Quasiseparable Matrices Research Articles

Numerical solution of the boundary value problems for the biharmonic equations via quasiseparable representations

AbstractThe paper incorporates new methods of numerical linear algebra for the approximation of the biharmonic equation with potential, namely, numerical solution of the Dirichlet problem for $$ \left( \frac{d}{dx}\right) ^4u(x)+c(x)u(x)=\phi (x),\quad 0<x<1. $$ d dx 4 u ( x ) + c ( x ) u ( x ) = ϕ ( x ) , 0 < x < 1 . High-order discrete finite difference operators are presented, constructed on the basis of discrete Hermitian derivatives, and the associated Discrete Biharmonic Operator (DBO). It is shown that the matrices associated with the discrete operator belong to a class of quasiseparable matrices of low rank matrices. The application of quasiseparable representation of rank structured matrices yields fast and stable algorithm for variable potentials c(x). Numerical examples corroborate the claim of high order accuracy of the algorithm, with optimal complexity O(N).

Efficient inversion of matrix ϕ-functions of low order

The paper is concerned with efficient numerical methods for solving a linear system ϕ(A)x=b, where ϕ(z) is a ϕ-function and A∈RN×N. In particular in this work we are interested in the computation of ϕ(A)−1b for the case where ϕ(z)=ϕ1(z)=ez−1z and ϕ(z)=ϕ2(z)=ez−1−zz2. Under suitable conditions on the spectrum of A we design fast algorithms for computing both ϕℓ(A)−1 and ϕℓ(A)−1b based on Newton's iteration and Krylov-type methods, respectively. Adaptations of these schemes for structured matrices are considered. In particular the cases of banded and more generally quasiseparable matrices are investigated. Numerical results are presented to show the effectiveness of our proposed algorithms.

The bisection eigenvalue method for unitary Hessenberg matrices via their quasiseparable structure

If $N_0$ is a normal matrix, then the Hermitian matrices $\\frac{1}{2}(N_0+N_0^*)$ and $\\frac{i}{2}(N_0^*-N_0)$have the same eigenvectors as $N_0$. Their eigenvalues are the real part and the imaginary part of the eigenvalues of $N_0$, respectively.If $N_0$ is unitary, then only the real part of each of its eigenvalues and the sign ofthe imaginary part is needed to completely determine the eigenvalue, sincethe sum of the squares of these two parts is known to be equal to $1$.Since a unitary upper Hessenberg matrix $U$ has a quasiseparable structure of order oneand we express the matrix $A=\frac{1}{2}(U+U^*)$ as quasiseparable matrix of order two , we can findthe real part of the eigenvalues and, when needed, a corresponding eigenvector $x$, by using techniquesthat have been established in the paper by Eidelman and Haimovici [Oper. Theory Adv. Appl., 271 (2018), pp.&nbsp;181&ndash;200].<br>We describe here a fast procedure, which takes only $1.7\%$ of the bisection method time, to find the signof the imaginary part.For instance, in the worst case only, we build one rowof the quasiseparable matrix $U$ and multiply it by a known eigenvector of$A$, as the main part of the procedure.This case occurs for our algorithm when amongthe $4$ numbers $\pm\cos t\pm i \sin t$ there are exactly $2$ eigenvalues andthey are opposite, so that we have to distinguish between the case $\lambda,-\lambda$and the case $\overline\lambda,-\overline\lambda$.<br>The performance of the developedalgorithm is illustrated by a series of numerical tests. The algorithm is more accurate and many times faster(when executed in Matlab) than forgeneral Hermitian matrices of quasiseparable order two, because the action of the quasiseparable generators,which are small matrices in the previous cited paper, can be replaced by scalars, most of them real numbers.

Structured condition numbers for Sylvester matrix equation with parameterized quasiseparable matrices

Structured condition numbers for Sylvester matrix equation with parameterized quasiseparable matrices

Structured generalized eigenvalue condition numbers for parameterized quasiseparable matrices

In this paper, when A and B are {1;1}-quasiseparable matrices, we consider the structured generalized relative eigenvalue condition numbers of the pair $$(A, \, B)$$ with respect to relative perturbations of the parameters defining A and B in the quasiseparable and the Givens-vector representations of these matrices. A general expression is derived for the condition number of the generalized eigenvalue problems of the pair $$(A,\, B)$$ , where A and B are any differentiable function of a vector of parameters with respect to perturbations of such parameters. Moreover, the explicit expressions of the corresponding structured condition numbers with respect to the quasiseparable and Givens-vector representation via tangents for $$\{1; 1\}$$ -quasiseparable matrices are derived. Our proposed condition numbers can be computed efficiently by utilizing the recursive structure of quasiseparable matrices. We investigate relationships between various condition numbers of structured generalized eigenvalue problem when A and B are {1;1}-quasiseparable matrices. Numerical results show that there are situations in which the unstructured condition number can be much larger than the structured ones.

Theoretical Modelling of Charge Transport Properties of Individual Single-Wall Carbon Nanotubes

Experimental projection of transport properties of semiconductor devices faces a challenge nowadays. As devices scale to nanometre scale range, the classical transport equations used in current device simulators can no longer be applied. Conversely, the use of a more accurate and better non-equilibrium green function (NEGF) is limited by the fact that it requires excessive quantum of memory and computational time, having quasi-separable matrices that are extremely convoluted to solve. This work exploits the Boltzmann Transport Equation (BTE) to assess the transport properties of carbon nanotubes. Previous works on solving the BTE have employed either a stochastic method or an approximate method, both of which do not possess the necessary properties for practical device applications. Therefore, this work represents the first direct theoretical solution of the BTE for one-dimensional carbon nanotubes that can be utilized for practical device applications. The complete spectrum of transport in CNTs extending from ohmic to high-field through ballistic transmission is examined to delineate plethora of transport properties. The transport for arbitrary values of the electric field is based on the BTE applied to experimental data on CNTs. In the limit of low field, the mobility expressions are obtained in terms of the mean free path (mfp) that is distinctly shorter than the length of the sample. The ohmic resistance is quantized a value of 6.453k-ohms consistent with experimental findings with transmission approaching unity as channel length shrinks below the carrier mfp. The emission of a quantum was observed to lower the saturation velocity that is independent of scattering and hence ballistic. Transition to ballistic domain was found to occur when the channel length is scaled below the ballistic limit that is shown to be the extended version of the long-channel mfp modulated by injections from the contacts, yet the mobility degrades. The mobility degradation is shown to be the cause of resistance quantum in the low-channel length limit. These findings are important in predicting the transport properties of low-dimensional CNTs.

Time and space efficient generators for quasiseparable matrices

The class of quasiseparable matrices is defined by the property that any submatrix entirely below or above the main diagonal has small rank, namely below a bound called the order of quasiseparability. These matrices arise naturally in solving PDE's for particle interaction with the Fast Multi-pole Method (FMM), or computing generalized eigenvalues. From these application fields, structured representations and algorithms have been designed in numerical linear algebra to compute with these matrices in time linear in the matrix dimension and either quadratic or cubic in the quasiseparability order. Motivated by the design of the general purpose exact linear algebra library LinBox, and by algorithmic applications in algebraic computing, we adapt existing techniques introduce novel ones to use quasiseparable matrices in exact linear algebra, where sub-cubic matrix arithmetic is available. In particular, we will show, the connection between the notion of quasiseparability and the rank profile matrix invariant, that we have introduced in 2015. It results in two new structured representations, one being a simpler variation on the hierarchically semiseparable storage, and the second one exploiting the generalized Bruhat decomposition. As a consequence, most basic operations, such as computing the quasiseparability orders, applying a vector, a block vector, multiplying two quasiseparable matrices together, inverting a quasiseparable matrix, can be at least as fast and often faster than previous existing algorithms.

On the sensitivity of generators for the QR factorization of quasiseparable matrices with total nonpositivity

In this paper, we provide a relatively robust representation for the QR factorization of quasiseparable matrices with total nonpositivity. This representation allows us to develop a structure-preserving perturbation analysis. Consequently, stronger perturbation bounds are obtained to show that its generators determine the factors Q and R to high relative accuracy, independent of any conventional condition number. This means that it is possible to accurately compute the QR factorization by operating on these generators.

Structured condition numbers for linear systems with parameterized quasiseparable coefficient matrices

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.

Structured eigenvalue condition numbers for parameterized quasiseparable matrices

The development of fast algorithms for performing computations with $$n\times n$$n×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 defining these matrices may suffer, 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 different 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 $$\{1;1\}$${1ź1}-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.

Quasiseparable Hessenberg reduction of real diagonal plus low rank matrices and applications

We present a novel algorithm to perform the Hessenberg reduction of an n×n matrix A of the form A=D+UV⁎ where D is diagonal with real entries and U and V are n×k matrices with k≤n. The algorithm has a cost of O(n2k) arithmetic operations and is based on the quasiseparable matrix technology. Applications are shown to solving polynomial eigenvalue problems and some numerical experiments are reported in order to analyze the stability of the approach.

The spectral connection matrix for classical orthogonal polynomials of a single parameter

In this paper we study the so-called connection problem of, given a polynomial expressed in the basis of one set of orthogonal polynomials, computing the coefficients with respect to a different set of orthogonal polynomials. We restrict our current study to the classical real orthogonal polynomials of the Hermite, Laguerre, and Gegenbauer (including Legendre) families.The computational tool for this work is the class of quasiseparable matrices. While the relationships between orthogonal polynomials and rank-structured matrices such as quasiseparable matrices are very well-known, in this paper we investigate a more recently considered relationship. We prove that, while the matrix that implements the desired connection is not itself quasiseparable, it is an eigenvector matrix of one that is quasiseparable. We suggest to refer to this structured matrix as the spectral connection matrix.Finally, we present a simple algorithm exploiting the computationally favorable properties of quasiseparable matrices to implement the desired change of basis. By exploiting the quasiseparable structure, this algorithm enjoys an order of magnitude reduction of complexity as compared to the simple method of inverting the connection matrix directly. While not the focus of the paper, some very preliminary numerical experimentation shows some positive indications that even with this reduction in complexity the accuracy of the resulting change of basis algorithm is comparable to that of inverting the connection matrix directly.

Stability of QR-based fast system solvers for a subclass of quasiseparable rank one matrices

The development of fast algorithms to perform computations with quasiseparable matrices has received a lot of attention in the last decade. Many different algorithms have been presented by several research groups all over the world. Despite this intense activity, to the best of our knowledge, there is no rounding error analysis published for these fast algorithms. In this paper, we present error analyses for two fast solvers of quasiseparable linear systems when they are applied on order one quasiseparable matrices that include the diagonal in the lower triangular rank structure. Both solvers are based on computing first the QR factorization of the coefficient matrix, and their error analyses require novel structured techniques for proving rigorously that only one of the considered algorithms is backward stable, while the other one is not. Two fundamental consequences of this work are: (i) users should employ with caution fast algorithms for quasiseparable matrices since they may be unstable; and (ii) a lot of work has to be done to identify which fast algorithms for quasiseparable matrices are backward stable among the large family available in the literature.

Nested Product Decomposition of Quasiseparable Matrices

This paper introduces a nested product decomposition of an $n\times n$ quasiseparable matrix $A$. The nested product is formed from small Householder transformations and very sparse matrices with only a few nonzeros isolated to a single column. The decomposition is computed by the direct application of Householder transformations and requires $O(n^2)$ operations. Once the matrix has been decomposed, matrix-vector products can be formed and linear systems can be solved with $O(n)$ operations. An error analysis shows that the algorithm is backward stable.

Differential qd Algorithm with Shifts for Rank-Structured Matrices

Although QR iterations dominate in eigenvalue computations, there are several important cases when alternative LR-type algorithms may be preferable, in particular, in the symmetric tridiagonal case where the differential qd algorithm with shifts (dqds) proposed by Fernando and Parlett enjoys often faster convergence while preserving high relative accuracy. In eigenvalue computations for rank-structured matrices, the QR algorithm is also a popular choice since, in the symmetric case, the rank structure is preserved. In the unsymmetric case, however, the QR algorithm destroys the rank structure and, hence, LR-type algorithms come in to play once again. In the current paper we adapt several variants of qd algorithms to quasi-separable matrices. Remarkably, one of them, when applied to Hessenberg matrices, becomes a direct generalization of the dqds algorithm for tridiagonal matrices. Therefore, it can be applied to such important matrices as companion and confederate and provides an alternative algorithm for finding roots of a polynomial represented in a basis of orthogonal polynomials. Results of preliminary numerical experiments are presented.

Representation of quasiseparable matrices using excluded sums and equivalent charges

A new parametric representation for the general quasiseparable matrix is derived, based on the ideas from the multipole method. It uses functional expansions and successive skeleton approximations, approximations, but finally is formulated in the matrix language. The number of parameters is linear in the dimension of the matrix and in the quasiseparable rank. Stable numerical algorithm is provided for the computation of parameters, defining the decomposition. Numerical examples illustrate the effectiveness of our approach.

Qd-Type Methods for Quasiseparable Matrices

In the last few years many numerical techniques for computing eigenvalues of structured rank matrices have been proposed. Most of them are based on QR iterations since, in the symmetric case, the rank structure is preserved and high accuracy is guaranteed. In the unsymmetric case, however, the QR algorithm destroys the rank structure, which is instead preserved if LR iterations are used. We consider a wide class of quasiseparable matrices which can be represented in terms of the same parameters involved in their Neville factorization. This class, if assumptions are made to prevent possible breakdowns, is closed under LR steps. Moreover, we propose an implicit shifted LR method with a linear cost per step, which resembles the qd method for tridiagonal matrices. We show that for totally nonnegative quasiseparable matrices the algorithm is stable and breakdowns cannot occur if the Laguerre shift, or other shift strategy preserving nonnegativity, is used. Computational evidence shows that good accuracy is obtained also when applied to symmetric positive definite matrices.

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.

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.

Green’s matrices

Our goal is to identify and understand matrices A that share essential properties of the unitary Hessenberg matrices M that are fundamental for Szegö’s orthogonal polynomials. Those properties include: (i) Recurrence relations connect characteristic polynomials {rk(x)} of principal minors of A. (ii) A is determined by generators (parameters generalizing reflection coefficients of unitary Hessenberg theory). (iii) Polynomials {rk(x)} correspond not only to A but also to a certain “CMV-like” five-diagonal matrix. (iv) The five-diagonal matrix factors into a product BC of block diagonal matrices with 2 × 2 blocks. (v) Submatrices above and below the main diagonal of A have rank 1. (vi) A is a multiplication operator in the appropriate basis of Laurent polynomials. (vii) Eigenvectors of A can be expressed in terms of those polynomials.Conditions (v) connects our analysis to the study of quasi-separable matrices. But the factorization requirement (iv) narrows it to the subclass of “Green’s matrices” that share Properties (i)–(vii).The key tool is “twist transformations” that provide 2n matrices all sharing characteristic polynomials of principal minors with A. One such twist transformation connects unitary Hessenberg to CMV. Another twist transformation explains findings of Fiedler who noticed that companion matrices give examples outside the unitary Hessenberg framework. We mention briefly the further example of a Daubechies wavelet matrix. Infinite matrices are included.