Rational Matrix Research Articles

An Approximate Transfer Function Model for a Double-Pipe Counter-Flow Heat Exchanger

The transfer functions G(s) for different types of heat exchangers obtained from their partial differential equations usually contain some irrational components which reflect quite well their spatio-temporal dynamic properties. However, such a relatively complex mathematical representation is often not suitable for various practical applications, and some kind of approximation of the original model would be more preferable. In this paper we discuss approximate rational transfer functions G^(s) for a typical thick-walled double-pipe heat exchanger operating in the counter-flow mode. Using the semi-analytical method of lines, we transform the original partial differential equations into a set of ordinary differential equations representing N spatial sections of the exchanger, where each nth section can be described by a simple rational transfer function matrix Gn(s), n=1,2,…,N. Their proper interconnection results in the overall approximation model expressed by a rational transfer function matrix G^(s) of high order. As compared to the previously analyzed approximation model for the double-pipe parallel-flow heat exchanger which took the form of a simple, cascade interconnection of the sections, here we obtain a different connection structure which requires the use of the so-called linear fractional transformation with the Redheffer star product. Based on the resulting rational transfer function matrix G^(s), the frequency and the steady-state responses of the approximate model are compared here with those obtained from the original irrational transfer function model G(s). The presented results show: (a) the advantage of the counter-flow regime over the parallel-flow one; (b) better approximation quality for the transfer function channels with dominating heat conduction effects, as compared to the channels characterized by the transport delay associated with the heat convection.

Matrix orthogonality in the plane versus scalar orthogonality in a Riemann surface

Abstract We consider a non-Hermitian matrix orthogonality on a contour in the complex plane. Given a diagonalizable and rational matrix valued weight, we show that the Christoffel–Darboux (CD) kernel, which is built in terms of matrix orthogonal polynomials, is equivalent to a scalar valued reproducing kernel of meromorphic functions in a Riemann surface. If this Riemann surface has genus $0$, then the matrix valued CD kernel is equivalent to a scalar reproducing kernel of polynomials in the plane. Interestingly, this scalar reproducing kernel is not necessarily a scalar CD kernel. As an application of our result, we show that the correlation kernel of certain doubly periodic lozenge tiling models admits a double contour integral representation involving only a scalar CD kernel. This simplifies a formula of Duits and Kuijlaars.

Structured strong linearizations of structured rational matrices

Structured rational matrices such as symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, and para-Hermitian rational matrices arise in many applications. Linearizations of rational matrices have been introduced recently for computing poles, eigenvalues, eigenvectors, minimal bases and minimal indices of rational matrices. For structured rational matrices, it is desirable to construct structure-preserving linearizations so as to preserve the symmetry in the eigenvalues and poles of the rational matrices. With a view to constructing structure-preserving linearizations of structured rational matrices, we propose a family of Fiedler-like pencils and show that the family of Fiedler-like pencils is a rich source of structure-preserving strong linearizations of structured rational matrices. We construct symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, skew-Hermitian, para-Hermitian and para-skew-Hermitian strong linearizations of a rational matrix when has the same structure. We also describe recovery of eigenvectors, minimal bases and minimal indices of from those of the linearizations of and show that the recovery is operation-free.

A Perron–Frobenius theorem for deciding matrix growth

Matrix interpretations are widely used in automated complexity analysis. Certifying such analyses boils down to determining the growth rate of An for a fixed non-negative rational matrix A. There exists a conceptually simple algorithm to determine the growth rate, but this algorithm has the disadvantage that it is based on algebraic number computations.In this work we present an even simpler algorithm to compute the growth rate. Its soundness is based on a variant of a Perron–Frobenius theorem that has been conjectured in earlier work. So far it only has been proven for small matrices, and here we present a proof for the general case.We further verify both the algorithm and the new Perron–Frobenius theorem in the proof assistant Isabelle/HOL, and integrate it into CeTA, a verified certifier for various properties, including complexity proofs. Because of the new results, CeTA no longer requires a verified implementation of algebraic numbers.

SIMETRISASI BENTUK KANONIK JORDAN

If the characteristic polynomial of a linear operator is completely factored in scalar field of then Jordan canonical form of can be converted to its rational canonical form of , and vice versa. If the characteristic polynomial of linear operator is not completely factored in the scalar field of ,then the rational canonical form of can still be obtained but not its Jordan canonical form matrix . In this case, the rational canonical form of can be converted to its Jordan canonical form by extending the scalar field of to Splitting Field of minimal polynomial of , thus forming the Jordan canonical form of over Splitting Field of . Conversely, converting the Jordan canonical form of over Splitting Field of to its rational canonical form uses symmetrization on the Jordan decomposition basis of so as to form a cyclic decomposition basis of which is then used to form the rational canonical matrix of

On minimal bases and indices of rational matrices and their linearizations

A complete theory of the relationship between the minimal bases and indices of rational matrices and those of their strong linearizations is presented. Such theory is based on establishing first the relationships between the minimal bases and indices of rational matrices and those of their polynomial system matrices under the classical minimality condition and certain additional conditions of properness. This is related to pioneering results obtained by Verghese, Van Dooren and Kailath in 1979-1980, which were the first proving results of this type. It is shown that the definitions of linearizations and strong linearizations do not guarantee any relationship between the minimal bases and indices of the linearizations and the rational matrices in general. In contrast, simple relationships are obtained for the family of strong block minimal bases linearizations, which can be used to compute minimal bases and indices of any rational matrix, including rectangular ones, via algorithms for pencils. These results extend the corresponding ones for other families of linearizations available in recent literature for square rational matrices.

FSID - A Frequency Weighted MIMO Frequency Domain Identification and Rational Matrix Approximation Method for Python, Julia and Matlab

An open source toolbox, FSID, implemented in the Python, Julia and MATLAB programming languages is described. The toolbox provides scripts which estimates linear multiinput multi-output state-space models from sample data using frequency-domain subspace algorithms. Algorithms which estimate models based on samples of the transfer function matrix as well as frequency domain input and output vectors are provided. The algorithms can be used for discrete-time models, continuous-time models as well as for approximation of rational matrices from samples corresponding to arbitrary points in the complex plane. The algorithms can handle frequency dependent weighting which enable to obtain approximate BLUE estimates. To reduce the computational complexity for the estimation algorithms, an accelerated algorithm is provided which evaluate the state-space realization of the transfer function matrix at arbitrary points.

Fast Randomized Non-Hermitian Eigensolvers Based on Rational Filtering and Matrix Partitioning

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 30 June 2020Accepted: 16 June 2021Published online: 13 September 2021Keywordsrational filtering, matrix partitioning, contour integral eigensolvers, non-Hermitian eigenvalue problems, randomized algorithmsAMS Subject Headings65F15, 15A18, 65F50Publication DataISSN (print): 1064-8275ISSN (online): 1095-7197Publisher: Society for Industrial and Applied MathematicsCODEN: sjoce3

Lignin as a MALDI matrix for small molecules: a proof of concept.

Driven by the interest in metabolomic studies and the progress of imaging techniques, small molecule analysis is booming, while it remains challenging to be realized by matrix-assisted laser desorption/ionization mass spectrometry (MALDI-MS). Herein, lignin, the second most abundant biomass in nature, was applied as a dual-ion-mode MALDI matrix for the first time to analyze small molecules. The low ionization efficiency and strong optical absorption properties make lignin a potential MALDI matrix in small molecule analysis. A total of 30 different small molecules were identified qualitatively and six kinds of representative molecules were detected quantitatively with a good linear response (R2 > 0.995). To verify the accuracy of our quantitative method in MALDI, myricitrin, a major bioactive component in Chinese bayberry, was analyzed in different cultivars and tissues. The myricitrin content in real samples detected by MALDI was highly consistent (R2 > 0.999) with that detected by high-performance liquid chromatography, thus indicating the applicability of the lignin matrix. Further characterization by ultraviolet and nuclear magnetic resonance spectroscopy was carried out to explain the possible mechanism of lignin as a matrix and provide more theories for a rational matrix design.

The convex invertible cone structure of positive real odd rational matrix functions

The convex invertible cone structure of positive real odd rational matrix functions

Generalized Spectral Characterization of Mixed Graphs

The spectral characterization of graphs is an important topic in spectral graph theory, which has been studied extensively in recent years. Unlike the undirected case, however, the spectral characterization of mixed graphs (digraphs) has received much less attention so far, which will be the main focus of this paper. A mixed graph $G$ is said to be strongly determined by its generalized Hermitian spectrum (abbreviated SHDGS), if, up to isomorphism, $G$ is the unique mixed graph that is cospectral with $G$ w.r.t. the generalized Hermitian spectrum.
 Let $G$ be a self-converse mixed graph of order $n$ with Hermitian adjacency matrix $A$ and let $W=[e,Ae,\ldots,A^{n-1}e]$ ($e$ is the all-one vector). Suppose that $2^{-\lfloor n/2\rfloor}\det W$ is \emph{norm-free} in $\mathbb{Z}[i]$ (i.e., for any Gaussian prime $p$, the norm $N(p)=p\bar{p}$ does not divide $2^{-\lfloor n/2\rfloor}\det W$). We conjecture that every such graph is SHDGS and prove that, for any mixed graph $H$ that is cospectral with $G$ w.r.t. the generalized Hermitian spectrum, there exists a Gaussian rational unitary matrix $U$ with $Ue=e$ such that $U^*A(G)U=A(H)$ and $(1+i)U$ is a Gaussian integral matrix. We have verified the conjecture in two extremal cases when $G$ is either an undirected graph or a self-converse oriented graph. Moreover, as consequences of our main results, we prove that all directed paths of even order are SHDGS. Analogous results are also obtained in the setting of \emph{restrictive} determination by generalized Hermitian spectrum (i.e., the spectral determination within the subset of all self-converse mixed graphs), which extends a recent result of the first author on the generalized spectral characterization of undirected graphs.

A Toeplitz-Like Operator with Rational Matrix Symbol Having Poles on the Unit Circle: Fredholm Properties

In this paper a definition is given for an unbounded Toeplitzlike operator with rational symbol which has poles on the unit circle. It is shown that the operator is Fredholm if and only if the symbol has no zeroes on the unit circle, and a formula for the index is given as well. Finally, a matrix representation of the operator is discussed.

Riemann-Hilbert Boundary Value Problem with Piecewise Constant Transition Function

We prove that, for any Fuchsian system of differential equations on the Riemann sphere, there exists a rational matrix function whose partial indices coincide with the splitting type of the canonical vector bundle induced from the Fuchsian system. From this, we obtain solution of the Riemann-Hilbert boundary value problem for piecewise constant matrix function in terms of holomorphic sections of vector bundle and calculate the partial indices of the problem.

On mesh geometries of root Coxeter orbits and mesh algorithms for corank two edge-bipartite signed graphs

Following a Coxeter spectral analysis problems for positive edge-bipartite graphs (signed multigraphs with a separation property) introduced in Simson (2013) [41] and Simson (2013) [42], we study analogous problems for loop-free corank two edge-bipartite graphs Δ=(Δ0,Δ1), i.e., for edge-bipartite graphs Δ, with n+2=|Δ0|≥3 vertices such that its rational symmetric Gram matrix GΔ:=12(GˇΔ+GˇΔtr)∈Mn+2(Q) is positive semi-definite of rank n. We study such connected edge-bipartite signed graphs (bigraphs, for short), up to the strong Gram congruence Δ≈ZΔ′, by means of the non-symmetric Gram matrix GˇΔ∈Mn+2(Z), the Coxeter matrix CoxΔ:=−GˇΔ⋅GˇΔ−tr∈Mn+2(Z), its complex spectrum speccΔ⊆C, and an associated simply laced Dynkin diagram DynΔ∈{An,Dn,E6,E7,E8}, with n≥1 vertices. It is known that if the congruence holds Δ≈ZΔ′ (i.e., there exists B∈Mn+2(Z) such that det⁡B=±1 and GˇΔ′=Btr⋅GˇΔ⋅B) then (speccΔ,DynΔ)=(speccΔ′,DynΔ′).The inverse implication is proved in [Fund. Inform. 152 (2017) 185–222] for all pairs of non-negative corank two bigraphs Δ, Δ′ without loops with n+2≤6 vertices. One of the main aims of the paper is to find algorithms that construct the set Cong(Δ,Δ′) of all Z-invertible matrices B∈Mn+2(Z) defining the congruence Δ≈ZΔ′, for any pair of corank two bigraphs Δ and Δ′ such that (speccΔ,DynΔ)=(speccΔ′,DynΔ′). We do it in case when Δ and Δ′ have at most six vertices in three steps: (i) to any such Δ, we associate algorithmically a ΦΔ-mesh translation quiver ΓΔ•=Γ(RˆΔ,ΦΔ) in Zn+2 (called ΦΔ-mesh geometry of roots of Δ, in the sense of Simson (2013) [42]), where ΦΔ:Zn+2→Zn+2, v↦v⋅CoxΔ, is the Coxeter transformation of Δ; (ii) we define an injective contravariant map h˜•:Cong(Δ,Δ′)→Iso(ΓΔ′•,ΓΔ•), where Iso(ΓΔ′•,ΓΔ•) is the set of all mesh translation quiver isomorphisms h˜B:ΓΔ′•→ΓΔ•, and (iii) a numerical algorithm computing a required matrix B∈Cong(Δ,Δ′) is constructed by means of the shape of the ΦΔ-mesh translation quiver ΓΔ•, its toroidal-tubular structure, and an analysis of admissible isomorphisms h˜B:ΓΔ′•→ΓΔ•.

Rational forms for block lower-triangular matrices

Linear time invariant (LTI) systems are often represented by rational forms or rational matrix functions. Such forms exhibit important properties of a system, but these properties are often thought to be due to time invariance. Are there such forms for linear time-variant (LTV) systems with comparable properties, generalizing the LTI case? One major obstacle to derive them has been the lack of a divisibility theory for general block lower-triangular matrices, in particular the lack of a Euclidean algorithm or, in its wake, Smith forms and Smith-McMillan forms. But do rational forms for matrices, generalizing the time-invariant properties, exist at all? It turns out they do. The theory presented here produces representations for block lower-triangular matrices as ratios of (block) staircase (or echelon) matrices, and shows how central properties such as co-prime factorization or Bezout identities hold. Instead of the Euclidean algorithm, the theory uses a technique derived from a paper of Paul Van Dooren, namely ‘dead-beat control’. The theory finds good applications in various domains (computational efficiency, determining pre-conditioners, control theory and model reduction via generalized forms of interpolation).

Interval matrices: Realization of ranks by rational matrices

Let α be a p×q interval matrix with p≥q and with the endpoints of all its entries in Q. We prove that, if α contains a rank-r real matrix with r∈{2,q−2,q−1,q}, then it contains a rank-r rational matrix.

Rational Transfer Function Model for a Double-Pipe Parallel-Flow Heat Exchanger

Transfer functions of typical heat exchangers, resulting from their partial differential equations, usually contain irrational functions which quite accurately describe the spatio-temporal nature of the processes occurring therein. However, such an accurate but complex mathematical representation is often not convenient from the practical point of view, and some approximation of the original model would be more useful. This paper discusses approximate rational transfer functions for a typical thick-walled double-pipe heat exchanger working in the parallel-flow configuration. Using the method of lines with the backward difference scheme, the original symmetric hyperbolic partial differential equations describing the heat transfer phenomena are transformed into a set of ordinary differential equations and expressed in the form of N subsystems representing spatial sections of the exchanger. Each section is described by a rational transfer function matrix and their cascade interconnection results in the overall approximation model expressed by a matrix of rational transfer functions of high order. Based on the rational transfer function representation, the frequency and steady-state responses of the approximate model are evaluated and compared with those resulting from its original irrational transfer function model. The presented results show better approximation quality for the “crossover” input–output channels where the in-domain heat conduction effects prevail as compared to the “straightforward” channels, where the transport delay associated with the heat convection dominates.

A Structure-Preserving Method for Positive Realness Problem in Control

In this paper the positive realness problem in control theory is studied. A numerical method is developed for verifying the positive realness of a given proper rational matrix H(s) for which H(s) + HT(−s) has purely imaginary zeros. The proposed method is only based on orthogonal transformations, it is structure-preserving and has a complexity which is cubic in the state dimension of H(s). Some examples are given to illustrate the performance of the proposed method.

Local linearizations of rational matrices with application to rational approximations of nonlinear eigenvalue problems

This paper presents a definition for local linearizations of rational matrices and studies their properties. This definition allows to introduce matrix pencils associated to a rational matrix that preserve its structure of zeros and poles in subsets of any algebraically closed field and also at infinity. This new theory of local linearizations captures and explains rigorously the properties of all the different pencils that have been used from the 1970's until 2020 for computing zeros, poles and eigenvalues of rational matrices. Particular attention is paid to those pencils that have appeared recently in the numerical solution of nonlinear eigenvalue problems through rational approximation.

Identifying Specific Combinations of Matrix Properties that Promote Controlled and Sustained Release of a Hydrophobic Drug from Electrospun Meshes.

Despite advances in the development of degradable polymers for drug delivery, effective translation of drug-loaded materials is often hindered due to a poor understanding of matrix property combinations that promote controlled and sustained release. In this study, we investigated the influence of dominant factors on the release of a hydrophobic glucocorticoid dexamethasone (DEX) from electrospun meshes. Polycaprolactone meshes released 98% of the drug within 24 h, while poly(l-lactide) meshes exhibited negligible release even after 28 days despite both polymers being slow-degrading. Differences in drug-polymer interactions and drug-polymer miscibility—but neither matrix degradation nor differences in bulk hydrophobicity—influenced DEX release from these semi-crystalline matrices. Poly(d,l-lactide-co-glycolide) 50:50 meshes possessing two different fiber diameters exhibited a sequential burst and sustained release, while poly(d,l-lactide-co-glycolide) 85:15 meshes cumulatively released 26% drug in a controlled manner. Although initial drug release from these matrices was driven by differences in matrix architecture and solid-state drug solubility, release toward the later stages was influenced by a combination of fiber swelling and matrix degradation as evidenced by gross and microstructural changes to the mesh network. We suggest that drug release from polymeric matrices can be better understood via investigation of critical matrix characteristics influencing release, as well as concomitant examination of drug-polymer interactions and miscibility. Our findings offer rational matrix design criteria to achieve controlled/extended drug release for promoting sustained biological responses.