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We discuss efficient computation of $f$-centralities and nonbacktracking centralities for time-evolving networks with nonnegative weights. We present a node-level formula for its combinatorially exact computation which proves to be more tractable than previously existing formulae at edge-level for dense networks. Additionally, we investigate the impact of the addition of a final time frame to such a time-evolving network, analyzing its effect on the resulting nonbacktracking Katz centrality. Finally, we demonstrate by means of computational experiments that the node-level formula presented is much more efficient for dense networks than the previously known edge-level formula. As a tool for our goals, in an appendix of the paper, we develop a spectral theory of matrices whose elements are vectors.

Let $R$ be a commutative ring and $g(t) \in R[t]$ a monic polynomial. The commutative ring of polynomials $f(C_g)$ in the companion matrix $C_g$ of $g(t)$, where $f(t)\in R[t]$, is called the Companion Ring of $g(t)$. Special instances include the rings of circulant matrices, skew-circulant matrices, pseudo-circulant matrices, or lower triangular Toeplitz matrices. When $R$ is an Elementary Divisor Domain, we develop new tools for computing the Smith forms of matrices in Companion Rings. In particular, we obtain a formula for the second last non-zero determinantal divisor, we provide an $f(C_g) \leftrightarrow g(C_f)$ swap theorem, and a composition theorem. When $R$ is a principal ideal domain we also obtain a formula for the number of non-unit invariant factors. By applying these to families of circulant matrices that arise as relation matrices of cyclically presented groups, in many cases we compute the groups' abelianizations. When the group is the fundamental group of a three dimensional manifold, this provides the homology of the manifold. In other cases we obtain lower bounds for the rank of the abelianisation and record consequences for finiteness or solvability of the group, or for the Heegaard genus of a corresponding manifold.

In this paper we study para-Hermitian rational matrices and the associated structured rational eigenvalue problem (REP). Para-Hermitian rational matrices are square rational matrices that are Hermitian for all $z$ on the unit circle that are not poles. REPs are often solved via linearization, that is, using matrix pencils associated to the corresponding rational matrix that preserve the spectral structure. Yet, non-constant polynomial matrices cannot be para-Hermitian. Therefore, given a para-Hermitian rational matrix $R(z)$, we instead construct a $*$-palindromic linearization for $(1+z)R(z)$, whose eigenvalues that are not on the unit circle preserve the symmetries of the zeros and poles of $R(z)$. This task is achieved via M\"{o}bius transformations. We also give a constructive method that is based on an additive decomposition into the stable and anti-stable parts of $R(z)$. Analogous results are presented for para-skew-Hermitian rational matrices, i.e., rational matrices that are skew-Hermitian upon evaluation on those points of the unit circle that are not poles.

We propose an extremely versatile approach to address a large family of matrix nearness problems, possibly with additional linear constraints. Our method is based on splitting a matrix nearness problem into two nested optimization problems, of which the inner one can be solved either exactly or cheaply, while the outer one can be recast as an unconstrained optimization task over a smooth real Riemannian manifold. We observe that this paradigm applies to many matrix nearness problems of practical interest appearing in the literature, thus revealing that they are equivalent in this sense to a Riemannian optimization problem. We also show that the objective function to be minimized on the Riemannian manifold can be discontinuous, thus requiring regularization techniques, and we give conditions for this to happen. Finally, we demonstrate the practical applicability of our method by implementing it for a number of matrix nearness problems that are relevant for applications and are currently considered very demanding in practice. Extensive numerical experiments demonstrate that our method often greatly outperforms its predecessors, including algorithms specifically designed for those particular problems., Comment: 31 pages, 7 figures, 3 tables

Rosenbrock's theorem on polynomial system matrices is a classical result in linear systems theory that relates the Smith-McMillan form of a rational matrix $G$ with the Smith forms of an irreducible polynomial system matrix $P$ giving rise to $G$ and of a submatrix of $P$. This theorem has been essential in the development of algorithms for computing the poles and zeros of a rational matrix via linearizations and generalized eigenvalue algorithms. In this paper, we extend Rosenbrock's theorem to system matrices $P$ with entries in an arbitrary elementary divisor domain $\mathfrak R$ and matrices $G$ with entries in the field of fractions of $\mathfrak R$. These are the most general rings where the involved Smith-McMillan and Smith forms both exist and, so, where the problem makes sense. Moreover, we analyze in detail what happens when the system matrix is not irreducible. Finally, we explore how Rosenbrock's theorem can be extended when the system matrix $P$ itself has entries in the field of fractions of the elementary divisor domain.

Katz centrality (and its limiting case, eigenvector centrality) is a frequently used tool to measure the importance of a node in a network, and to rank the nodes accordingly. One reason for its popularity is that Katz centrality can be computed very efficiently when the network is sparse, i.e., having only $O(n)$ edges between its $n$ nodes. While sparsity is common in practice, in some applications one faces the opposite situation of a very dense network, where only $O(n)$ potential edges are missing with respect to a complete graph. We explain why and how, even for very dense networks, it is possible to efficiently compute the ranking stemming from Katz centrality for unweighted graphs, possibly directed and possibly with loops, by working on the complement graph. Our approach also provides an interpretation, regardless of sparsity, of "Katz centrality with negative parameter" as usual Katz centrality on the complement graph. For weighted graphs, we provide instead an approximation method that is based on removing sufficiently many edges from the network (or from its complement), and we give sufficient conditions for this approximation to provide the correct ranking. We include numerical experiments to illustrate the advantages of the proposed approach.

We investigate an application of network centrality measures to portfolio optimization, by generalizing the method in [Pozzi, Di Matteo and Aste, \emph{Spread of risks across financial markets: better to invest in the peripheries}, Scientific Reports 3:1665, 2013], that however had significant limitations with respect to the state of the art in network theory. In this paper, we systematically compare many possible variants of the originally proposed method on S\&P 500 stocks. We use daily data from twenty-seven years as training set and their following year as test set. We thus select the best network-based methods according to different viewpoints including for instance the highest Sharpe Ratio and the highest expected return. We give emphasis in new centrality measures and we also conduct a thorough analysis, which reveals significantly stronger results compared to those with more traditional methods. According to our analysis, this graph-theoretical approach to investment can be used successfully by investors with different investment profiles leading to high risk-adjusted returns.

Given a square pencil $A+ \lambda B$, where $A$ and $B$ are $n\times n$ complex (resp. real) matrices, we consider the problem of finding the singular complex (resp. real) pencil nearest to it in the Frobenius distance. This problem is known to be very difficult, and the few algorithms available in the literature can only deal efficiently with pencils of very small size. We show that the problem is equivalent to minimizing a certain objective function $f$ over the Riemannian manifold $SU(n) \times SU(n)$ (resp. $SO(n) \times SO(n)$ if the nearest real singular pencil is sought), where $SU(n)$ denotes the special unitary group (resp. $SO(n)$ denotes the special orthogonal group). This novel perspective is based on the generalized Schur form of pencils, and yields competitive numerical methods, by pairing it with { algorithms} capable of doing optimization on { Riemannian manifolds. We propose one algorithm that directly minimizes the (almost everywhere, but not everywhere, differentiable) function $f$, as well as a smoothed alternative and a third algorithm that is smooth and can also solve the problem} of finding a nearest singular pencil with a specified minimal index. We provide numerical experiments that show that the resulting methods allow us to deal with pencils of much larger size than alternative techniques, yielding candidate minimizers of comparable or better quality. In the course of our analysis, we also obtain a number of new theoretical results related to the generalized Schur form of a (regular or singular) square pencil and to the minimal index of a singular square pencil whose nullity is $1$.

It is known that the generating function associated with the enumeration of non-backtracking walks on finite graphs is a rational matrix-valued function of the parameter; such function is also closely related to graph-theoretical results such as Ihara's theorem and the zeta function on graphs. In [P. Grindrod, D. J. Higham, V. Noferini, The deformed graph Laplacian and its application to network centrality analysis, SIAM J. Matrix Anal. Appl. 39(1), 310--341, 2018], the radius of convergence of the generating function was studied for simple (i.e., undirected, unweighted and with no loops) graphs, and shown to depend on the number of cycles in the graph. In this paper, we use technologies from the theory of polynomial and rational matrices to greatly extend these results by studying the radius of convergence of the corresponding generating function for general, possibly directed and/or weighted, graphs. We give an analogous characterization of the radius of convergence for directed unweighted graphs, showing that it depends on the number of cycles in the undirectization of the graph. For weighted graphs, we provide for the first time an exact formula for the radius of convergence, improving a previous result that exhibited a lower bound. Finally, we consider also backtracking-downweighted walks on unweighted digraphs, and we prove a version of Ihara's theorem in that case.

We define a compact local Smith-McMillan form of a rational matrix $R(\lambda)$ as the diagonal matrix whose diagonal elements are the nonzero entries of a local Smith-McMillan form of $R(\lambda)$. We show that a recursive rank search procedure, applied to a block-Toeplitz matrix built on the Laurent expansion of $R(\lambda)$ around an arbitrary complex point $\lambda_0$, allows us to compute a compact local Smith-McMillan form of that rational matrix $R(\lambda)$ at the point $\lambda_0$, provided we keep track of the transformation matrices used in the rank search. It also allows us to recover the root polynomials of a polynomial matrix and root vectors of a rational matrix, at an expansion point $\lambda_0$. Numerical tests illustrate the promising performance of the resulting algorithm.

We revisit the concept of a minimal basis through the lens of the theory of modules over a commutative ring $R$. We first review the conditions for the existence of a basis for submodules of $R^n$ where $R$ is a B\'{e}zout domain. Then, we define the concept of invertible basis of a submodule of $R^n$ and, when $R$ is an elementary divisor domain, we link it to the Main Theorem of [G. D. Forney Jr., SIAM J. Control 13, 493--520, 1975]. Over an elementary divisor domain, the submodules admitting an invertible basis are precisely the free pure submodules of $R^n$. As an application, we let $\Omega \subseteq \mathbb{C}$ be either a connected compact set or a connected open set, and we specialize to $R=\mathcal{A}$, the ring of functions that are analytic on $\Omega$. We show that, for any matrix $A(z) \in \mathcal{A}^{m \times n}$, $\mathrm{ker} \ A(z) \cap \mathcal{A}^n$ is a free $\mathcal{A}$-module and admits an invertible basis, or equivalently a basis that is full rank upon evaluation at any $\lambda \in \Omega$. Finally, given $\lambda \in \Omega$, we use invertible bases to define and study maximal sets of root vectors at $\lambda$ for $A(z)$. This in particular allows us to define eigenvectors also for analytic matrices that do not have full column rank.

Given a possibly singular matrix polynomial $P(z)$, we study how the eigenvalues, eigenvectors, root polynomials, minimal indices, and minimal bases of the pencils in the vector space $\mathbb{DL}(P)$ introduced in Mackey, Mackey, Mehl, and Mehrmann [SIAM J. Matrix Anal. Appl. 28(4), 971-1004, 2006] are related to those of $P(z)$. If $P(z)$ is regular, it is known that those pencils in $\mathbb{DL}(P)$ satisfying the generic assumptions in the so-called eigenvalue exclusion theorem are strong linearizations for $P(z)$. This property and the block-symmetric structure of the pencils in $\mathbb{DL}(P)$ have made these linearizations among the most influential for the theoretical and numerical treatment of structured regular matrix polynomials. However, it is also known that, if $P(z)$ is singular, then none of the pencils in $\mathbb{DL}(P)$ is a linearization for $P(z)$. In this paper, we prove that despite this fact a generalization of the eigenvalue exclusion theorem holds for any singular matrix polynomial $P(z)$ and that such a generalization allows us to recover all the relevant quantities of $P(z)$ from any pencil in $\mathbb{DL}(P)$ satisfying the eigenvalue exclusion hypothesis. Our proof of this general theorem relies heavily in the representation of the pencils in $\mathbb{DL} (P)$ via B\'{e}zoutians by Nakatsukasa, Noferini and Townsend [SIAM J. Matrix Anal. Appl. 38(1), 181-209, 2015].

We study the eigendecompositions of para-Hermitian matrices $H(z)$, that is, matrix-valued functions that are analytic and Hermitian on the unit circle $S^1 \subset \mathbb C$. In particular, we fill existing gaps in the literature and prove the existence of a decomposition $H(z)=U(z)D(z)U(z)^P$ where, for all $z \in S^1$, $U(z)$ is unitary, $U(z)^P=U(z)^*$ is its conjugate transpose, and $D(z)$ is real diagonal; moreover, $U(z)$ and $D(z)$ are analytic functions of $w=z^{1/N}$ for some positive integer $N$, and $U(z)^P$ is the so-called para-Hermitian conjugate of $U(z)$. This generalizes the celebrated theorem of Rellich for matrix-valued functions that are analytic and Hermitian on the real line. We also show that there also exists a decomposition $H(z)=V(z)C(z)V(z)^P$ where $C(z)$ is pseudo-circulant, $V(z)$ is unitary and both are analytic in $z$. We argue that, in fact, a version of Rellich's theorem can be stated for matrix-valued function that are analytic and Hermitian on any line or any circle on the complex plane. Moreover, we extend these results to para-Hermitian matrices whose entries are Puiseux series (that is, on the unit circle they are analytic in $w$ but possibly not in $z$). Finally, we discuss the implications of our results on the singular value decomposition of a matrix whose entries are $S^1$-analytic functions of $w$, and on the sign characteristics associated with unimodular eigenvalues of $*$-palindromic matrix polynomials.

In this paper we revisit the greatest common right divisor (GCRD) extraction from a set of polynomial matrices $P_i(\lambda)\in \F[\la]^{m_i\times n}$, $i=1,\ldots,k$ with coefficients in a generic field $\F$, and with common column dimension $n$. We give necessary and sufficient conditions for a matrix $G(\la)\in \F[\la]^{\ell\times n}$ to be a GCRD using the Smith normal form of the $m \times n$ compound matrix $P(\lambda)$ obtained by concatenating $P_i(\lambda)$ vertically, where $m=\sum_{i=1}^k m_i$. We also describe the complete set of degrees of freedom for the solution $G(\la)$, and we link it to the Smith form and Hermite form of $P(\la)$. We then give an algorithm for constructing a particular minimum rank solution for this problem when $\F=\C$ or $\R$, using state-space techniques. This new method works directly on the coefficient matrices of $P(\la)$, using orthogonal transformations only. The method is based on the staircase algorithm, applied to a particular pencil derived from a generalized state-space model of $P(\la)$.

We explore the concept of eigenvalue avoidance, which is well understood for real symmetric and Hermitian matrices, for other classes of structured matrices. We adopt a differential geometric perspective and study the generic behaviour of the eigenvalues of regular and injective curves $t \in ]a,b[ \mapsto A(t) \in \mathcal{N} $ where $\mathcal{N}$ is a smooth real Riemannian submanifold of either $\mathbb{R}^{n \times n}$ or $\mathbb{C}^{n \times n}$. We focus on the case where $\mathcal{N}$ corresponds to some class of (real or complex) structured matrices including skew-symmetric, skew-Hermitian, orthogonal, unitary, banded symmetric, banded Hermitian, banded skew-symmetric, and banded skew-Hermitian. We argue that for some structures eigenvalue avoidance always happens, whereas for other structures this may depend on the parity of the size, on the numerical value of the multiple eigenvalue, and possibly on the value of determinant. As a further application of our tools we also study singular value avoidance for unstructured matrices in $\mathbb{R}^{m \times n}$ or $\mathbb{C}^{m \times n}$.

Zeros of rational transfer function matrices $R(\lambda)$ are the eigenvalues of associated polynomial system matrices $P(\lambda)$, under minimality conditions. In this paper we define a structured condition number for a simple eigenvalue $\lambda_0$ of a (locally) minimal polynomial system matrix $P(\lambda)$, which in turn is a simple zero $\lambda_0$ of its transfer function matrix $R(\lambda)$. Since any rational matrix can be written as the transfer function of a polynomial system matrix, our analysis yield a structured perturbation theory for simple zeros of rational matrices $R(\lambda)$. To capture all the zeros of $R(\lambda)$, regardless of whether they are poles or not, we consider the notion of root vectors. As corollaries of the main results, we pay particular attention to the special case of $\lambda_0$ being not a pole of $R(\lambda)$ since in this case the results get simpler and can be useful in practice. We also compare our structured condition number with Tisseur's unstructured condition number for eigenvalues of matrix polynomials, and show that the latter can be unboundedly larger. Finally, we corroborate our analysis by numerical experiments., Comment: 20 pages, 6 figures

The notion of root polynomials of a polynomial matrix $P(\lambda)$ was 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]. In this paper, we extend such a systematic approach to general rational matrices $R(\lambda)$, 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 $R(\lambda)$, without any need to assume that $R(\lambda)$ 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 $R(\lambda)$ 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 $\lambda_0$. If $\lambda_0$ is also a pole, then it is necessary to apply a preprocessing step that removes the pole while making it possible to recover the 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.

We extend the notion of nonbacktracking walks from unweighted graphs to graphs whose edges have a nonnegative weight. Here the weight associated with a walk is taken to be the product over the weights along the individual edges. We give two ways to compute the associated generating function, and corresponding node centrality measures. One method works directly on the original graph and one uses a line graph construction followed by a projection. The first method is more efficient, but the second has the advantage of extending naturally to time-evolving graphs. Computational results are also provided.

We analyse the recovery of different roles in a network modelled by a directed graph, based on the so-called Neighbourhood Pattern Similarity approach. Our analysis uses results from random matrix theory to show that when assuming 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 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.

We study walk-based centrality measures for time-ordered network sequences. For the case of standard dynamic walk-counting, we show how to derive and compute centrality measures induced by analytic functions. We also prove that dynamic Katz centrality, based on the resolvent function, has the unique advantage of allowing computations to be performed entirely at the node level. We then consider two distinct types of backtracking and develop a framework for capturing dynamic walk combinatorics when either or both is disallowed.

We use results concerning the Smith forms of circulant matrices to identify when cyclically presented groups have free abelianisation and so can be Labelled Oriented Graph (LOG) groups. We generalize a theorem of Odoni and Cremona to show that for a fixed defining word, whose corresponding representer polynomial has an irreducible factor that is not cyclotomic and not equal to $\pm t$, there are at most finitely many $n$ for which the corresponding $n$-generator cyclically presented group has free abelianisation. We classify when Campbell and Robertson's generalized Fibonacci groups $H(r,n,s)$ are LOG groups and when the Sieradski groups are LOG groups. We prove that amongst Johnson and Mawdesley's groups of Fibonacci type, the only ones that can be LOG groups are Gilbert-Howie groups $H(n,m)$. We conjecture that if a Gilbert-Howie group is a LOG group, then it is a Sieradski group, and prove this in certain cases (in particular, for fixed $m$, the conjecture can only be false for finitely many $n$). We obtain necessary conditions for a cyclically presented group to be a connected LOG group in terms of the representer polynomial and apply them to the Prishchepov groups.

We state some widely satisfied hypotheses, depending only on two functions $g$ and $h$, under which the composition of a stable algorithm for $g$ and a stable algorithm for $h$ is a stable algorithm for the composition $g \circ h$.

Tropical roots of tropical polynomials have been previously studied and used to localize roots of classical polynomials and eigenvalues of matrix polynomials. We extend the theory of tropical roots from tropical polynomials to tropical Laurent series. Our proposed definition ensures that, as in the polynomial case, there is a bijection between tropical roots and slopes of the Newton polygon associated with the tropical Laurent series. We show that, unlike in the polynomial case, there may be infinitely many tropical roots; moreover, there can be at most two tropical roots of infinite multiplicity. We then apply the new theory by relating the inner and outer radii of convergence of a classical Laurent series to the behavior of the sequence of tropical roots of its tropicalization. Finally, as a second application, we discuss localization results both for roots of scalar functions that admit a local Laurent series expansion and for nonlinear eigenvalues of regular matrix valued functions that admit a local Laurent series expansion.

We study the empirical spectral distribution (ESD) for complex n x n matrix polynomials of degree k. We obtain exact formulae for the almost sure limit of the ESD in two distinct scenarios: (1) n -> \infty with k constant and (2) k -> \infty with n bounded by O(k^P) for some P>0. The main tools used are the logarithmic potential of some measure related to the matrix polynomial, and some classical estimates on the singular values of full random matrices with i.i.d. entries., Comment: arXiv admin note: text overlap with arXiv:2005.07501

We develop a complete theory for the combinatorics of walk-counting on a directed graph in the case where each backtracking step is downweighted by a given factor. By deriving expressions for the associated generating functions, we also obtain linear systems for computing centrality measures in this setting. In particular, we show that backtrack-downweighted Katz-style network centrality can be computed at the same cost as standard Katz. Studying the limit of this centrality measure at its radius of convergence also leads to a new expression for backtrack-downweighted eigenvector centrality that generalizes previous work to the case where directed edges are present. The new theory allows us to combine advantages of standard and nonbacktracking cases, avoiding localization while accounting for tree-like structures. We illustrate the behaviour of the backtrack-downweighted centrality measure on both synthetic and real networks.

We define the empirical spectral distribution (ESD) of a random matrix polynomial with invertible leading coefficient, and we study it for complex $n \times n$ Gaussian monic matrix polynomials of degree $k$. We obtain exact formulae for the almost sure limit of the ESD in two distinct scenarios: (1) $n \rightarrow \infty$ with $k$ constant and (2) $k \rightarrow \infty$ with $n$ constant. The main tool for our approach is the replacement principle by Tao, Vu and Krishnapur. Along the way, we also develop some auxiliary results of potential independent interest: we slightly extend a result by B\"{u}rgisser and Cucker on the tail bound for the norm of the pseudoinverse of a non-zero mean matrix, and we obtain several estimates on the singular values of certain structured random matrices., Comment: 21 pages, 2 figures

We study the problem of finding the nearest $\Omega$-stable matrix to a certain matrix $A$, i.e., the nearest matrix with all its eigenvalues in a prescribed closed set $\Omega$. Distances are measured in the Frobenius norm. An important special case is finding the nearest Hurwitz or Schur stable matrix, which has applications in systems theory. We describe a reformulation of the task as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices. The problem can then be solved using standard methods from the theory of Riemannian optimization. The resulting algorithm is remarkably fast on small-scale and medium-scale matrices, and returns directly a Schur factorization of the minimizer, sidestepping the numerical difficulties associated with eigenvalues with high multiplicity.

A standard approach to compute the roots of a univariate polynomial is to compute the eigenvalues of an associated \emph{confederate} matrix instead, such as, for instance the companion or comrade matrix. The eigenvalues of the confederate matrix can be computed by Francis's QR algorithm. Unfortunately, even though the QR algorithm is provably backward stable, mapping the errors back to the original polynomial coefficients can still lead to huge errors. However, the latter statement assumes the use of a non-structure exploiting QR algorithm. In [J. Aurentz et al., Fast and backward stable computation of roots of polynomials, SIAM J. Matrix Anal. Appl., 36(3), 2015] it was shown that a structure exploiting QR algorithm for companion matrices leads to a structured backward error on the companion matrix. The proof relied on decomposing the error into two parts: a part related to the recurrence coefficients of the basis (monomial basis in that case) and a part linked to the coefficients of the original polynomial. In this article we prove that the analysis can be extended to other classes of comrade matrices. We first provide an alternative backward stability proof in the monomial basis using structured QR algorithms; our new point of view shows more explicitly how a structured, decoupled error on the confederate matrix gets mapped to the associated polynomial coefficients. This insight reveals which properties must be preserved by a structure exploiting QR algorithm to end up with a backward stable algorithm. We will show that the previously formulated companion analysis fits in this framework and we will analyze in more detail Jacobi polynomials (Comrade matrices) and Chebyshev polynomials (Colleague matrices).

We study the Smith forms of matrices of the form $f(C_g)$ where $f(t),g(t)\in R[t]$, $C_g$ is the companion matrix of the (monic) polynomial $g(t)$, and $R$ is an elementary divisor domain. Prominent examples of such matrices are circulant matrices, skew-circulant matrices, and triangular Toeplitz matrices. In particular, we reduce the calculation of the Smith form of the matrix $f(C_g)$ to that of the matrix $F(C_G)$, where $F,G$ are quotients of $f(t),g(t)$ by some common divisor. This allows us to express the last non-zero determinantal divisor of $f(C_g)$ as a resultant. A key tool is the observation that a matrix ring generated by $C_g$ -- the companion ring of $g(t)$ -- is isomorphic to the polynomial ring $Q_g=R[t]/$. We relate several features of the Smith form of $f(C_g)$ to the properties of the polynomial $g(t)$ and the equivalence classes $[f(t)]\in Q_g$. As an application we let $f(t)$ be the Alexander polynomial of a torus knot and $g(t)=t^n-1$, and calculate the Smith form of the circulant matrix $f(C_g)$. By appealing to results concerning cyclic branched covers of knots and cyclically presented groups, this provides the homology of all Brieskorn manifolds $M(r,s,n)$ where $r,s$ are coprime., Comment: 20 pages

We propose a new approach to the theory of conditioning for numerical analysis problems for which both classical and stochastic perturbation theory fail to predict the observed accuracy of computed solutions. To motivate our ideas, we present examples of problems that are discontinuous at a given input and have infinite classical and stochastic condition number, but where the solution is still computed to machine precision without relying on structured algorithms. Stimulated by the failure of classical and stochastic perturbation theory in capturing such phenomena, we define and analyse a weak worst-case and a weak stochastic condition number. This new theory is a more powerful predictor of the accuracy of computations than existing tools, especially when the worst-case and the expected sensitivity of a problem to perturbations of the input is not finite. We apply our analysis to the computation of simple eigenvalues of matrix polynomials, including the more difficult case of singular matrix polynomials. In addition, we show how the weak condition numbers can be estimated in practice., Comment: 24 pages, 3 figures

Sylvester's law of inertia states that the number of positive, negative and zero eigenvalues of Hermitian matrices is preserved under congruence transformations. The same is true of generalized Hermitian definite eigenvalue problems, in which the two matrices are allowed to undergo different congruence transformations, but not for the indefinite case. In this paper we investigate the possible change in inertia under congruence for generalized Hermitian indefinite eigenproblems, and derive sharp bounds that show the inertia of the two individual matrices often still provides useful information about the eigenvalues of the pencil, especially when one of the matrices is almost definite. A prominent application of the original Sylvester's law is in finding the number of eigenvalues in an interval. Our results can be used for estimating the number of real eigenvalues in an interval for generalized indefinite and nonlinear eigenvalue problems., Comment: 19 pages

We revisit the landmark paper [D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann, SIAM J. Matrix Anal. Appl., 28 (2006), pp.~971--1004] and, by viewing matrices as coefficients for bivariate polynomials, we provide concise proofs for key properties of linearizations for matrix polynomials. We also show that every pencil in the double ansatz space is intrinsically connected to a B\'{e}zout matrix, which we use to prove the eigenvalue exclusion theorem. In addition our exposition allows for any polynomial basis and for any field. The new viewpoint also leads to new results. We generalize the double ansatz space by exploiting its algebraic interpretation as a space of B\'{e}zout pencils to derive new linearizations with potential applications in the theory of structured matrix polynomials. Moreover, we analyze the conditioning of double ansatz space linearizations in the important practical case of a Chebyshev basis., Comment: 19 pages

Hidden-variable resultant methods are a class of algorithms for solving multidimensional polynomial rootfinding problems. In two dimensions, when significant care is taken, they are competitive practical rootfinders. However, in higher dimensions they are known to miss zeros, calculate roots to low precision, and introduce spurious solutions. We show that the hidden variable resultant method based on the Cayley (Dixon or B\'ezout) matrix is inherently and spectacularly numerically unstable by a factor that grows exponentially with the dimension. We also show that the Sylvester matrix for solving bivariate polynomial systems can square the condition number of the problem. In other words, two popular hidden variable resultant methods are numerically unstable, and this mathematically explains the difficulties that are frequently reported by practitioners. Regardless of how the constructed polynomial eigenvalue problem is solved, severe numerical difficulties will be present. Along the way, we prove that the Cayley resultant is a generalization of Cramer's rule for solving linear systems and generalize Clenshaw's algorithm to an evaluation scheme for polynomials expressed in a degree-graded polynomial basis., Comment: 24 pages, 1 figure