΔ′ and DynΔ = DynΔ′. A complete classification of connected non-negative loop-free edge-bipartite graphs Δ with at most six vertices of corank two, up to the ℤ-congruence Δ ≈ℤ Δ′, is also given. A complete list of representatives of the ℤ-congruence classes of all connected non-negative edge-bipartite graphs of corank two with with at most 6 vertices is constructed; it consists of 1, 2, 2 and 8 edge-bipartite graphs of corank two with 3, 4, 5 and 6 vertices, respectively.

Wiener–Hopf indices of unitary functions on the unit circle in terms of realizations and related results on Toeplitz operators

We provide new formulas for the Wiener–Hopf factorization indices of a rational matrix function R which has neither poles nor zeros on the unit circle. In addition, we recover recent results on the Fredholm characteristics of the Toeplitz operator with symbol R via the method of matricial coupling. Furthermore, we present an alternative formula for the index in terms of the Fourier coefficients of R.

Rational factorizations of completely positive matrices

In this note it is proved that every rational matrix which lies in the interior of the cone of completely positive matrices also has a rational cp-factorization.

On the solution of a rational matrix equation arising in G-networks

We consider the problem of solving a rational matrix equation arising in the solution of G-networks. We propose and analyze two numerical methods: a fixed point iteration and the Newton–Raphson method. The fixed point iteration is shown to be globally convergent with linear convergence rate, while the Newton method is shown to have a local convergence, with quadratic convergence rate. Numerical experiments show the effectiveness of the proposed methods.

Quaternionic Wiener Algebras, Factorization and Applications

We define an almost periodic extension of the Wiener algebras in the quaternionic setting and prove a Wiener-Levy type theorem for it, as well as extending the theorem to the matrix-valued case. We prove a Wiener-Hopf factorization theorem for the quaternionic matrix-valued Wiener algebras (discrete and continuous) and explore the connection to the Riemann-Hilbert problem in that setting. As applications, we characterize solvability of two classes of quaternionic functional equations and give an explicit formula for the canonical factorization of quaternionic rational matrix functions via realization.

Nonnegative Matrix Factorization Requires Irrationality

Nonnegative matrix factorization (NMF) is the problem of decomposing a given nonnegative $n \times m$ matrix $M$ into a product of a nonnegative $n \times d$ matrix $W$ and a nonnegative $d \times m$ matrix $H$. A longstanding open question, posed by Cohen and Rothblum in 1993, is whether a rational matrix $M$ always has an NMF of minimal inner dimension $d$ whose factors $W$ and $H$ are also rational. We answer this question negatively, by exhibiting a matrix for which $W$ and $H$ require irrational entries.

Design of non-rational MIMO feedback systems with inputs and outputs satisfying certain bounding conditions by using rational approximants

When a feedback system has components described by non-rational transfer functions, a standard practice in designing such a system is to replace the non-rational functions with rational approximants and then carry out the design with the approximants by means of a method that copes with rational systems. In order to ensure that the design carried out with the approximants still provides satisfactory results for the original system, a criterion of approximation should be explicitly taken into account in the design formulation. This paper derives such a criterion for multi-input multi-output (MIMO) feedback systems whose design objective is to ensure that the absolute values of every error and every controller output components always stay within prescribed bounds whenever the inputs satisfy certain bounding conditions. The obtained criterion generalizes a known result which was derived for single-input single-output (SISO) systems; furthermore, for a given rational approximant matrix, it is expressed as a set of inequalities that can be solved in practice. Finally, a controller for a binary distillation column is designed by using the criterion in conjunction with the method of inequalities. The numerical results clearly demonstrate that the usefulness of the criterion in obtaining a design solution for the original system.

BuApplication of Multidimensional Scaling (MDS) Structural Analysis: A Case Study on Dog Food Products

Objectives: In the modern marketing strategy, MDS (Multidimentional Scaling) can be an important core technique carving a company’s product or brand in the differentiated position in comparison with competitors. In order to extract efficient information from the data and to summarize these relationships among objects from the dissimilarity or similarity data, MDS is the typical method. MDS is one of the multivariate analysis methods in terms of statistical analysis method. Methods/Statistical Analysis: On this study, seven functional properties of dog foods are selected to apply MDS. The dog foods highly ranked in sales in Korean domestic dog food market are classified into five classes such as organic, holistic, super-premium, premium, and grocery. Five products per each class except of the premium class (four products on the premium class) are extracted. In order to visualize a two-dimensional positioning map, the statistical software (SPSS 22+) and 24 products are used. By using the Euclidean distance matrix, the analysis of 24 products is mapped on the twodimensional positioning map. According to a similarity result, the similarity at low values less than 1.0 turned up on the rational distance matrix. The products of the high similarity were a group of the organic class and the holistic class. And the grocery class also showed that domestic products mostly consisted of its class. Findings: The outcome of this research can be that the Korean brands’ super premium class with slightly high main nutrients can be viewed as having the highest competitiveness from the efficient preemption of market gap. Improvements/Applications: The results are applied to build a competitive brand positioning strategy to cope with the market changes in the domestic dog food market.

Recovery of eigenvectors of rational matrix functions from Fiedler-like linearizations

Linearization is a standard method often used when dealing with matrix polynomials. Recently, the concept of linearization has been extended to rational matrix functions and Fiedler-like matrix pencils for rational matrix functions have been constructed. A linearization L(λ) of a rational matrix function G(λ) does not necessarily guarantee a simple way of recovering eigenvectors of G(λ) from those of L(λ). We show that Fiedler-like pencils of G(λ) allow an easy operation free recovery of eigenvectors of G(λ), that is, eigenvectors of G(λ) are recovered from eigenvectors of Fiedler-like pencils of G(λ) without performing any arithmetic operations. We also consider Fiedler-like pencils of the Rosenbrock system polynomial S(λ) associated with an LTI system Σ in state-space form (SSF) and show that the Fiedler-like pencils allow operation free recovery of eigenvectors of S(λ). The eigenvectors of S(λ) are the invariant zero directions of the LTI system Σ.

Some Properties of a Class of RLCM Active Networks Over F(z)

In order to explore the properties depending only on structure such as structural controllability, the structured matrix (SM), and the rational function matrix (RFM) in multi-parameters (RFM) were used to describe the coefficient matrices of systems and networks. In this paper, Definitions 3.3 and 3.4 are given. On the basis of RLCM passive networks, some properties (separability, reducibility, and controllability) of a class of RLCM active networks over [Formula: see text] are studied. Some criteria are obtained. Since these criteria depend only on network structures, their applications are straightforward. Finally, two illustrative examples are given.

On rational matrix exact covering systems of [formula omitted] and its applications to Ramanujan's forty identities

For a nonsingular integer matrix B, the set of cosets of the quotient module Zn/BZn forms an exact covering system (ECS) of Zn. In this paper, we use the Smith normal form to obtain another type of matrix ECS with rational entries which we call rational matrix ECS. Using rational matrix ECS of Z2, we prove eight identities in Ramanujan's list of forty identities for the Rogers–Ramanujan functions, as well as some other identities.

Skew‐selfadjoint Dirac systems with rational rectangular Weyl functions: explicit solutions of direct and inverse problems and integrable wave equations

In this paper we study direct and inverse problems for discrete and continuous skew‐selfadjoint Dirac systems with rectangular (possibly non‐square) pseudo‐exponential potentials. For such a system the Weyl function is a strictly proper rational matrix function and any strictly proper rational matrix function appears in this way. In fact, extending earlier results, given a strictly proper rational matrix function we present an explicit procedure to recover the corresponding potential using techniques from mathematical system and control theory. We also introduce and study a nonlinear generalized discrete Heisenberg magnet model, extending earlier results for the isotropic case. A large part of the paper is devoted to the related discrete systems of which the pseudo‐exponential potential depends on an additional continuous time parameter. Our technique allows us to obtain explicit solutions for the generalized discrete Heisenberg magnet model and evolution of the Weyl functions.

Linearizations for Rational Matrix Functions and Rosenbrock System Polynomials

Our aim in this paper is twofold: First, for solving rational eigenproblems we introduce linearizations of rational matrix functions and propose a framework for their constructions via minimal realizations. We also introduce Fiedler-like pencils for rational matrix functions and show that the Fiedler-like pencils are linearizations of the rational matrix functions. Second, for computing zeros of a linear time-invariant system $\Sigma$ in state-space form, we introduce a trimmed structured linearization, which we refer to as a Rosenbrock linearization, of the Rosenbrock system polynomial $\mathcal{S}(\lambda)$ associated with $\Sigma.$ Also we introduce Fiedler-like pencils for $\mathcal{S}(\lambda),$ describe an algorithm for their constructions, and show that the Fiedler-like pencils are Rosenbrock linearizations of the system polynomial $\mathcal{S}(\lambda).$

New interpolation solution and application in system modeling and optimal control with prescribed distance to instability

SummaryWe present a system theoretic interpretation of a two‐sided interpolation problem with a stable rational matrix U (interpolant) without constraints on its norm. It is known that all solutions U of that problem can be expressed as U = Uh+Up, where Uh ranges in the set of all solutions of the associated homogeneous problem, and Up is a particular solution. We present a new solution for Up and prove that it is actually the minimal ‐norm interpolant in the set of all interpolants.We apply these results in system modeling and in optimal control of one‐block plants, with a prescribed bound on the distance to instability of the closed‐loop system. The applications are illustrated by examples. Interesting connections to the augmented basic interpolation problem, to Nehari's problem, and to the stability of one‐block plants with multiple unstable invariant zeros are given. Copyright © 2015 John Wiley & Sons, Ltd.

Barycentric Rational Interpolation Iteration Collocation Method for Solving Nonlinear Vibration Problems

In this article, a powerful computational methodology, named as barycentric rational interpolation iteration collocation method (BRICM), for obtaining the numerical solutions of nonlinear vibration problems is presented. The nonlinear vibration problems are governed by initial-value problems of nonlinear differential equations. Given an initial guess value of the unknown function, the nonlinear differential equations can be transformed into linear differential equations. By applying barycentric rational interpolation and differential matrix, the linearized differential equation is discretized into algebraic equations in the matrix form. The latest solution of nonlinear differential equation is obtained by solving the algebraic equations. The numerical solution of nonlinear vibration problem can be calculated by iteration method under given control precision. Then, the velocity and acceleration can be obtained by differential matrix of barycentric rational interpolation, and the period of nonlinear vibration is also computed by BRICM. Some examples of nonlinear vibration demonstrate the proposed methodological advantages of effectiveness, simple formulations, and high precision.