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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

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

Let $\cL = (\cL_1,\cL_2)$ be a list consisting of a sublist $\cL_1$ of powers of irreducible (monic) scalar polynomials over an algebraically closed field $\FF$, and a sublist $\cL_2$ of nonnegative integers. For an arbitrary such list $\cL$, we give easily verifiable necessary and sufficient conditions for $\cL$ to be the list of elementary divisors and minimal indices of some $T$-palindromic quadratic matrix polynomial with entries in the field $\FF$. For $\cL$ satisfying these conditions, we show how to explicitly construct a $T$-palindromic quadratic matrix polynomial having $\cL$ as its structural data; that is, we provide a $T$-palindromic quadratic realization of $\cL$. Our construction of $T$-palindromic realizations is accomplished by taking a direct sum of low bandwidth $T$-palindromic blocks, closely resembling the Kronecker canonical form of matrix pencils. An immediate consequence of our in-depth study of the structure of $T$-palindromic quadratic polynomials is that all even grade $T$-palindromic matrix polynomials have a $T$-palindromic strong quadratification. Finally, using a particular M\"{o}bius transformation, we show how all of our results can be easily extended to quadratic matrix polynomials with $T$-even structure.

The structural data of any rational matrix R(\lambda ), i.e., the structural indices of its poles and zeros together with the minimal indices of its left and right nonespaces, is known to satisfy a simple condition involving certain sums of these indices. This fundamental constraint was first proved by Van Dooren in 1978; here we refer to this result as the rational index sum theorem. An analogous result for polynomial matrices has been independently discovered (and rediscovered) several times in the past three decades. In this paper we clarify the connection between these two seemingly different index sum theorems, describe a little bit of the history of their development, and discuss their curious apparent unawareness of each other. Finally, we use the connection between these results to solve a fundamental inverse problem for rational matrices---for which lists \scrL of prescribed structural data does there exist some rational matrix R(\lambda ) that realizes exactly the list \scrL ? We show that Van Dooren's condition is the only constraint on rational realizability; that is, a list \scrL is the structural data of some rational matrix R(\lambda ) if and only if \scrL satisfies the rational index sum condition.

We show that the set of $n \times n$ complex symmetric matrix pencils of rank at most $r$ is the union of the closures of $\lfloor r/2\rfloor +1$ sets of matrix pencils with some, explicitly described, complete eigenstructures. As a consequence, these are the generic complete eigenstructures of $n \times n$ complex symmetric matrix pencils of rank at most $r$. We also show that these closures correspond to the irreducible components of the set of $n\times n$ symmetric matrix pencils with rank at most $r$ when considered as an algebraic set., 15 pages

We develop a complete and rigorous theory of root polynomials of arbitrary matrix polynomials, i.e., either regular or singular, and study how these vector polynomials are related to the spectral properties of matrix polynomials. We pay particular attention to the so-called maximal sets of root polynomials and prove that they carry complete information about the eigenvalues (finite or infinite) of matrix polynomials and that they are related to the matrices that transform any matrix polynomial into its Smith form. In addition, we describe clearly, for the first time in the literature, the extremality properties of such maximal sets and identify some of them whose vectors have minimal grade. Once the main theoretical properties of root polynomials have been established, the interaction of root polynomials with three problems that have attracted considerable attention in the literature is analyzed. More precisely, we study the change of root polynomials under rational transformations, or reparametrizations, of matrix polynomials, the recovery of the root polynomials of a matrix polynomial from those of its most important linearizations, and the relationship between the root polynomials of two dual pencils. We emphasize that for the case of regular matrix polynomials all the results in this paper can be translated into the language of Jordan chains, as a consequence of the well known relationship between root polynomials and Jordan chains. Therefore, a number of open problems are also solved for Jordan chains of regular matrix polynomials. We also briefly discuss how root polynomials can be used to define eigenvectors and eigenspaces for singular matrix polynomials. Supported by “Ministerio de Economía, Industria y Competitividad of Spain” and “Fondo Europeo de Desarrollo Regional (FEDER) of EU” through grants MTM2015-65798-P and MTM2017-90682-REDT

The standard way of solving the polynomial eigenvalue problem associated with a matrix polynomial is to embed the matrix coefficients of the polynomial into a matrix pencil, transforming the problem into an equivalent generalized eigenvalue problem. Such pencils are known as linearizations. Many of the families of linearizations for matrix polynomials available in the literature are extensions of the so-called family of Fiedler pencils. These families are known as generalized Fiedler pencils, Fiedler pencils with repetition, and generalized Fiedler pencils with repetition—or Fiedler-like pencils for simplicity. The goal of this work is to unify the Fiedler-like pencils approach with the more recent one based on strong block minimal bases pencils introduced in F.M. Dopico et al. (2017) [17] . To this end, we introduce a family of pencils that we have named extended block Kronecker pencils, whose members are, under some generic nonsingularity conditions, strong block minimal bases pencils, and show that, with the exception of the non-proper generalized Fiedler pencils, all Fiedler-like pencils belong to this family modulo permutations. As a consequence of this result, we obtain a much simpler theory for Fiedler-like pencils than the one available so far. Moreover, we expect this simplification to allow for further developments in the theory of Fiedler-like pencils such as global or local backward error analyses and eigenvalue conditioning analyses of polynomial eigenvalue problems solved via Fiedler-like linearizations.

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

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 show that the set of $m \times m$ complex skew-symmetric matrix polynomials of odd grade $d$, i.e., of degree at most $d$, and (normal) rank at most $2r$ is the closure of the single set of matrix polynomials with the certain, explicitly described, complete eigenstructure. This complete eigenstructure corresponds to the most generic $m \times m$ complex skew-symmetric matrix polynomials of odd grade $d$ and rank at most $2r$. In particular, this result includes the case of skew-symmetric matrix pencils ($d=1$)., 21 pages

We determine the generic complete eigenstructures for $n \times n$ complex symmetric matrix polynomials of odd grade $d$ and rank at most $r$. More precisely, we show that the set of $n \times n$ complex symmetric matrix polynomials of odd grade $d$, i.e., of degree at most $d$, and rank at most $r$ is the union of the closures of the $\lfloor rd/2\rfloor+1$ sets of symmetric matrix polynomials having certain, explicitly described, complete eigenstructures. Then, we prove that these sets are open in the set of $n \times n$ complex symmetric matrix polynomials of odd grade $d$ and rank at most $r$. In order to prove the previous results, we need to derive necessary and sufficient conditions for the existence of symmetric matrix polynomials with prescribed grade, rank, and complete eigenstructure, in the case where all their elementary divisors are different from each other and of degree $1$. An important remark on the results of this paper is that the generic eigenstructures identified in this work are completely different from the ones identified in previous works for unstructured and skew-symmetric matrix polynomials with bounded rank and fixed grade larger than one, because the symmetric ones include eigenvalues while the others not. This difference requires to use new techniques.

M��bius transformations have been used in numerical algorithms for computing eigenvalues and invariant subspaces of structured generalized and polynomial eigenvalue problems (PEPs). These transformations convert problems with certain structures arising in applications into problems with other structures and whose eigenvalues and invariant subspaces are easily related to the ones of the original problem. Thus, an algorithm that is efficient and stable for some particular structure can be used for solving efficiently another type of structured problem via an adequate M��bius transformation. A key question in this context is whether these transformations may change significantly the conditioning of the problem and the backward errors of the computed solutions, since, in that case, their use may lead to unreliable results. We present the first general study on the effect of M��bius transformations on the eigenvalue condition numbers and backward errors of approximate eigenpairs of PEPs. By using the homogeneous formulation of PEPs, we are able to obtain two clear and simple results. First, we show that, if the matrix inducing the M��bius transformation is well conditioned, then such transformation approximately preserves the eigenvalue condition numbers and backward errors when they are defined with respect to perturbations of the matrix polynomial which are small relative to the norm of the polynomial. However, if the perturbations in each coefficient of the matrix polynomial are small relative to the norm of that coefficient, then the corresponding eigenvalue condition numbers and backward errors are preserved approximately by the M��bius transformations induced by well-conditioned matrices only if a penalty factor, depending on those coefficients, is moderate. It is important to note that these simple results are no longer true if a non-homogeneous formulation is used., 33 pages, 3 figures

In this work, we propose a new method, termed as R-CORK, for the numerical solution of large-scale rational eigenvalue problems, which is based on a linearization and on a compact decomposition of the rational Krylov subspaces corresponding to this linearization. R-CORK is an extension of the compact rational Krylov method (CORK) introduced very recently by Van Beeumen et al. to solve a family of non-linear eigenvalue problems that can be expressed and linearized in certain particular ways and which include arbitrary polynomial eigenvalue problems, but not arbitrary rational eigenvalue problems. The R-CORK method exploits the structure of the linearized problem by representing the Krylov vectors in a compact form in order to reduce the cost of storage, resulting in a method with two levels of orthogonalization. The first level of orthogonalization works with vectors of the same size that the original problem, and the second level works with vectors of size much smaller than the original problem. Since vectors of the size of the linearization are never stored or orthogonalized, R-CORK is more efficient from the point of views of memory and orthogonalization than the classical rational Krylov method applied directly to the linearization. Taking into account that the R-CORK method is based on a classical rational Krylov method, to implement implicit restarting is also possible and we show how to do it in a memory efficient way. Finally, some numerical examples are included in order to show that the R-CORK method performs satisfactorily in practice.

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.

Over any field F every square matrix A can be factored into the product of two symmetric matrices as A = S1 . S2 with S_i = S_i^T ∈ F^(n,n) and either factor can be chosen nonsingular, as was discovered by Frobenius in 1910. Frobenius’ symmetric matrix factorization has been lying almost dormant for a century. The first successful method for computing matrix symmetrizers, i.e., symmetric matrices S such that SA is symmetric, was inspired by an iterative linear systems algorithm of Huang and Nong (2010) in 2013 [29, 30]. The resulting iterative algorithm has solved this computational problem over R and C, but at high computational cost. This paper develops and tests another linear equations solver, as well as eigen- and principal vector or Schur Normal Form based algorithms for solving the matrix symmetrizer problem numerically. Four new eigendata based algorithms use, respectively, SVD based principal vector chain constructions, Gram-Schmidt orthogonalization techniques, the Arnoldi method, or the Schur Normal Form of A in their formulations. They are helped by Datta’s 1973 method that symmetrizes unreduced Hessenberg matrices directly. The eigendata based methods work well and quickly for generic matrices A and create well conditioned matrix symmetrizers through eigenvector dyad accumulation. But all of the eigen based methods have differing deficiencies with matrices A that have ill-conditioned or complicated eigen structures with nontrivial Jordan normal forms. Our symmetrizer studies for matrices with ill-conditioned eigensystems lead to two open problems of matrix optimization. This research was partially supported by the Ministerio de Economía y Competitividad of Spain through the research grant MTM2012-32542. Publicado

The standard way of solving numerically a polynomial eigenvalue problem (PEP) is to use a linearization and solve the corresponding generalized eigenvalue problem (GEP). In addition, if the PEP possesses one of the structures arising very often in applications, then the use of a linearization that preserves such structure combined with a structured algorithm for the GEP presents considerable numerical advantages. Block-symmetric linearizations have proven to be very useful for constructing structured linearizations of structured matrix polynomials. In this scenario, we analyze the eigenvalue condition numbers and backward errors of approximated eigenpairs of a block symmetric linearization that was introduced by Fiedler (Linear Algebra Appl 372:325–331, 2003) for scalar polynomials and generalized to matrix polynomials by Antoniou and Vologiannidis (Electron J Linear Algebra 11:78–87, 2004). This analysis reveals that such linearization has much better numerical properties than any other block-symmetric linearization analyzed so far in the literature, including those in the well known vector space $$\mathbb {DL}(P)$$ of block-symmetric linearizations. The main drawback of the analyzed linearization is that it can be constructed only for matrix polynomials of odd degree, but we believe that it will be possible to extend its use to even degree polynomials via some strategies in the near future.

In this paper, we consider the different eigenvalue condition numbers for matrix polynomials used in the literature and we compare them. One of these condition numbers is a generalization of the Wilkinson condition number for the standard eigenvalue problem. This number has the disadvantage of only being defined for finite eigenvalues. In order to give a unified approach to all the eigenvalues of a matrix polynomial, both finite and infinite, two (homogeneous) condition numbers have been defined in the literature. In their definition, very different approaches are used. One of the main goals of this note is to show that, when the matrix polynomial has a moderate degree, both homogeneous numbers are essentially the same and one of them provides a geometric interpretation of the other. We also show how the homogeneous condition numbers compare with the "Wilkinson-like" eigenvalue condition number and how they extend this condition number to zero and infinite eigenvalues., 19 pages, 1 figure

We construct a new family of strong linearizations of rational matrices considering the polynomial part of them expressed in a basis that satisfies a three term recurrence relation. For this purpose, we combine the theory developed by Amparan et al., MIMS EPrint 2016.51, and the new linearizations of polynomial matrices introduced by Fa{\ss}bender and Saltenberger, Linear Algebra Appl., 525 (2017). In addition, we present a detailed study of how to recover eigenvectors of a rational matrix from those of its linearizations in this family. We complete the paper by discussing how to extend the results when the polynomial part is expressed in other bases, and by presenting strong linearizations that preserve the structure of symmetric or Hermitian rational matrices. A conclusion of this work is that the combination of the results in this paper with those in Amparan et al., MIMS EPrint 2016.51, allows us to use essentially all the strong linearizations of polynomial matrices developed in the last fifteen years to construct strong linearizations of any rational matrix by expressing such matrix in terms of its polynomial and strictly proper parts., Comment: 40 pages, no figures

This paper defines for the first time strong linearizations of arbitrary rational matrices, studies in depth properties and diferent characterizations of such linear matrix pencils, and develops infinitely many examples of strong linearizations that can be explicitly and easily constructed from a minimal state-space realization of the strictly proper part of the considered rational matrix and the coefficients of the polynomial part. As a consequence, the results in this paper establish a rigorous foundation for the numerical computation of the complete structure of zeros and poles, both finite and at infinity, of any rational matrix by applying any well known backward stable algorithm for generalized eigenvalue problems to any of the strong linearizations explicitly constructed in this work. Since the results of this paper require to use several concepts that are not standard in matrix computations, a considerable effort has been done to make the paper as self-contained as possible.

The set POL d , r m × n of m × n complex matrix polynomials of grade d and (normal) rank at most r in a complex ( d + 1 ) m n dimensional space is studied. For r = 1 , … , min ⁡ { m , n } − 1 , we show that POL d , r m × n is the union of the closures of the r d + 1 sets of matrix polynomials with rank r, degree exactly d, and explicitly described complete eigenstructures. In addition, for the full-rank rectangular polynomials, i.e. r = min ⁡ { m , n } and m ≠ n , we show that POL d , r m × n coincides with the closure of a single set of the polynomials with rank r, degree exactly d, and the described complete eigenstructure. These complete eigenstructures correspond to generic m × n matrix polynomials of grade d and rank at most r.

Given a matrix polynomial P ( λ ) = ∑ i = 0 k λ i A i of degree k , where A i are n × n matrices with entries in a field F , the development of linearizations of P ( λ ) that preserve whatever structure P ( λ ) might posses has been a very active area of research in the last decade. Most of the structure-preserving linearizations of P ( λ ) discovered so far are based on certain modifications of block-symmetric linearizations. The block-symmetric linearizations of P ( λ ) available in the literature fall essentially into two classes: linearizations based on the so-called Fiedler pencils with repetition, which form a finite family, and a vector space of dimension k of block-symmetric pencils, called DL ( P ) , such that most of its pencils are linearizations. One drawback of the pencils in DL ( P ) is that none of them is a linearization when P ( λ ) is singular. In this paper we introduce new vector spaces of block-symmetric pencils, most of which are strong linearizations of P ( λ ) . The dimensions of these spaces are O ( n 2 ) , which, for n ≥ k , are much larger than the dimension of DL ( P ) . When k is odd, many of these vector spaces contain linearizations also when P ( λ ) is singular. The coefficients of the block-symmetric pencils in these new spaces can be easily constructed as k × k block-matrices whose n × n blocks are of the form 0, ± α I n , ± α A i , or arbitrary n × n matrices, where α is an arbitrary nonzero scalar.

The set of m x n singular matrix pencils with normal rank at most r is an algebraic set with r + 1 irreducible components. These components are the closure of the orbits (under strict equivalence) of r 1 matrix pencils which are in Kronecker canonical form. In this paper, we provide a new explicit description of each of these irreducible components which is a parametrization of each component. Therefore one can explicitly construct any pencil in each of these components. The new description of each of these irreducible components consists of the sum of r rank-1 matrix pencils, namely, a column polynomial vector of degree at most 1 times a row polynomial vector of degree at most 1, where we impose one of these two vectors to have degree zero. The number of row vectors with zero degree determines each irreducible component. (C) 2017 Elsevier Inc. All rights reserved.

The aim of the present paper is to analyze the behavior of Fiedler companion matrices in the polynomial root-finding problem from the point of view of conditioning of eigenvalues. More precisely, we compare: (a) the condition number of a given root of a monic polynomial p(z) with the condition number of as an eigenvalue of any Fiedler matrix of p(z), (b) the condition number of as an eigenvalue of an arbitrary Fiedler matrix with the condition number of as an eigenvalue of the classical Frobenius companion matrices, and (c) the pseudozero sets of p(z) and the pseudospectra of any Fiedler matrix of p(z). We prove that, if the coefficients of the polynomial p(z) are not too large and not all close to zero, then the conditioning of any root of p(z) is similar to the conditioning of as an eigenvalue of any Fiedler matrix of p(z). On the contrary, when p(z) has some large coefficients, or they are all close to zero, the conditioning of as an eigenvalue of any Fiedler matrix can be arbitrarily much larger than its conditioning as a root of p(z) and, moreover, when p(z) has some large coefficients there can be two different Fiedler matrices such that the ratio between the condition numbers of as an eigenvalue of these two matrices can be arbitrarily large. Finally, we relate asymptotically the pseudozero sets of p(z) with the pseudospectra of any given Fiedler matrix of p(z), and the pseudospectra of any two Fiedler matrices of p(z). This work was partially supported by the Ministerio de Economía y Competitividad of Spain through Grants MTM-2012-32542, MTM2015-68805-REDT and MTM2015-65798-P, and by the Engineering and Physical Sciences Research Council Grant EP/I005293 (Javier Pérez).

The development of strong linearizations preserving whatever structure a matrix polynomial might possess has been a very active area of research in the last years, since such linearizations are the starting point of numerical algorithms for computing eigenvalues of structured matrix polynomials with the properties imposed by the considered structure. In this context, Hermitian matrix polynomials are one of the most important classes of matrix polynomials arising in applications and their real eigenvalues are of great interest. The sign characteristic is a set of signs attached to these real eigenvalues which is crucial for determining the behavior of systems described by Hermitian matrix polynomials and, therefore, it is desirable to develop linearizations that preserve the sign characteristic of these polynomials, but, at present, only one such linearization is known. In this paper, we present a complete characterization of all the Hermitian strong linearizations that preserve the sign characteristic of a given Hermitian matrix polynomial and identify several families of such linearizations that can be constructed very easily from the coefficients of the polynomial.

In this paper, strong relative perturbation bounds are developed for a number of linear algebra problems involving diagonally dominant matrices. The key point is to parameterize diagonally dominant matrices using their off-diagonal entries and diagonally dominant parts and to consider small relative componentwise perturbations of these parameters. This allows us to obtain new relative perturbation bounds for the inverse, the solution to linear systems, the symmetric indefinite eigenvalue problem, the singular value problem, and the nonsymmetric eigenvalue problem. These bounds are much stronger than traditional perturbation results, since they are independent of either the standard condition number or the magnitude of eigenvalues/singular values. Together with previously derived perturbation bounds for the LDU factorization and the symmetric positive definite eigenvalue problem, this paper presents a complete and detailed account of relative structured perturbation theory for diagonally dominant matrices. This research was partially supported by the Ministerio de Economía y Competitividad of Spain under grant MTM2012-32542. Publicado

Low-rank structured matrices have attracted much attention in the last decades, since they arise in many applications and all share the fundamental property that can be represented by O(n)$\mathcal {O}(n)$ parameters, where n×n is the size of the matrix. This property has allowed the development of fast algorithms for solving numerically many problems involving low-rank structured matrices by performing operations on the parameters describing the matrices, instead of directly on the matrix entries. Among these problems, the solution of linear systems of equations is probably the most basic and relevant one. Therefore, it is important to measure, via structured computable condition numbers, the relative sensitivity of the solutions of linear systems with low-rank structured coefficient matrices with respect to relative perturbations of the parameters representing such matrices, since this sensitivity determines the maximum accuracy attainable by fast algorithms and allows us to decide which set of parameters is the most convenient from the point of view of accuracy. To develop and analyze such condition numbers is the main goal of this paper. To this purpose, a general expression is obtained for the condition number of the solution of a linear system of equations whose coefficient matrix is any differentiable function of a vector of parameters with respect to perturbations of such parameters. Since there are many different classes of low-rank structured matrices and many different types of parameters describing them, it is not possible to cover all of them in a single work. Therefore, the general expression of the condition number is particularized to the important case of {1,1}-quasiseparable matrices and to the quasiseparable and the Givens-vector representations, in order to obtain explicit expressions of the corresponding two condition numbers that can be estimated in O(n) operations. In addition, detailed theoretical and numerical comparisons of these two condition numbers between themselves, and with respect to unstructured condition numbers, are provided, which show that there are situations in which the unstructured condition number is much larger than the structured ones, but that the opposite never happens. The approach presented in this manuscript can be generalized to other classes of low-rank structured matrices and parameterizations.

Least squares problems min(x) parallel to b - Ax parallel to(2) where the matrix A is an element of C-mXn (m >= n) has some particular structure arise frequently in applications. Polynomial data fitting is a well-known instance of problems that yield highly structured matrices, but many other examples exist. Very often, structured matrices have huge condition numbers kappa(2)(A) = parallel to A parallel to(2) parallel to A(dagger)parallel to(2) (A(dagger) is the Moore-Penrose pseudoinverse of A) and therefore standard algorithms fail to compute accurate minimum 2-norm solutions of least squares problems. In this work, we introduce a framework that allows us to compute minimum 2-norm solutions of many classes of structured least squares problems accurately, i.e., with errors parallel to(x) over cap (0) - x(0)parallel to(2)/parallel to x(0)parallel to(2) = O(u), where u is the unit roundoff, independently of the magnitude of kappa(2)(A) for most vectors b. The cost of these accurate computations is O(n(2)m) flops, i.e., roughly the same cost as standard algorithms for least squares problems. The approach in this work relies in computing first an accurate rank-revealing decomposition of A, an idea that has been widely used in recent decades to compute, for structured ill-conditioned matrices, singular value decompositions, eigenvalues, and eigenvectors in the Hermitian case and solutions of linear systems with high relative accuracy. In order to prove that accurate solutions are computed, a new multiplicative perturbation theory of the least squares problem is needed. The results presented in this paper are valid for both full rank and rank deficient problems and also in the case of underdetermined linear systems (m < n). Among other types of matrices, the new method applies to rectangular Cauchy, Vandermonde, and graded matrices, and detailed numerical tests for Cauchy matrices are presented. This work was supported by the Ministerio de Economía y Competitividad of Spain through grants MTM-2009-09281, MTM-2012-32542 (Ceballos, Dopico, and Molera) and MTM2010-18057 (Castro-González). Publicado

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 describe the generic change of the partial multiplicities at a given eigenvalue $\lambda_0$ of a regular matrix pencil $A_0+\lambda A_1$ under perturbations with low normal rank. More precisely, if the pencil $A_0+\lambda A_1$ has exactly $g$ nonzero partial multiplicities at $\lambda_0$, then for most perturbations $B_0+\lambda B_1$ with normal rank $r

A little known property of a pair of eigenvectors (column and row) of a real tridiagonal matrix is presented. With its help we can define necessary and sufficient conditions for the unique real tridiagonal matrix for which an approximate pair of complex eigenvectors is exact. Similarly, we can designate the unique real tridiagonal matrix for which two approximate real eigenvectors, with different real eigenvalues, are also exact. We close with an illustration that these unique “backward error” matrices are sensitive to small rounding errors in certain partial sums which play a key role in determining the matrices., Ministerio de Economía y Competitividad of Spain through research grant MTM2012-32542, info:eu-repo/semantics/publishedVersion

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.

Bounds for the variation of the Moore–Penrose inverse of general matrices under multiplicative perturbations are presented. Their advantages with respect to classical bounds under additive perturbations and with respect to other bounds under multiplicative perturbations available in the literature are carefully studied and established. Closely connected to these developments a complete multiplicative perturbation theory for least squares problems, valid for perturbations of any size, is also presented, improving in this way recent multiplicative perturbation bounds which are valid only to first order in the size of the perturbations. The results in this paper are mainly based on exact expressions of the perturbed Moore–Penrose inverse in terms of the unperturbed one, the perturbation matrices, and certain orthogonal projectors. Such feature makes the new results amenable to be generalized in the future to linear operators in infinite dimensional spaces.

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

The development of fast algorithms for performing computations with n x n low-rank structured matrices has been a very active area of research during the last two decades, as a consequence of the numerous applications where these matrices arise. The key ideas behind these fast algorithms are that low-rank structured matrices can be described in terms of O(n) parameters and that these algorithms operate on the parameters instead on the matrix entries. Therefore, the sensitivity of any computed quantity should be measured with respect to the possible variations that the parameters de ning these matrices may su er, since this determines the maximum accuracy of a given fast computation. In other words, it is necessary to develop condition numbers with respect to parameters for di erent magnitudes and classes of low-rank structured matrices, but, as far as we know, this has not yet been accomplished in any case. In this paper, we derive structured relative eigenvalue condition numbers for the important class of low-rank structured matrices known as {1;1}-quasiseparable matrices with respect to relative perturbations of the parameters in the quasiseparable and in the Givens-vector representations of these matrices, and we provide fast algorithms for computing them. Comparisons among the new structured condition numbers and the unstructured one are also presented, as well as numerical experiments showing that the structured condition numbers can be small in situations where the unstructured one is huge. In addition, the approach presented in this paper is general and may be extended to other problems and classes of low-rank structured matrices. This research was partially supported by Ministerio de Economía y Competitividad of Spain through grant MTM2012-32542. Publicado

The matrix Sylvester equation for congruence, or T-Sylvester equation, has recently attracted considerable attention as a consequence of its close relation to palindromic eigenvalue problems. The theory concerning T-Sylvester equations is rather well understood and there are stable and e cient numerical algorithms which solve these equations for small- to medium-sized matrices. However, developing numerical algorithms for solving large-scale T-Sylvester equations still remains an open problem. In this paper, we present several projection algorithms based on di erent Krylov spaces for solving this problem when the right-hand side of the T-Sylvester equation is a low-rank matrix. The new algorithms have been extensively tested, and the reported numerical results show that they work very well in practice, o ering a clear guidance on which algorithm is the most convenient in each situation. This work has been supported by Ministerio de Economía y Competitividad of Spain through grant MTM2012-32542. Publicado

The proceeding at: Joint ALAMA-GAMM/ANLA 2014 Meeting, took place 2014, July 14-16, in Barcelona (Spain). We present necessary and su cient conditions for the existence of a matrix polynomial when its degree, its nite and in nite elementary divisors, and its left and right minimal indices are prescribed. These conditions hold for arbitrary in nite elds and are determined mainly by the \index sum theorem", which is a fundamental relationship between the rank, the degree, the sum of all partial multiplicities, and the sum of all minimal indices of any matrix polynomial. The proof developed for the existence of such polynomial is constructive and, therefore, solves a very general inverse problem for matrix polynomials with prescribed complete eigenstructure. This result allows us to x the problem of the existence of (l)-ifications of a given matrix polynomial, as well as to determine all their possible sizes and eigenstructures. This research was partially supported by the Ministerio de Economía y Competitividad of Spain through grant MTM-2012-32542 and by the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. Publicado

The concept of linearization is fundamental for theory, applications, and spectral computations related to matrix polynomials. However, recent research on several important classes of structured matrix polynomials arising in applications has revealed that the strategy of using linearizations to develop structure-preserving numerical algorithms that compute the eigenvalues of structured matrix polynomials can be too restrictive, because some structured polynomials do not have any linearization with the same structure. This phenomenon strongly suggests that linearizations should sometimes be replaced by other low degree matrix polynomials in applied numerical computations. Motivated by this fact, we introduce equivalence relations that allow the possibility of matrix polynomials (with coefficients in an arbitrary field) to be equivalent, with the same spectral structure, but have different sizes and degrees. These equivalence relations are directly modeled on the notion of linearization, and consequently inherit the simplicity, applicability, and most relevant properties of linearizations; simultaneously, though, they are much more flexible in the possible degrees of equivalent polynomials. This flexibility allows us to define in a unified way the notions of quadratification and l-ification, to introduce the concept of companion form of arbitrary degree, and to provide concrete and simple examples of these notions that generalize in a natural and smooth way the classical first and second Frobenius companion forms. The properties of l-ifications are studied in depth; in this process a fundamental result on matrix polynomials, the “Index Sum Theorem”, is recovered and extended to arbitrary fields. Although this result is known in the systems theory literature for real matrix polynomials, it has remained unnoticed by many researchers. It establishes that the sum of the (finite and infinite) partial multiplicities, together with the (left and right) minimal indices of any matrix polynomial is equal to the rank times the degree of the polynomial. The “Index Sum Theorem” turns out to be a key tool for obtaining a number of significant results: on the possible sizes and degrees of l-ifications and companion forms, on the minimal index preservation properties of companion forms of arbitrary degree, as well as on obstructions to the existence of structured companion forms for structured matrix polynomials of even degree. This paper presents many new results, blended together with results already known in the literature but extended here to the most general setting of matrix polynomials of arbitrary sizes and degrees over arbitrary fields. Therefore we have written the paper in an expository and self-contained style that makes it accessible to a wide variety of readers.

The proceeding at: 6th Conference on Structured Numerical Linear and Multilinear Algebra: Analysis, Algorithms and Applications (SLA 2014), took place at 2014, Septembe 8-12, in Kalamata (Grece), Computing roots of scalar polynomials as the eigenvalues of Frobenius companion matrices using backward stable eigenvalue algorithms is a classical approach. The introduction of new families of companion matrices allows for the use of other matrices in the root-finding problem. In this paper, we analyze the backward stability of polynomial root-finding algorithms via Fiedler companion matrices. In other words, given a polynomial p(z), the question is to determine whether the whole set of computed eigenvalues of the companion matrix, obtained with a backward stable algorithm for the standard eigenvalue problem, are the set of roots of a nearby polynomial or not. We show that, if the coefficients of p(z) are bounded in absolute value by a moderate number, then algorithms for polynomial root-finding using Fiedler matrices are backward stable, and Fiedler matrices are as good as the Frobenius companion matrices. This allows us to use Fiedler companion matrices with favorable structures in the polynomial root-finding problem. However, when some of the coefficients of the polynomial are large, Fiedler companion matrices may produce larger backward errors than Frobenius companion matrices, although in this case neither Frobenius nor Fiedler matrices lead to backward stable computations. To prove this we obtain explicit expressions for the change, to first order, of the characteristic polynomial coefficients of Fielder matrices under small perturbations. We show that, for all Fiedler matrices except the Frobenius ones, this change involves quadratic terms in the coefficients of the characteristic polynomial of the original matrix, while for the Frobenius matrices it only involves linear terms. We present extensive numerical experiments that support these theoretical results. The effect of balancing these matrices is also investigated. This work has been supported by the Ministerio de Economía y Competitividad of Spain through grant MTM2012-32542 Publicado

Several matrix norms of the classical Frobenius companion matrices of a monic polynomial p(z) have been used in the literature to obtain simple lower and upper bounds on the absolute values of the roots lambda of p(z). Recently, M. Fiedler (2003) [9] has introduced a new family of companion matrices of p(z) that has received considerable attention and it is natural to investigate if matrix norms of Fiedler companion matrices may be used to obtain new and sharper lower and upper bounds on vertical bar lambda vertical bar. The development of such bounds requires first to know simple expressions for some relevant matrix norms of Fiedler matrices and we obtain them in the case of the 1- and infinity-matrix norms. With these expressions at hand, we will show that norms of Fiedler matrices produce many new bounds, but that none of them improves significatively the classical bounds obtained from the Frobenius companion matrices. However, we will prove that if the norms of the inverses of Fiedler matrices are used, then another family of new bounds on vertical bar lambda vertical bar is obtained and some of the bounds in this family improve significatively the bounds coming from the Frobenius companion matrices for certain polynomials. This work has been supported by the Ministerio de Economía y Competitividad of Spain through grant MTM2012-32542.

This work introduces a new perturbation bound for the L factor of the LDU factorization of (row) diagonally dominant matrices computed via the column diagonal dominance pivoting strategy. This strategy yields L and U factors which are always well-conditioned and, so, the LDU factorization is guaranteed to be a rank-revealing decomposition. The new bound together with those for the D and U factors in [F. M. Dopico and P. Koev, Numer. Math., 119 (2011), pp. 337– 371] establish that if diagonally dominant matrices are parameterized via their diagonally dominant parts and off-diagonal entries, then tiny relative componentwise perturbations of these parameters produce tiny relative normwise variations of L and U and tiny relative entrywise variations of D when column diagonal dominance pivoting is used. These results will allow us to prove in a follow-up work that such perturbations also lead to strong perturbation bounds for many other problems involving diagonally dominant matrices. Research supported in part by Ministerio de Economía y Competitividad of Spain under grant MTM2012-32542. Publicado

The proceeding at: The International Symposium "The Alan Turing Legacy" held in Madrid (Spain) in October 23-24, 2012. This symposium was organized and funded by the Real Academia de Ciencias Exactas, Físicas y Naturales of Spain and Fundación Ramón Areces. The solution of a system of linear equations is by far the most important problem in Applied Mathematics. It is important both in itself and because it is an intermediate step in many other important problems. Gaussian elimination is nowadays the standard method for solving this problem numerically on a computer and it was the first numerical algorithm to be subjected to rounding error analysis. In 1948, Alan Turing published a remarkable paper on this topic: "Rounding-off errors in matrix processes" (Quart. J. Mech. Appl. Math. 1, pp. 287-308). In this paper, Turing formulated Gaussian elimination as the matrix LU factorization and introduced the "condition number of a matrix", both of them fundamental notions of modern Numerical Analysis. In addition, Turing presented an error analysis of Gaussian elimination for general matrices that deeply influenced the spirit of the definitive analysis developed by James Wilkinson in 1961. Alan Turing's work on Gaussian elimination appears in a fascinating period for modern Numerical Analysis. Other giants of Mathematics, as John von Neumann, Herman Goldstine, and Harold Hotelling were also working in the mid-1940s on Gaussian elimination. The goal of these researchers was to find an efficient and reliable method for solving systems of linear equations in modern "automatic computers". At that time, it was not clear at all whether Gaussian elimination was a right choice or not. The purpose of this paper is to revise, at an introductory level, the contributions of Alan Turing and other authors to the error analysis of Gaussian elimination, the historical context of these contributions, and their influence on modern Numerical Analysis. La resolución de sistemas de ecuaciones lineales es sin duda el problema más importante en Matemática Aplicada. Es importante en sí mismo y también porque es un paso intermedio en la resolución de muchos otros problemas de gran relevancia. La eliminación Gaussiana es hoy en día el método estándar para resolver este problema en un ordenador y, además, fue el primer algoritmo numérico para el que se realizó un análisis de errores de redondeo. En 1948, Alan Turing publicó un artículo de gran relevancia sobre este tema: “Rounding-off errors in matrix processes” (Quart. J. Mech. Appl. Math. 1, pp. 287-308). En este artículo, Turing formuló la eliminación Gaussiana en términos de la factorización LU de una matriz e introdujo la noción de número de condición de una matriz, que son dos de las nociones más fundamentales del Análisis Numérico moderno. Además, Turing presentó un análisis de errores de la eliminación Gaussiana para matrices generales que influyó profundamente en el espíritu del análisis de errores definitivo desarrollado por Wilkinson en 1961. El trabajo de Alan Turing sobre la eliminación Gaussiana aparece en un periodo fascinante del Análisis Numérico moderno. Otros gigantes de las matemáticas como John von Neumann, Herman Goldstine y Harold Hotelling también realizaron investigaciones sobre la eliminación Gaussiana en la década de 1940-50. El objetivo de estos investigadores era encontrar un método eficiente y fiable para resolver sistemas de ecuaciones lineales en los ordenadores modernos que estaban desarrollándose por entonces. En aquella época, no estaba claro en absoluto si utilizar la eliminación Gaussiana era una elección adecuada o no. El propósito de este artículo es revisar, a nivel básico, las contribuciones realizadas por Alan Turing y otros investigadores al análisis de errores de la eliminación Gaussiana, el contexto histórico de esas contribuciones y su influencia en el Análisis Numérico moderno. This work was partially supported by the Ministerio de Economía y Competitividad of Spain through grant MTM-2009-09281. Publicado