An Implicit Q Theorem for Hessenberg-like Matrices

The implicit Q theorem for Hessenberg matrices is a widespread and powerful theorem. It is used in the development of, for example, implicit QR algorithms to compute the eigendecomposition of Hessenberg matrices. Moreover it can also be used to prove the essential uniqueness of orthogonal similarity transformations of matrices to Hessenberg form. The theorem is also valid for symmetric tridiagonal matrices, proving thereby also in the symmetric case its power. Currently there is a growing interest to so-called semiseparable matrices. These matrices can be considered as the inverses of tridiagonal matrices. In a similar way, one can consider Hessenberg-like matrices as the inverses of Hessenberg matrices. In this paper, we formulate and prove an implicit Q theorem for the class of Hessenberg-like matrices. We introduce the notion of strongly unreduced Hessenberg-like matrices and also a method for transforming matrices via orthogonal transformations to this form is proposed. Moreover, as the theorem is valid for Hessenberg-like matrices it is also valid for symmetric semiseparable matrices. The research was partially supported by the Research Council K.U.Leuven, project OT/00/16 (SLAP: Structured Linear Algebra Package), by the Fund for Scientific Research–Flanders (Belgium), projects G.0078.01 (SMA: Structured Matrices and their Applications), G.0176.02 (ANCILA: Asymptotic aNalysis of the Convergence behavior of Iterative methods in numerical Linear Algebra), G.0184.02 (CORFU: Constructive study of Orthogonal Functions) and G.0455.0 (RHPH: Riemann-Hilbert problems, random matrices and Padé-Hermite approximation), and by the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister’s Office for Science, Technology and Culture, project IUAP V-22 (Dynamical Systems and Control: Computation, Identification & Modelling). This research was partially supported by by MIUR, grant number 2004015437 (third author). The scientific responsibility rests with the authors.     

Linear complexity algorithm for semiseparable matrices

A new linear complexity algorithm for general nonsingular semiseparable matrices is presented. For symmetric matrices whose semiseparability rank equals to 1 this algorithm leads to an explicit formula for the inverse matrix.Supported in part by the NSF Grant DMS 9306357     

Algorithms for inversion of diagonal plus semiseparable operator matrices

LetH be a Hilbert space andRHH be a bounded linear operator represented by an operator matrix which is a sum of a diagonal and of a semiseparable type of order one operator matrices. We consider three methods for solution of the operator equationRx=y. The obtained results yields new algorithms for solution of integral equations and for inversion of matrices.     

On common invariant cones for families of matrices

The existence and construction of common invariant cones for families of real matrices is considered. The complete results are obtained for 2×2 matrices (with no additional restrictions) and for families of simultaneously diagonalizable matrices of any size. Families of matrices with a shared dominant eigenvector are considered under some additional conditions.     

Orthogonal similarity transformation of a symmetric matrix into a diagonal-plus-semiseparable one with free choice of the diagonal

In this paper we describe an orthogonal similarity transformation for transforming arbitrary symmetric matrices into a diagonal-plus-semiseparable matrix, where we can freely choose the diagonal. Very recently an algorithm was proposed for transforming arbitrary symmetric matrices into similar semiseparable ones. This reduction is strongly connected to the reduction to tridiagonal form. The class of semiseparable matrices can be considered as a subclass of the diagonalplus- semiseparable matrices. Therefore we can interpret the proposed algorithm here as an extension of the reduction to semiseparable form. A numerical experiment is performed comparing thereby the accuracy of this reduction algorithm with respect to the accuracy of the traditional reduction to tridiagonal form, and the reduction to semiseparable form. The experiment indicates that all three reduction algorithms are equally accurate. Moreover it is shown in the experiments that asymptotically all the three approaches have the same complexity, i.e. that they have the same factor preceding the n³ term in the computational complexity. Finally we illustrate that special choices of the diagonal create a specific convergence behavior. The research was partially supported by the Research Council K.U.Leuven, project OT/05/40 (Large rank structured matrix computations), by the Fund for Scientific Research–Flanders (Belgium), projects G.0078.01 (SMA: Structured Matrices and their Applications), G.0176.02 (ANCILA: Asymptotic aNalysis of the Convergence behavior of Iterative methods in numerical Linear Algebra), G.0184.02 (CORFU: Constructive study of Orthogonal Functions) and G.0455.0 (RHPH: Riemann-Hilbert problems, random matrices and Padé-Hermite approximation), and by the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister's Office for Science, Technology and Culture, project IUAP V-22 (Dynamical Systems and Control: Computation, Identification & Modelling). The scientific responsibility rests with the authors.     

A quasiseparable approach to five-diagonal CMV and Fiedler matrices

Recent work in the characterization of structured matrices in terms of characteristic polynomials of principal submatrices is furthered in this paper. Some classical classes of matrices with quasiseparable structure include tridiagonal (related to real orthogonal polynomials) and banded matrices, unitary Hessenberg matrices (related to Szegö polynomials), and semiseparable matrices, as well as others. Hence working with the class of quasiseparable matrices provides new results which generalize and unify classical results.Previous work has focused on characterizing (H,1)-quasiseparable matrices, matrices with order-one quasiseparable structure that are also upper Hessenberg. In this paper, the authors introduce the concept of a twist transformation, and use such transformations to explain the relationship between (H,1)-quasiseparable matrices and the subclass of (1,1)-quasiseparable matrices (without the upper Hessenberg restriction) which are related to the same systems of polynomials. These results generalize the discoveries of Cantero, Fiedler, Kimura, Moral and Velázquez of five-diagonal matrices related to Horner and Szegö polynomials in the context of quasiseparable matrices.     

On the convergence properties of the orthogonal similarity transformations to tridiagonal and semiseparable (plus diagonal) form

In this paper, we will compare the convergence properties of three basic reduction methods, by placing them in a general framework. It covers the reduction to tridiagonal, semiseparable and semiseparable plus diagonal form. These reductions are often used as the first step in the computation of the eigenvalues and/or eigenvectors of arbitrary matrices. In this way, the calculation of the eigenvalues using, for example, the QR-algorithm reduces in complexity. First we will investigate the convergence properties of these three reduction algorithms. It will be shown that for the partially reduced matrices at step k of any of these reduction algorithms, the lower right k × k (already reduced) sub-block will have the Lanczos–Ritz values, w.r.t. a certain starting vector. It will also be shown that the reductions to semiseparable and to semiseparable plus diagonal form have an extra convergence behavior a special type of subspace iteration is performed on the lower right k × k submatrix, which contains these Ritz-values. Secondly we look in more detail at the behavior of the involved subspace iteration. It will be shown that the reduction method can be interpreted as a nested type of multi-shift iteration. Theoretical results will be presented, making it possible to predict the convergence behavior of these reduction algorithms. Also a theoretical bound on the convergence rate is presented. Finally we illustrate by means of numerical examples, how it is possible to tune the convergence behavior such that it can become a powerful tool for certain applications.     

REPRINT OF: A quasiseparable approach to five-diagonal CMV and Fiedler matrices

Recent work in the characterization of structured matrices in terms of characteristic polynomials of principal submatrices is furthered in this paper. Some classical classes of matrices with quasiseparable structure include tridiagonal (related to real orthogonal polynomials) and banded matrices, unitary Hessenberg matrices (related to Szegö polynomials), and semiseparable matrices, as well as others. Hence working with the class of quasiseparable matrices provides new results which generalize and unify classical results.Previous work has focused on characterizing (H,1)-quasiseparable matrices, matrices with order-one quasiseparable structure that are also upper Hessenberg. In this paper, the authors introduce the concept of a twist transformation, and use such transformations to explain the relationship between (H,1)-quasiseparable matrices and the subclass of (1,1)-quasiseparable matrices (without the upper Hessenberg restriction) which are related to the same systems of polynomials. These results generalize the discoveries of Cantero, Fiedler, Kimura, Moral and Velázquez of five-diagonal matrices related to Horner and Szegö polynomials in the context of quasiseparable matrices.     

An implicit QR algorithm for symmetric semiseparable matrices

The QR algorithm is one of the classical methods to compute the eigendecomposition of a matrix. If it is applied on a dense n × n matrix, this algorithm requires O(n³) operations per iteration step. To reduce this complexity for a symmetric matrix to O(n), the original matrix is first reduced to tridiagonal form using orthogonal similarity transformations. In the report (Report TW360, May 2003) a reduction from a symmetric matrix into a similar semiseparable one is described. In this paper a QR algorithm to compute the eigenvalues of semiseparable matrices is designed where each iteration step requires O(n) operations. Hence, combined with the reduction to semiseparable form, the eigenvalues of symmetric matrices can be computed via intermediate semiseparable matrices, instead of tridiagonal ones. The eigenvectors of the intermediate semiseparable matrix will be computed by applying inverse iteration to this matrix. This will be achieved by using an O(n) system solver, for semiseparable matrices. A combination of the previous steps leads to an algorithm for computing the eigenvalue decompositions of semiseparable matrices. Combined with the reduction of a symmetric matrix towards semiseparable form, this algorithm can also be used to calculate the eigenvalue decomposition of symmetric matrices. The presented algorithm has the same order of complexity as the tridiagonal approach, but has larger lower order terms. Numerical experiments illustrate the complexity and the numerical accuracy of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.     

Set-theoretic defining equations of the tangential variety of the Segre variety

We prove a set-theoretic version of the Landsberg-Weyman Conjecture on the defining equations of the tangential variety of a Segre product of projective spaces. We introduce and study the concept of exclusive rank. For the proof of this conjecture, we use a connection to the author’s previous work and re-express the tangential variety as the variety of principal minors of symmetric matrices that have exclusive rank no more than 1. We discuss applications to semiseparable matrices, tensor rank versus border rank, context-specific independence models and factor analysis models.     

Fast algorithms for Toeplitz and Hankel matrices

The paper gives a self-contained survey of fast algorithms for solving linear systems of equations with Toeplitz or Hankel coefficient matrices. It is written in the style of a textbook. Algorithms of Levinson-type and Schur-type are discussed. Their connections with triangular factorizations, Padè recursions and Lanczos methods are demonstrated. In the case in which the matrices possess additional symmetry properties, split algorithms are designed and their relations to butterfly factorizations are developed.     

Fast algorithms for hierarchically semiseparable matrices

Semiseparable matrices and many other rank‐structured matrices have been widely used in developing new fast matrix algorithms. In this paper, we generalize the hierarchically semiseparable (HSS) matrix representations and propose some fast algorithms for HSS matrices. We represent HSS matrices in terms of general binary HSS trees and use simplified postordering notation for HSS forms. Fast HSS algorithms including new HSS structure generation and HSS form Cholesky factorization are developed. Moreover, we provide a new linear complexity explicit ULV factorization algorithm for symmetric positive definite HSS matrices with a low‐rank property. The corresponding factors can be used to solve the HSS systems also in linear complexity. Numerical examples demonstrate the efficiency of the algorithms. All these algorithms have nice data locality. They are useful in developing fast‐structured numerical methods for large discretized PDEs (such as elliptic equations), integral equations, eigenvalue problems, etc. Some applications are shown. Copyright © 2009 John Wiley & Sons, Ltd.     

New fast divide‐and‐conquer algorithms for the symmetric tridiagonal eigenvalue problem

In this paper, two accelerated divide‐and‐conquer (ADC) algorithms are proposed for the symmetric tridiagonal eigenvalue problem, which cost O(N²r) flops in the worst case, where N is the dimension of the matrix and r is a modest number depending on the distribution of eigenvalues. Both of these algorithms use hierarchically semiseparable (HSS) matrices to approximate some intermediate eigenvector matrices, which are Cauchy‐like matrices and are off‐diagonally low‐rank. The difference of these two versions lies in using different HSS construction algorithms, one (denoted by ADC1) uses a structured low‐rank approximation method and the other (ADC2) uses a randomized HSS construction algorithm. For the ADC2 algorithm, a method is proposed to estimate the off‐diagonal rank. Numerous experiments have been carried out to show their stability and efficiency. These algorithms are implemented in parallel in a shared memory environment, and some parallel implementation details are included. Comparing the ADCs with highly optimized multithreaded libraries such as Intel MKL, we find that ADCs could be more than six times faster for some large matrices with few deflations. Copyright © 2016 John Wiley & Sons, Ltd.     

Accurate computations with Said-Ball-Vandermonde matrices

A generalization of the Vandermonde matrices which arise when the power basis is replaced by the Said-Ball basis is considered. When the nodes are inside the interval (0,1), then those matrices are strictly totally positive. An algorithm for computing the bidiagonal decomposition of those Said-Ball-Vandermonde matrices is presented, which allows us to use known algorithms for totally positive matrices represented by their bidiagonal decomposition. The algorithm is shown to be fast and to guarantee high relative accuracy. Some numerical experiments which illustrate the good behaviour of the algorithm are included.     

Unitarily achievable zero patterns and traces of words in A and A

Our primary objective is to identify a natural and substantial problem about unitary similarity on arbitrary complex matrices: which 0-patterns may be achieved for any given n-by-n complex matrix via some unitary similarity of it. To this end, certain restrictions on “achievable” 0-patterns are mentioned, both positional and, more important, on the maximum number of achievable 0’s. Prior results fitting this general question are mentioned, as well as the “first” unresolved pattern (for 3-by-3 matrices!). In the process a recent question is answered.A closely related additional objective is to mention the best known bound for the maximum length of words necessary for the application of Specht’s theorem about which pairs of complex matrices are unitarily similar, which seems not widely known to matrix theorists. In the process, we mention the number of words necessary for small size matrices.     

Vector least-squares solutions for coupled singular matrix equations

The weighted least-squares solutions of coupled singular matrix equations are too difficult to obtain by applying matrices decomposition. In this paper, a family of algorithms are applied to solve these problems based on the Kronecker structures. Subsequently, we construct a computationally efficient solutions of coupled restricted singular matrix equations. Furthermore, the need to compute the weighted Drazin and weighted Moore–Penrose inverses; and the use of Tian's work and Lev-Ari's results are due to appearance in the solutions of these problems. The several special cases of these problems are also considered which includes the well-known coupled Sylvester matrix equations. Finally, we recover the iterative methods to the weighted case in order to obtain the minimum D-norm G-vector least-squares solutions for the coupled Sylvester matrix equations and the results lead to the least-squares solutions and invertible solutions, as a special case.     

A small note on the scaling of symmetric positive definite semiseparable matrices

In this paper we will adapt a known method for diagonal scaling of symmetric positive definite tridiagonal matrices towards the semiseparable case. Based on the fact that a symmetric, positive definite tridiagonal matrix satisfies property A, one can easily construct a diagonal matrix such that has the lowest condition number over all matrices , for any choice of diagonal matrix . Knowing that semiseparable matrices are the inverses of tridiagonal matrices, one can derive similar properties for semiseparable matrices. Here, we will construct the optimal diagonal scaling of a semiseparable matrix, based on a new inversion formula for semiseparable matrices. Some numerical experiments are performed. In a first experiment we compare the condition numbers of the semiseparable matrices before and after the scaling. In a second numerical experiment we compare the scalability of matrices coming from the reduction to semiseparable form and matrices coming from the reduction to tridiagonal form. *The research was partially supported by the Research Council K.U. Leuven, project OT/00/16 (SLAP: Structured Linear Algebra Package), by the Fund for Scientific Research–Flanders (Belgium), projects G.0078.01 (SMA: Structured Matrices and their Applications), G.0176.02 (ANCILA: Asymptotic aNalysis of the Convergence behavior of Iterative methods in numerical Linear Algebra), G.0184.02 (CORFU: Constructive study of Orthogonal Functions) and G.0455.0 (RHPH: Riemann–Hilbert problems, random matrices and Padé–Hermite approximation), and by the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister's Office for Science, Technology and Culture, project IUAP V-22 (Dynamical Systems and Control: Computation, Identification & Modelling). The scientific responsibility rests with the authors. The second author participates in the SCCM program, Gates 2B, Stanford University, CA, USA and is also partially supported by the NSF. The first author visited the second one with a grant by the Fund for Scientific Research–Flanders (Belgium).     

How real is your matrix?

For scalars there is essentially just one way to define reality, real part and to measure nonreality. In this paper various ways of defining respective concepts for complex-entried matrices are considered. In connection with this, products of circulant and diagonal matrices often appear and algorithms to approximate additively and multiplicatively with them are devised. Multiplicative structures have applications, for instance, in diffractive optics, preconditioning and fast Fourier expansions.