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A well known method to solve the Polynomial Eigenvalue Problem (PEP) is via linearization. That is, transforming the PEP into a generalized linear eigenvalue problem with the same spectral information and solving such linear problem with some of the eigenvalue algorithms available in the literature. Linearizations of matrix polynomials are usually defined using unimodular transformations. In this paper we establish a connection between the standard definition of linearization for matrix polynomials introduced by Gohberg, Lancaster and Rodman and the notion of polynomial system matrix introduced by Rosenbrock. This connection gives new techniques to show that a matrix pencil is a linearization of the corresponding matrix polynomial arising in a PEP.

In this paper we consider the computation of the modified moments for the system of Laguerre polynomials on the real semiaxis with the Hermite weight. These moments can be used for the computation of integrals with the Hermite weight for the real semiaxis, either via product rules or via the construction of the n-point Gaussian quadrature rule with respect to this nonstandard weight using the Chebyshev algorithm. We propose a new computational method based on the construction of the null-space of a rectangular matrix derived from the three--term recurrence relation of the system of orthonormal Laguerre polynomials. It is shown that the proposed algorithm computes the modified moments with high accuracy and linear complexity. Numerical examples illustrate the effectiveness of the proposed method.

Funding Information: The authors thank the anonymous reviewers for useful comments. G.B. and V.N. are supported by an Academy of Finland grant (Suomen Akatemian päätös 331240). G.B. thanks the Alfred Kordelinin säätiö for the financial support under Grant No. 210122. P.V.D. is supported by an Aalto Science Institute Visitor Programme. Publisher Copyright: © 2022 American Physical Society. We analyze the recovery of different roles in a network modeled by a directed graph, based on the so-called Neighborhood Pattern Similarity approach. Our analysis uses results from random matrix theory to show that, when assuming that the graph is generated as a particular stochastic block model with Bernoulli probability distributions for the different blocks, then the recovery is asymptotically correct when the graph has a sufficiently large dimension. Under these assumptions there is a sufficient gap between the dominant and dominated eigenvalues of the similarity matrix, which guarantees the asymptotic correct identification of the number of different roles. We also comment on the connections with the literature on stochastic block models, including the case of probabilities of order log(n)/n where n is the graph size. We provide numerical experiments to assess the effectiveness of the method when applied to practical networks of finite size.

We prove that we can always construct strongly minimal linearizations of an arbitrary rational matrix from its Laurent expansion around the point at infinity, which happens to be the case for polynomial matrices expressed in the monomial basis. If the rational matrix has a particular self-conjugate structure we show how to construct strongly minimal linearizations that preserve it. The structures that are considered are the Hermitian and skew-Hermitian rational matrices with respect to the real line, and the para-Hermitian and para-skew-Hermitian matrices with respect to the imaginary axis. We pay special attention to the construction of strongly minimal linearizations for the particular case of structured polynomial matrices. The proposed constructions lead to efficient numerical algorithms for constructing strongly minimal linearizations. The fact that they are valid for {\em any} rational matrix is an improvement on any other previous approach for constructing other classes of structure preserving linearizations, which are not valid for any structured rational or polynomial matrix. The use of the recent concept of strongly minimal linearization is the key for getting such generality., 29 pages

International audience; This paper addresses the problem of stability for general two-dimensional (2D) discrete-time and continuous-discrete time Lyapunov systems, where the linear matrix inequalities (LMI's) approach is applied to derive a new sufficient condition for the asymptotic stability.

Funding Information: Supported by an Academy of Finland grant (Suomen Akatemian päätös 331240).Supported by an Aalto Science Institute Visitor Programme. Publisher Copyright: © 2022 The Author(s) The notion of root polynomials of a polynomial matrix P(λ) was thoroughly studied in Dopico and Noferini (2020) [6]. In this paper, we extend such a systematic approach to general rational matrices R(λ), possibly singular and possibly with coalescent pole/zero pairs. We discuss the related theory for rational matrices with coefficients in an arbitrary field. As a byproduct, we obtain sensible definitions of eigenvalues and eigenvectors of a rational matrix R(λ), without any need to assume that R(λ) has full column rank or that the eigenvalue is not also a pole. Then, we specialize to the complex field and provide a practical algorithm to compute them, based on the construction of a minimal state space realization of the rational matrix R(λ) and then using the staircase algorithm on the linearized pencil to compute the null space as well as the root polynomials in a given point λ0. If λ0 is also a pole, then it is necessary to apply a preprocessing step that removes the pole while making it possible to recoverthe root vectors of the original matrix: in this case, we study both the relevant theory (over a general field) and an algorithmic implementation (over the complex field), still based on minimal state space realizations.

This paper presents a definition for local linearizations of rational matrices and studies their properties. This definition allows us 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. Moreover, such definition includes, as particular cases, other definitions that have been used previously in the literature. In this way, 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 2019 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., 49 pages

In this paper we revisit the problem of performing a $QZ$ step with a so-called ‘perfect shift’, which is an ‘exact’ eigenvalue of a given regular pencil $\lambda B-A$ in unreduced Hessenberg triangular form. In exact arithmetic, the $QZ$ step moves that eigenvalue to the bottom of the pencil, while the rest of the pencil is maintained in Hessenberg triangular form, which then yields a deflation of the given eigenvalue. But in finite precision the $QZ$ step gets ‘blurred’ and precludes the deflation of the given eigenvalue. In this paper we show that when we first compute the corresponding eigenvector to sufficient accuracy, then the $QZ$ step can be constructed using this eigenvector, so that the deflation is also obtained in finite precision. An important application of this technique is the computation of the index of a system of differential algebraic equations, since an exact deflation of the infinite eigenvalues is needed to impose correctly the algebraic constraints of such differential equations.

We study different representations of a given rational transfer function that represents a passive (or positive real) discrete-time system. When the system is subject to perturbations, passivity or stability may be lost. To make the system robust, we use the freedom in the representation to characterize and construct optimally robust representations in the sense that the distance to non-passivity is maximized with respect to an appropriate matrix norm. We link this construction to the solution set of certain linear matrix inequalities defining passivity of the transfer function. We present an algorithm to compute a nearly optimal representation using an eigenvalue optimization technique. We also briefly consider the problem of finding the nearest passive system to a given non-passive one.

Publisher Copyright: © 2022 Informa UK Limited, trading as Taylor & Francis Group. We introduce a new family of linearizations of rational matrices, which we call block full rank linearizations. The theory of block full rank linearizations is useful as it establishes very simple criteria to determine if a pencil is a linearization of a rational matrix in a target set or in the whole underlying field, by using rank conditions. Block full rank linearizations allow us to recover locally information about zeros and poles. To recover the pole-zero information at infinity, we will define the grade of the new block full rank linearizations as linearizations at infinity and the notion of degree of a rational matrix will be used. Moreover, the eigenvectors of a rational matrix associated with its eigenvalues in a target set can be obtained from the eigenvectors of its block full rank linearizations in that set. This new family of linearizations generalizes and includes the structures appearing in most of the linearizations for rational matrices constructed in the literature. As example, we study thestructure and properties of the linearizations in [P. Lietaert et al., Automatic rational approximation and linearization of nonlinear eigenvalue problems, 2021].

This paper studies generic and perturbation properties inside the linear space of $m\times (m+n)$ polynomial matrices whose rows have degrees bounded by a given list $d_1, \ldots, d_m$ of natural numbers, which in the particular case $d_1 = \cdots = d_m = d$ is just the set of $m\times (m+n)$ polynomial matrices with degree at most $d$. Thus, the results in this paper extend to a much more general setting the results recently obtained in [Van Dooren & Dopico, Linear Algebra Appl. (2017), http://dx.doi.org/10.1016/j.laa.2017.05.011] only for polynomial matrices with degree at most $d$. Surprisingly, most of the properties proved in [Van Dooren & Dopico, Linear Algebra Appl. (2017)], as well as their proofs, remain to a large extent unchanged in this general setting of row degrees bounded by a list that can be arbitrarily inhomogeneous provided the well-known Sylvester matrices of polynomial matrices are replaced by the new trimmed Sylvester matrices introduced in this paper. The following results are presented, among many others, in this work: (1) generically the polynomial matrices in the considered set are minimal bases with their row degrees exactly equal to $d_1, \ldots , d_m$, and with right minimal indices differing at most by one and having a sum equal to $\sum_{i=1}^{m} d_i$, and (2), under perturbations, these generic minimal bases are robust and their dual minimal bases can be chosen to vary smoothly., arXiv admin note: text overlap with arXiv:1612.03793

We revisit the notion of root polynomials, thoroughly studied in [F. Dopico and V. Noferini, Root polynomials and their role in the theory of matrix polynomials, Linear Algebra Appl. 584:37--78, 2020] for general polynomial matrices, and show how they can efficiently be computed in the case of matrix pencils. The staircase algorithm implicitly computes so-called zero directions, as defined in [P. Van Dooren, Computation of zero directions of transfer functions, Proceedings IEEE 32nd CDC, 3132--3137, 1993]. However, zero directions generally do not provide the correct information on partial multiplicities and minimal indices. These indices are instead provided by two special cases of zero directions, namely, root polynomials and vectors of a minimal basis of the pencil. We show how to extract, starting from the block triangular pencil that the staircase algorithm computes, both a minimal basis and a maximal set of root polynomials in an efficient manner. Moreover, we argue that the accuracy of the computation of the root polynomials can be improved by making use of iterative refinement.

In this paper we analyze an indirect approach, called the Neighborhood Pattern Similarity approach, to solve the so-called role extraction problem of a large-scale graph. The method is based on the preliminary construction of a node similarity matrix which allows in a second stage to group together, with an appropriate clustering technique, the nodes that are assigned to have the same role. The analysis builds on the notion of ideal graphs where all nodes with the same role, are also structurally equivalent.

Community detection plays an important role in understanding and exploiting the structure of complex systems. Many algorithms have been developed for community detection using modularity maximization or other techniques. In this paper, we formulate the community detection problem as a constrained nonsmooth optimization problem on the compact Stiefel manifold. A Riemannian projected proximal gradient method is proposed and used to solve the problem. To the best of our knowledge, this is the first attempt to use Riemannian optimization for community detection problem. Numerical experimental results on synthetic benchmarks and real-world networks show that our algorithm is effective and outperforms several state-of-art algorithms.

In this paper we show how to construct diagonal scalings for arbitrary matrix pencils $\lambda B-A$, in which both $A$ and $B$ are complex matrices (square or nonsquare). The goal of such diagonal scalings is to "balance" in some sense the row and column norms of the pencil. We see that the problem of scaling a matrix pencil is equivalent to the problem of scaling the row and column sums of a particular nonnegative matrix. However, it is known that there exist square and nonsquare nonnegative matrices that can not be scaled arbitrarily. To address this issue, we consider an approximate embedded problem, in which the corresponding nonnegative matrix is square and can always be scaled. The new scaling methods are then based on the Sinkhorn-Knopp algorithm for scaling a square nonnegative matrix with total support to be doubly stochastic or on a variant of it. In addition, using results of U. G. Rothblum and H. Schneider (1989), we give simple sufficient conditions on the zero pattern for the existence of diagonal scalings of square nonnegative matrices to have any prescribed common vector for the row and column sums. We illustrate numerically that the new scaling techniques for pencils improve the accuracy of the computation of their eigenvalues., Comment: 35 pages, 1 figure

In this paper we describe how to swap two 2 × 2 blocks in a real Schur form and a generalized real Schur form. We pay special attention to the numerical stability of the method. We also illustrate the stability of our approach by a series of numerical tests. ispartof: Journal Of Computational And Applied Mathematics vol:373 ispartof: location:GREECE, Kalamata status: published

In this paper, we study the identification problem of strictly passive systems from frequency response data. We present a simple construction approach based on the Mayo–Antoulas generalized realization theory that automatically yields a port-Hamiltonian realization for every strictly passive system with simple spectral zeros. Furthermore, we discuss the construction of a frequency-limited port-Hamiltonian realization. We illustrate the proposed method by means of several examples.

When computing the eigenstructure of matrix pencils associated with the passivity analysis of perturbed port-Hamiltonian descriptor system using a structured generalized eigenvalue method, one should make sure that the computed spectrum satisfies the symmetries that corresponds to this structure and the underlying physical system. We perform a backward error analysis and show that for matrix pencils associated with port-Hamiltonian descriptor systems and a given computed eigenstructure with the correct symmetry structure there always exists a nearby port-Hamiltonian descriptor system with exactly that eigenstructure. We also derive bounds for how near this system is and show that the stability radius of the system plays a role in that bound.

We introduce a new family of strong linearizations of matrix polynomials---which we call "block Kronecker pencils"---and perform a backward stability analysis of complete polynomial eigenproblems. These problems are solved by applying any backward stable algorithm to a block Kronecker pencil, such as the staircase algorithm for singular pencils or the QZ algorithm for regular pencils. This stability analysis allows us to identify those block Kronecker pencils that yield a computed complete eigenstructure which is exactly that of a slightly perturbed matrix polynomial. The global backward error analysis in this work presents for the first time the following key properties: it is a rigurous analysis valid for finite perturbations (i.e., it is not a first order analysis), it provides precise bounds, it is valid simultaneously for a large class of linearizations, and it establishes a framework that may be generalized to other classes of linearizations. These features are related to the fact that block Kronecker pencils are a particular case of the new family of "strong block minimal bases pencils", which are robust under certain perturbations and, so, include certain perturbations of block Kronecker pencils. We hope that this robustness property will allow us to extend the results in this paper to other contexts., this article supersedes MIMS Eprint 2016.34

We present an efficient algorithm to compute the H ∞ norm of a fractional system. The algorithm is based on the computation of level sets of the maximum singular value of the transfer function, as a function of frequency. Numerical examples are given to illustrate the new method.

Polynomial minimal bases of rational vector subspaces are a classical concept that plays an important role in control theory, linear systems theory, and coding theory. It is a common practice to arrange the vectors of any minimal basis as the rows of a polynomial matrix and to call such matrix simply a minimal basis. Very recently, minimal bases, as well as the closely related pairs of dual minimal bases, have been applied to a number of problems that include the solution of general inverse eigenstructure problems for polynomial matrices, the development of new classes of linearizations and l-ifications of polynomial matrices, and backward error analyses of complete polynomial eigenstructure problems solved via a wide class of strong linearizations. These new applications have revealed that although the algebraic properties of minimal bases are rather well understood, their robustness and the behavior of the corresponding dual minimal bases under perturbations have not yet been explored in the literature, as far as we know. Therefore, the main purpose of this paper is to study in detail when a minimal basis M ( λ ) is robust under perturbations, i.e., when all the polynomial matrices in a neighborhood of M ( λ ) are minimal bases, and, in this case, how perturbations of M ( λ ) change its dual minimal bases. In order to study such problems, a new characterization of whether or not a polynomial matrix is a minimal basis in terms of a finite number of rank conditions is introduced and, based on it, we prove that polynomial matrices are generically minimal bases with very specific properties. In addition, some applications of the results of this paper are discussed.

We construct optimally robust port-Hamiltonian realizations of a given rational transfer function that represents a passive system. We show that the realization with a maximal passivity radius is a normalized port-Hamiltonian one. Its computation is linked to a particular solution of a linear matrix inequality that defines passivity of the transfer function, and we provide an algorithm to construct this optimal solution. We also consider the problem of finding the nearest passive system to a given non-passive one and provide a simple but suboptimal solution., 18 pages 2 figures

The computation of the eigenvalue decomposition of symmetric matrices is one of the most investigated problems in numerical linear algebra. For a matrix of moderate size, the customary procedure is to reduce it to a symmetric tridiagonal one by means of an orthogonal similarity transformation and then compute the eigendecomposition of the tridiagonal matrix. Recently, Malyshev and Dhillon have proposed an algorithm for deflating the tridiagonal matrix, once an eigenvalue has been computed. Starting from the aforementioned algorithm, in this manuscript we develop a procedure for computing an eigenvector of a symmetric tridiagonal matrix, once its associate eigenvalue is known. We illustrate the behavior of the proposed method with a number of numerical examples.

The standard way of solving a polynomial eigenvalue problem associated with a matrix polynomial starts by embedding the matrix coefficients of the polynomial into a matrix pencil, known as a strong linearization. This process transforms the problem into an equivalent generalized eigenvalue problem. However, there are some situations in which is more convenient to replace linearizations by other low degree matrix polynomials. This has motivated the idea of a strong l-ification of a matrix polynomial, which is a matrix polynomial of degree at most l having the same finite and infinite elementary divisors, and the same numbers of left and right minimal indices as the original matrix polynomial. We present in this work a novel method for constructing strong l-ifications of matrix polynomials of size m × n and grade d when l d , and l divides nd or md. This method is based on a family called “strong block minimal bases matrix polynomials”, and relies heavily on properties of dual minimal bases. We show how strong block minimal bases l-ifications can be constructed from the coefficients of a given matrix polynomial P ( λ ) . We also show that these l-ifications satisfy many desirable properties for numerical applications: they are strong l-ifications regardless of whether P ( λ ) is regular or singular, the minimal indices of the l-ifications are related to those of P ( λ ) via constant uniform shifts, and eigenvectors and minimal bases of P ( λ ) can be recovered from those of any of the strong block minimal bases l-ifications. In the special case where l divides d, we introduce a subfamily of strong block minimal bases matrix polynomials named “block Kronecker matrix polynomials”, which is shown to be a fruitful source of companion l-ifications.

We study how small perturbations of general matrix polynomials may change their elementary divisors and minimal indices by constructing the closure hierarchy (stratification) graphs of matrix polynomials’ orbits and bundles. To solve this problem, we construct the stratification graphs for the first companion Fiedler linearization of matrix polynomials. Recall that the first companion Fiedler linearization as well as all the Fiedler linearizations is matrix pencils with particular block structures. Moreover, we show that the stratification graphs do not depend on the choice of Fiedler linearization which means that all the spaces of the matrix polynomial Fiedler linearizations have the same geometry (topology). This geometry coincides with the geometry of the space of matrix polynomials. The novel results are illustrated by examples using the software tool StratiGraph extended with associated new functionality., VR E0485301, eSSENCE

The generalized Schur algorithm is a powerful tool allowing to compute classical decompositions of matrices, such as the Q R and L U factorizations. When applied to matrices with particular structures, the generalized Schur algorithm computes these factorizations with a complexity of one order of magnitude less than that of classical algorithms based on Householder or elementary transformations. In this manuscript, we describe the main features of the generalized Schur algorithm. We show that it helps to prove some theoretical properties of the R factor of the Q R factorization of some structured matrices, such as symmetric positive definite Toeplitz and Sylvester matrices, that can hardly be proven using classical linear algebra tools. Moreover, we propose a fast implementation of the generalized Schur algorithm for computing the rank of Sylvester matrices, arising in a number of applications. Finally, we propose a generalized Schur based algorithm for computing the null-space of polynomial matrices.

We present an algorithm for computing a symmetric rank-revealing decomposition of a symmetric n × n matrix A, as defined in the work of Hansen & Yalamov [9] : we factorize the original matrix into a product A = Q M Q T , with Q orthogonal and M symmetric and in block form, with one of the blocks containing the dominant information of A, such as its largest eigenvalues. Moreover, the matrix M is constructed in a form that is easy to update when adding to A a symmetric rank-one matrix or when appending a row and, symmetrically, a column to A: the cost of such an updating is O ( n 2 ) floating point operations. The proposed algorithm is based on the block anti-triangular form of the original matrix M, as introduced by the authors in [11] . Via successive orthogonal similarity transformations this form is then updated to a new form A = Q ˆ M ˆ Q ˆ T , whereby the first k rows and columns of M ˆ have elements bounded by a given threshold τ and the remaining bottom right part of M ˆ is maintained in block anti-triangular form. The updating transformations are all orthogonal, guaranteeing the backward stability of the algorithm, and the algorithm is very economical when the near rank deficiency is detected in some of the anti diagonal elements of the block anti-triangular form. Numerical results are also given showing the reliability of the proposed algorithm.

We propose an extended Lanczos bidiagonalization algorithm for finding a low rank approximation of a given matrix. We show that this method can yield better low-rank approximations than standard Lanczos bidiagonalization algorithm, without increasing the cost too much. We also describe a partial reorthogonalization process that can be used to maintain an adequate level of orthogonality of the Lanczos vectors in order to produce accurate low-rank approximations. We demonstrate the effectiveness and applicability of our algorithm for a number of applications.

This paper presents an algorithm that solves optimization problems on a matrix manifold M ⊆ R m × n with an additional rank inequality constraint. The algorithm resorts to well-known Riemannian optimization schemes on fixed-rank manifolds, combined with new mechanisms to increase or decrease the rank. The convergence of the algorithm is analyzed and a weighted low-rank approximation problem is used to illustrate the efficiency and effectiveness of the algorithm.

In this paper we revisit the problem of performing a QR step on an unreduced Hessenberg matrix H when we know an ``exact"" eigenvalue \lambda 0 of H. In exact arithmetic, this eigenvalue will appear on the diagonal of the transformed Hessenberg matrix H\~ and will be decoupled from the remaining part of the Hessenberg matrix, thus resulting in a deflation. But it is well known that in finite-precision arithmetic the so-called perfect shift can get blurred and that the eigenvalue \lambda 0 can then not be deflated and/or is perturbed significantly. In this paper, we develop a new strategy for computing such a QR step so that the deflation is almost always successful. We also show how to extend this technique to double QR steps with complex conjugate shifts

We discuss the problem of robust representations of stable and passive transfer functions in particular coordinate systems, and focus in particular on the so-called port-Hamiltonian representations. Such representations are typically far from unique and the degrees of freedom are related to the solution set of the so-called Kalman-Yakubovich-Popov linear matrix inequality (LMI). In this paper we analyze robustness measures for the different possible representations and relate it to quality functions defined in terms of the eigenvalues of the matrix associated with the LMI. In particular, we look at the analytic center of this LMI. From this, we then derive inequalities for the passivity radius of the given model representation.

In this paper, we consider the problem of factorizing the n×n matrix Jn of all ones into the n×n binary matrices. We show that under some conditions on the factors, these are isomorphic to a row permutation of a De Bruijn matrix. Moreover, we consider in particular the binary roots of Jn, i.e. the binary solutions to Am=Jn. On the one hand, we prove that any binary root with minimum rank is isomorphic to a row permutation of a De Bruijn matrix whose row permutation is represented by a block diagonal matrix. On the other hand, we partially solve Hoffman's open problem of characterizing the binary solutions to A2=Jn by providing a characterization of the binary solutions to A2=Jn with minimum rank. Finally, we provide a class of roots which are isomorphic to a De Bruijn matrix.

Modern phylogenomic analyses often result in large collections of phylogenetic trees representing uncertainty in individual gene trees, variation across genes, or both. Extracting phylogenetic signal from these tree sets can be challenging, as they are difficult to visualize, explore, and quantify. To overcome some of these challenges, we have developed TreeScaper, an application for tree set visualization as well as the identification of distinct phylogenetic signals. GUI and command-line versions of TreeScaper and a manual with tutorials can be downloaded from https://github.com/whuang08/TreeScaper/releases TreeScaper is distributed under the GNU General Public License.

We present a framework for the construction of linearizations for scalar and matrix polynomials based on dual bases which, in the case of orthogonal polynomials, can be described by the associated recurrence relations. The framework provides an extension of the classical linearization theory for polynomials expressed in non-monomial bases and allows to represent polynomials expressed in product families, that is as a linear combination of elements of the form φᵢ(λ) ψⱼ(λ), where {φᵢ(λ)} and {ψⱼ(λ)} can either be polynomial bases or polynomial families which satisfy some mild assumptions. We show that this general construction can be used for many different purposes. Among them, we show how to linearize sums of polynomials and rational functions expressed in different bases. As an example, this allows to look for intersections of functions interpolated on different nodes without converting them to the same basis. We then provide some constructions for structured linearizations for ★-even and ★-palindromic matrix polynomials. The extensions of these constructions to ★-odd and ★-antipalindromic of odd degree is discussed and follows immediately from the previous results. ispartof: SIAM Journal on Matrix Analysis and Applications vol:38 issue:1 pages:188-216 status: published

In this paper we revisit the problem of finding an orthogonal similarity transformation that puts an $n\times n$ matrix $A$ in a block upper-triangular form that reveals its Jordan structure at a particular eigenvalue $\lambda_0$. The obtained form in fact reveals the dimensions of the null spaces of $(A-\lambda_0 I)^i$ at that eigenvalue via the sizes of the leading diagonal blocks, and from this the Jordan structure at $\lambda_0$ is then easily recovered. The method starts from a Hessenberg form that already reveals several properties of the Jordan structure of $A$. It then updates the Hessenberg form in an efficient way to transform it to a block-triangular form in ${\cal O}(mn^2)$ floating point operations, where $m$ is the total multiplicity of the eigenvalue. The method only uses orthogonal transformations and is backward stable. We illustrate the method with a number of numerical examples.

Positive definite matrices factor into A = L L T (Cholesky). Symmetric indefinite matrices need a symmetric middle factor in A = L P L T . Then A and P have the same inertia (eigenvalues of the same sign). We construct P through elimination, so the inertias agree for all leading minors of A and P. When restricting P to be a variant of a symmetric permutation in which diagonal 1's can be replaced by 0's or −1's, it is unique.

We propose a fast algorithm to calculate the $H_{\infty}$-norm of a transfer matrix. The method builds on a well-known relationship between singular values of the transfer function and pure imaginary eigenvalues of a certain Hamiltonian matrix. Using this property we construct a two-parameter eigenvalue problem, where, in the generic case, the critical value corresponds to a two-dimensional Jordan block. We use the implicit determinant method which replaces the need for eigensolves by the solution of linear systems, a technique recently used in [M. A. Freitag and A. Spence, Linear Algebra Appl., 435 (2011), pp. 3189--3205] for finding the distance to instability. In this paper the method takes advantage of the structured linear systems that arise within the algorithm to obtain efficient solves using the staircase reduction. We give numerical examples and compare our method to the algorithm proposed in [N. Guglielmi, M. Gürbüzbalaban, and M. L. Overton, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 709--737].

We show that perturbations of polynomial matrices of full normal-rank can be analyzed viathe study of perturbations of companion form linearizations of such polynomial matrices.It is proved that a full normal-rank polynomial matrix has the same structural elements asits right (or left) linearization. Furthermore, the linearized pencil has a special structurethat can be taken into account when studying its stratification. This yields constraintson the set of achievable eigenstructures. We explicitly show which these constraints are.These results allow us to derive necessary and sufficient conditions for cover relationsbetween two orbits or bundles of the linearization of full normal-rank polynomial matrices.The stratification rules are applied to and illustrated on two artificial polynomial matricesand a half-car passive suspension system with four degrees of freedom.

Mobile phone datasets allow for the analysis of human behavior on an unprecedented scale. The social network, temporal dynamics and mobile behavior of mobile phone users have often been analyzed independently from each other using mobile phone datasets. In this article, we explore the connections between various features of human behavior extracted from a large mobile phone dataset. Our observations are based on the analysis of communication data of 100000 anonymized and randomly chosen individuals in a dataset of communications in Portugal. We show that clustering and principal component analysis allow for a significant dimension reduction with limited loss of information. The most important features are related to geographical location. In particular, we observe that most people spend most of their time at only a few locations. With the help of clustering methods, we then robustly identify home and office locations and compare the results with official census data. Finally, we analyze the geographic spread of users' frequent locations and show that commuting distances can be reasonably well explained by a gravity model., Comment: 16 pages, 12 figures

In this paper, we analyze an algorithm to compute a low-rank approximation of the similarity matrix S introduced by Blondel et al. in [1]. This problem can be reformulated as an optimization problem of a continuous function Φ(S)=tr(STM2(S)) where S is constrained to have unit Frobenius norm, and M2 is a non-negative linear map. We restrict the feasible set to the set of matrices of unit Frobenius norm with either k nonzero identical singular values or at most k nonzero (not necessarily identical) singular values. We first characterize the stationary points of the associated optimization problems and further consider iterative algorithms to find one of them. We analyze the convergence properties of our algorithm and prove that accumulation points are stationary points of Φ(S). We finally compare our method in terms of speed and accuracy to the full rank algorithm proposed in [1].

We start by introducing a new class of structured matrix polynomials, namely, the class of M-A-structured matrix polynomials, to provide a common framework for many classes of structured matrix polynomials that are important in applications: the classes of (skew-)symmetric, (anti-) palindromic, and alternating matrix polynomials. Then, we introduce the families of M-A-structured strong block minimal bases pencils and of M-A-structured block Kronecker pencils, which are particular examples of block minimal bases pencils recently introduced by Dopico, Lawrence, Perez and Van Dooren, and show that any M-A-structured odd-degree matrix polynomial can be strongly linearized via an M-A-structured block Kronecker pencil. Finally, for the classes of (skew-)symmetric, (anti-)palindromic, and alternating odd-degree matrix polynomials, the M-A-structured framework allows us to perform a global and structured backward stability analysis of complete structured polynomial eigenproblems, regular or singular, solved by applying to a M-A-structured block Kronecker pencil a structurally backward stable algorithm that computes its complete eigenstructure, like the palindromic-QR algorithm or the structured versions of the staircase algorithm. This analysis allows us to identify those M-A-structured block Kronecker pencils that yield a computed complete eigenstructure which is the exact one of a slightly perturbed structured matrix polynomial. These pencils include (modulo permutations) the well-known block-tridiagonal and block-anti-tridiagonal structure-preserving linearizations. Our analysis incorporates structure to the recent (unstructured) backward error analysis performed for block Kronecker linearizations by Dopico, Lawrence, Perez and Van Dooren, and share with it its key features, namely, it is a rigorous analysis valid for finite perturbations, i.e., it is not a first order analysis, it provides precise bounds, and it is valid simultaneously for a large class of structure-pre

We provide an algorithm for constructing strong l-ifications of a given matrix polynomial P ( λ ) of degree d and size m × n using only the coefficients of the polynomial and the solution of linear systems of equations. A strong l-ification of P ( λ ) is a matrix polynomial of degree l having the same finite and infinite elementary divisors, and the same numbers of left and right minimal indices as the original matrix polynomial P ( λ ) . All explicit constructions of strong l-ifications introduced so far in the literature have been limited to the case where l divides d, though recent results on the inverse eigenstructure problem for matrix polynomials show that more general constructions are possible. Based on recent developments on dual polynomial minimal bases, we present a general construction of strong l-ifications for wider choices of the degree l, namely, when l divides one of nd or md (and d ≥ l ). In the case where l divides nd (respectively, md), the strong l-ifications we construct allow us to easily recover the minimal indices of P ( λ ) . In particular, we show that they preserve the left (resp., right) minimal indices of P ( λ ) , and the right (resp., left) minimal indices of the l-ification are the ones of P ( λ ) increased by d − l (each). Moreover, in the particular case l divides d, the new method provides a companion l-ification that resembles very much the companion l-ifications already known in the literature.

Minimal bases of rational vector spaces are a well-known and important tool in systems theory. If minimal bases for two subspaces of rational n-space are displayed as the rows of polynomial matrices Z1(λ)_(k x n) and Z2(λ)_(m x n), respectively, then z_1and z_2are said to be dual minimal bases if the subspaces have complementary dimension, i.e., k+m=n, and Z_1 (λ) Z_2^T (λ)=0. In other words, each z_j (λ) provides a minimal basis for the nullspace of the other. It has long been known that for any dual minimal bases z_1 (λ) and z_2 (λ), the row degree sums of Z1 and Z2 are the same. In this paper we show that this is the only constraint on the row degrees, thus characterizing the possible row degrees of dual minimal bases. The proof is constructive, making extensive use of a new class of sparse, structured polynomial matrices that we have baptized zigzag matrices. Another application of these polynomial zigzag matrices is the constructive solution of the following inverse problem for minimal indices { given a list of left and right minimal indices and a desired degree d, does there exist a completely singular matrix polynomial (i.e., a matrix polynomial with no elementary divisors whatsoever) of degree d having exactly the prescribed minimal indices? We show that such a matrix polynomial exists if and only if d divides the sum of the minimal indices. The constructed realization is simple, and explicitly displays the desired minimal indices in a fashion analogous to the classical Kronecker canonical form of singular pencils. Supported by National Science Foundation grant DMS-1016224, and by Ministerio de Economía y Competitividad of Spain through grant MTM2012-32542. Publicado

We consider here the problem of tracking the dominant eigenspace of an indefinite matrix by updating recursively a rank k approximation of the given matrix. The tracking uses a window of the given matrix, which increases at every step of the algorithm. Therefore, the rank of the approximation increases also, and hence a rank reduction of the approximation is needed to retrieve an approximation of rank k. In order to perform the window adaptation and the rank reduction in an efficient manner, we make use of a new anti-triangular decomposition for indefinite matrices. All steps of the algorithm only make use of orthogonal transformations, which guarantees the stability of the intermediate steps. We also show some numerical experiments to illustrate the performance of the tracking algorithm.