Rational Matrix Research Articles

Realizations of Infinite Products, Ruelle Operators and Wavelet Filters

Using the system theory notion of state-space realization of matrix-valued rational functions, we describe the Ruelle operator associated with wavelet filters. The resulting realization of infinite products of rational functions have the following four features: (1) It is defined in an infinite-dimensional complex domain. (2) Starting with a realization of a single rational matrix-function $$M$$ , we show that a resulting infinite product realization obtained from $$M$$ takes the form of an (infinite-dimensional) Toeplitz operator with the symbol that is a reflection of the initial realization for $$M$$ . (3) Starting with a subclass of rational matrix functions, including scalar-valued ones corresponding to low-pass wavelet filters, we obtain the corresponding infinite products that realize the Fourier transforms of generators of $$\mathbf L_2(\mathbb R)$$ wavelets. (4) We use both the realizations for $$M$$ and the corresponding infinite product to obtain a matrix representation of the Ruelle-transfer operators used in wavelet theory. By “matrix representation” we refer to the slanted (and sparse) matrix which realizes the Ruelle-transfer operator under consideration.

Rational self-affine tiles

An integral self-affine tile is the solution of a set equation A T = ⋃ d ∈ D ( T + d ) \mathbf {A} \mathcal {T} = \bigcup _{d \in \mathcal {D}} (\mathcal {T} + d) , where A \mathbf {A} is an n × n n \times n integer matrix and D \mathcal {D} is a finite subset of Z n \mathbb {Z}^n . In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices A ∈ Q n × n \mathbf {A} \in \mathbb {Q}^{n \times n} . We define rational self-affine tiles as compact subsets of the open subring R n × ∏ p K p \mathbb {R}^n\times \prod _\mathfrak {p} K_\mathfrak {p} of the adèle ring A K \mathbb {A}_K , where the factors of the (finite) product are certain p \mathfrak {p} -adic completions of a number field K K that is defined in terms of the characteristic polynomial of A \mathbf {A} . Employing methods from classical algebraic number theory, Fourier analysis in number fields, and results on zero sets of transfer operators, we establish a general tiling theorem for these tiles. We also associate a second kind of tile with a rational matrix. These tiles are defined as the intersection of a (translation of a) rational self-affine tile with R n × ∏ p { 0 } ≃ R n \mathbb {R}^n \times \prod _\mathfrak {p} \{0\} \simeq \mathbb {R}^n . Although these intersection tiles have a complicated structure and are no longer self-affine, we are able to prove a tiling theorem for these tiles as well. For particular choices of the digit set D \mathcal {D} , intersection tiles are instances of tiles defined in terms of shift radix systems and canonical number systems. This enables us to gain new results for tilings associated with numeration systems.

Finite and infinite structures of rational matrices: a local approach

The structure of a rational matrix is given by its Smith-McMillan invariants. Some properties of the Smith-McMillan invariants of rational matrices with elements in different principal ideal domains are presented: In the ring of polynomials in one indeterminate (global structure), in the local ring at an irreducible polynomial (local structure), and in the ring of proper rational functions (infinite structure). Furthermore, the change of the finite (global and local) and infinite structures is studied when performing a Mobius transformation on a rational matrix. The results are applied to define an equivalence relation in the set of polynomial matrices, with no restriction on size, for which a complete system of invariants are the finite and infinite elementary divisors.

Spectral Clustering with Local Projection Distance Measurement

Constructing a rational affinity matrix is crucial for spectral clustering. In this paper, a novel spectral clustering via local projection distance measure (LPDM) is proposed. In this method, the Local-Projection-Neighborhood (LPN) is defined, which is a region between a pair of data, and other data in the LPN are projected onto the straight line among the data pairs. Utilizing the Euclidean distance between projective points, the local spatial structure of data can be well detected to measure the similarity of objects. Then the affinity matrix can be obtained by using a new similarity measurement, which can squeeze or widen the projective distance with the different spatial structure of data. Experimental results show that the LPDM algorithm can obtain desirable results with high performance on synthetic datasets, real-world datasets, and images.

Compact Rational Krylov Methods for Nonlinear Eigenvalue Problems

We propose a new uniform framework of compact rational Krylov (CORK) methods for solving large-scale nonlinear eigenvalue problems $A(\lambda) x = 0$. For many years, linearizations were used for solving polynomial and rational eigenvalue problems. On the other hand, for the general nonlinear case, $A(\lambda)$ can first be approximated by a (rational) matrix polynomial and then a convenient linearization is used. However, the major disadvantage of linearization-based methods is the growing memory and orthogonalization costs with the iteration count, i.e., in general they are proportional to the degree of the polynomial. Therefore, the CORK family of rational Krylov methods exploits the structure of the linearization pencils by using a generalization of the compact Arnoldi decomposition. In this way, the extra memory and orthogonalization costs due to the linearization of the original eigenvalue problem are negligible for large-scale problems. Furthermore, we prove that each CORK step breaks down into an orthogonalization step of the original problem dimension and a rational Krylov step on small matrices. We also briefly discuss implicit restarting of the CORK method and how to exploit low rank structure. The CORK method is illustrated with two large-scale examples.

Criterion of Approximation for Designing 2 × 2 Feedback Systems with Inputs Satisfying Bounding Conditions

A common practice in designing a feedback system with a non-rational transfer matrix is to replace the non-rational matrix with an appropriate rational approximant during the design process so that reliable and effcient computational tools for rational systems can be utilized. Consequently, a criterion of approximation is required to ensure that the controller obtained from the approximant still provides satisfactory results for the original system. This paper derives such a criterion for the case of two-input two-output feedback systems in which the design objective is to ensure that the errors and the controller outputs always stay within prescribed bounds whenever the inputs satisfy certain bounding conditions. For a given rational approximant matrix, the criterion is expressed as a set of inequalities that can be solved in practice. It will be seen that the criterion generalizes a known result for single-input single-output systems. Finally, a controller for a binary distillation column is designed by using the criterion in conjunction with the method of inequalities.

Kalman–Yakubovič–Popov lemma for descriptor systems

For a given descriptor realization of para-hermitian rational matrix Π(s), we present a generalization of Kalman–Yakubovič–Popov lemma, i.e. necessary and sufficient conditions for Π≥0 on the imaginary axis, in terms of an inequality with constant matrices. The result is quite general, since Π can have poles and zeros on the extended imaginary axis, hence the nonstrict inequality Π(jω)≥0, ω∈R can hold, instead of the strict inequality. Π can be singular. The descriptor realization is required to be only impulse controllable and controllable (or stabilizable and detΠ≠0). A spectral factorization of Π is given, by the above mentioned constant matrices. Three consequences of the generalized KYP lemma, and an illustrative numerical example are given.

High Temperature Cylinder Sleeve Design Research Self-Lubricating Materials

Cylinder Liner design innovation goal is energy conservation. As the main friction parts engine cylinder liner, reducing friction energy is energy saving basic requirements. Materials research cylinder sleeve is one of the main cylinder liner saving research. Through the piston ring and cylinder liner surface friction dual material hard phase, self-lubricating phase, toughening phase analysis, select compatibility, high strength, heat resistance, good high temperature performance, ease of manufacture, price rational matrix material and an appropriate proportion of nanoscale hard material, self-lubricating materials, ductile materials, using appropriate methods cladding and processing, design developed high-temperature self-lubricating cylinder sets of advanced materials. The latest international high temperature, wear-resistant, self-lubricating materials research: modern nanoα-Al2O3+Ni-base alloy composite materials, Ti2B/Fe metal-ceramic composites, metals and ceramics NiCr-Cr3C2 particles CaF2 self-lubricating composite alloy powder material, NiCr/Cr3C2-WS2 self-lubricating wear-resistant materials were studied and found to TiC, Al2O3 is hard reinforcing phase, CaF self-lubricating phase, NiCr/TiC eutectic toughening phase, high-temperature self-lubricating wear-resistant nanocomposite, and Cr18Ni9 class alloy as base material, by laser cladding method enables conventional cylinder liners manufacturing technological breakthroughs, has practical value.

Main Q-eigenvalues and generalized Q-cospectrality of graphs

Let QG denote the signless Laplacian matrix of a graph G. An eigenvalue μ of QG is said to be a main Q-eigenvalue of G if μ has an eigenvector which is not orthogonal to an all-ones vector e. We give some basic properties of main Q-eigenvalues. For a graph G of order n, G is called Q-controllable if G has n distinct main Q-eigenvalues. We show that a graph H is generalized Q-cospectral with a Q-controllable G if and only if H is Q-controllable and there exists a unique rational orthogonal matrix R such that Re = e, QH = R⊤QGR.

On Solvents of Matrix Polynomials

This paper is concerned with the construction of right and left solvents (also called block roots) of a matrix polynomial from latent roots and vectors. It addresses also the important case of the existence of a complete set of block roots. Solvents do not always exist, so conditions for the existence of such solvents are discussed. The inverse of a matrix polynomial is obtained as a particular case of the block partial fraction expansion of a related rational matrix. It involves the knowledge of a complete set of solvents and the computation of the inverse of a block Vandermonde matrix. Numerical examples are given to illustrate the two results.

Sign patterns of rational matrices with large rank

Let A be a real matrix. The term rank of A is the smallest number t of lines (that is, rows or columns) needed to cover all the nonzero entries of A. We prove a conjecture of Li et al. stating that, if the rank of A exceeds t−3, there is a rational matrix with the same sign pattern and rank as those of A. We point out a connection of the problem discussed with the Kapranov rank function of tropical matrices, and we show that the statement fails to hold in general if the rank of A does not exceed t−3.

Singular Degree of a Rational Matrix Pseudodifferential Operator

In our previous work we studied minimal fractional decompositions of a rational matrix pseudodifferential operator: H=A/B, where A and B are matrix differential operators, and B is non-degenerate of minimal possible degree deg(B). In the present paper we introduce the singular degree sdeg(H)=deg(B), and show that for an arbitrary rational expression H=sum_a (A^a_1)/(B^a_1)...(A^a_n)/(B^a_n), we have that sdeg(H) is less than or equal to sum_{a,i} deg(B^a_i). If the equality holds, we call such an expression minimal. We study the properties of the singular degree and of minimal rational expressions. These results are important for the computations involved in the Lenard-Magri scheme of integrability.

State Space Formulas for a Suboptimal Rational Leech Problem I: Maximum Entropy Solution

For the strictly positive case (the suboptimal case) the maximum entropy solution X to the Leech problem G(z)X(z) = K(z) and \({\|X\|_\infty={\rm sup}_{|z| \leq 1}\| X(z )\| \leq 1}\), with G and K stable rational matrix functions, is proved to be a stable rational matrix function. An explicit state space realization for X is given, and \({\| X \|_\infty}\) turns out to be strictly less than one. The matrices involved in this realization are computed from the matrices appearing in a state space realization of the data functions G and K. A formula for the entropy of X is also given.

Survey and Review

The Survey and Review article in this issue is “The Exponentially Convergent Trapezoidal Rule,” by Nick Trefethen and André Weideman. It deals with a fundamental and classical issue in numerical analysis---approximating an integral. By focusing on one deceptively simple method, the authors are able to combine a lively, historical overview with an insightful, up-to-date coverage of recent results. We learn about the work of Euler, Gauss, Maclaurin, and Poisson, and some of the 20th century pioneers of numerical analysis. We also learn about high-precision computation, treatment of singularities, choice of contours to integrate along, Gauss and Clenshaw--Curtis quadrature, Padé approximation, and a “quadrature formula that doesn't even integrate constants right” (see footnote number 15). The first half of the article looks at the mathematics behind the well-known geometric convergence of the trapezoidal rule. Here the two fundamental routes to a proof are (a) exploiting Taylor series and aliasing, and (b) introducing a function that has simple poles at the summation points in order to apply residue calculus. The second half deals with applications where the trapezoidal rule has proved useful, including contour integration, rational approximation, problems with endpoint singularities, inverse Laplace transforms, integral equations, zero finding, special functions, and the numerical solution of parabolic PDEs. Readers familiar with the work of the authors will not be surprised to learn that this article deftly illustrates the power of complex analysis, even for studying problems that appear to live on the real line. The survey also uses the trapezoidal rule as a means to connect ideas in rational approximation, Laplace transforms, Cauchy integrals, sampling theory, interpolation theory, and matrix functions. As the authors make clear, there are many alternatives to the humble trapezoidal rule, but the method deserves attention because of - its ease of use; - its powerful convergence properties, which can be understood through relatively straightforward analysis; and - its utility in generating explicit approximation formulas. Further, the method is shown to be an excellent vehicle for demonstrating deep connections between a wide range of research topics in applied and computational mathematics.

Autogenerator-based modelling framework for development of strategic games simulations: rational pigs game extended.

When considering strategic games from the conceptual perspective that focuses on the questions of participants' decision-making rationality, the very issues of modelling and simulation are rarely discussed. The well-known Rational Pigs matrix game has been relatively intensively analyzed in terms of reassessment of the logic of two players involved in asymmetric situations as gluttons that differ significantly by their attributes. This paper presents a successful attempt of using autogenerator for creating the framework of the game, including the predefined scenarios and corresponding payoffs. Autogenerator offers flexibility concerning the specification of game parameters, which consist of variations in the number of simultaneous players and their features and game objects and their attributes as well as some general game characteristics. In the proposed approach the model of autogenerator was upgraded so as to enable program specification updates. For the purpose of treatment of more complex strategic scenarios, we created the Rational Pigs Game Extended (RPGE), in which the introduction of a third glutton entails significant structural changes. In addition, due to the existence of particular attributes of the new player, “the tramp,” one equilibrium point from the original game is destabilized which has an influence on the decision-making of rational players.

An inversion-free method for finding positive definite solution of a rational matrix equation.

A new iterative scheme has been constructed for finding minimal solution of a rational matrix equation of the form X + A*X −1 A = I. The new method is inversion-free per computing step. The convergence of the method has been studied and tested via numerical experiments.

FEAST As A Subspace Iteration Eigensolver Accelerated By Approximate Spectral Projection

The calculation of a segment of eigenvalues and their corresponding eigenvectors of a Hermitian matrix or matrix pencil has many applications. A new density-matrix-based algorithm has been proposed recently and a software package FEAST has been developed. The density-matrix approach allows FEAST's implementation to exploit a key strength of modern computer architectures, namely, multiple levels of parallelism. Consequently, the software package has been well received, especially in the electronic structure community. Nevertheless, theoretical analysis of FEAST has lagged. For instance, the FEAST algorithm has not been proven to converge. This paper offers a detailed numerical analysis of FEAST. In particular, we show that the FEAST algorithm can be understood as an accelerated subspace iteration algorithm in conjunction with the Rayleigh--Ritz procedure. The novelty of FEAST lies in its accelerator, which is a rational matrix function that approximates the spectral projector onto the eigenspace in question. Analysis of the numerical nature of this approximate spectral projector and the resulting subspaces generated in the FEAST algorithm establishes the algorithm's convergence. This paper shows that FEAST is resilient against rounding errors and establishes properties that can be leveraged to enhance the algorithm's robustness. Finally, we propose an extension of FEAST to handle non-Hermitian problems and suggest some future research directions.

Combinatorics of Matrix Factorizations and Integrable Systems

We study relations between the eigenvectors of rational matrix functions on the Riemann sphere. Our main result is that for a subclass of functions that are products of two elementary blocks it is possible to represent these relations in a combinatorial–geometric way using a diagram of a cube. In this representation, vertices of the cube represent eigenvectors, edges are labeled by differences of locations of zeroes and poles of the determinant of our matrix function, and each face corresponds to a particular choice of a coordinate system on the space of such functions. Moreover, for each face this labeling encodes, in a neat and efficient way, a generating function for the expressions of the remaining four eigenvectors that label the opposing face of the cube in terms of the coordinates represented by the chosen face. The main motivation behind this work is that when our matrix is a Lax matrix of a discrete integrable system, such generating functions can be interpreted as Lagrangians of the system, and a choice of a particular face corresponds to a choice of the direction of the motion.

Generalized Spectral Characterization of Graphs Revisited

A graph $G$ is said to be determined by its generalized spectrum (DGS for short) if for any graph $H$, $H$ and $G$ are cospectral with cospectral complements implies that $H$ is isomorphic to $G$. Wang and Xu (2006) gave some methods for determining whether a family of graphs are DGS. In this paper, we shall review some of the old results and present some new ones along this line of research.More precisely, let $A$ be the adjacency matrix of a graph $G$, and let $W=[e,Ae,\cdots,A^{n-1}e]$ ($e$ is the all-one vector) be its walk-matrix. Denote by $\mathcal{G}_n$ the set of all graphs on $n$ vertices with $\det(W)\neq 0$. We define a large family of graphs $$\mathcal{F}_n=\{G\in{\mathcal{G}_n}|\frac{\det(W)}{2^{\lfloorn/2\rfloor}}\mbox{is square-free and }2^{\lfloorn/2\rfloor+1}\not|\det(W)\}$$ (which may have positive density among all graphs, as suggested by some numerical experiments). The main result of the paper shows that for any graph $G\in {\mathcal{F}_n}$, if there is a rational orthogonal matrix $Q$ with $Qe=e$ such that $Q^TAQ$ is a (0,1)-matrix, then $2Q$ must be an integral matrix (and hence, $Q$ has well-known structures). As a consequence, we get the conclusion that almost all graphs in $\mathcal{F}_n$ are DGS.

4-Phenyl-α-cyanocinnamic Acid Amide: Screening for a Negative Ion Matrix for MALDI-MS Imaging of Multiple Lipid Classes

Matrix-assisted laser desorption/ionization imaging mass spectrometry (MALDI-IMS) has become a method of choice in lipid analysis, as it provides localization information for defined lipids that is not readily accessible with nonmass spectrometric methods. Most current MALDI matrices have been found empirically. Nevertheless, preferential matrix properties for many analyte classes are poorly understood and may differ between lipid classes. We used rational matrix design and semiautomated screening for the discovery of new matrices suitable for MALDI-IMS of lipids. Utilizing Smartbeam- and nitrogen lasers for MALDI, we systematically compared doubly substituted α-cyanocinnamic acid derivatives (R(1)-CCA-R(2)) with respect to their ability to serve as negative ion matrix for various brain lipids. We identified 4-phenyl-α-cyanocinnamic acid amide (Ph-CCA-NH2) as a novel negative ion matrix that enables analysis and imaging of various lipid classes by MALDI-MS. We demonstrate that Ph-CCA-NH2 displays superior sensitivity and reproducibility compared to matrices commonly employed for lipids. A relatively small number of background peaks and good matrix suppression effect could make Ph-CCA-NH2 a widely applicable tool for lipid analysis.