Semiseparable Matrices Research Articles

A New Algorithm for the Nonequispaced Fast Fourier Transform on the Rotation Group

We develop an approximate algorithm to efficiently calculate the discrete Fourier transform on the rotation group $\text{SO}(3)$. Our method needs $\mathcal{O}(L^3 \log L \log (1/\varepsilon) + \log^3 (1/\varepsilon)\,Q)$ arithmetic operations for a degree-$L$ transform at $Q$ nodes free of choice and with desired accuracy $\varepsilon$. Our main contribution is a method that allows us to replace finite expansions in Wigner-$d$ functions of arbitrary orders with those of low orders. It is based on new insights into the structure of related semiseparable matrices. This enables us to employ an established divide-and-conquer algorithm for symmetric semiseparable eigenproblems together with the fast multipole method to achieve an efficient algorithm.

A Fast Randomized Algorithm for Computing a Hierarchically Semiseparable Representation of a Matrix

Randomized sampling has recently been proven a highly efficient technique for computing approximate factorizations of matrices that have low numerical rank. This paper describes an extension of such techniques to a wider class of matrices that are not themselves rank-deficient but have off-diagonal blocks that are; specifically, the class of so-called hierarchically semiseparable (HSS) matrices. HSS matrices arise frequently in numerical analysis and signal processing, particularly in the construction of fast methods for solving differential and integral equations numerically. The HSS structure admits algebraic operations (matrix-vector multiplications, matrix factorizations, matrix inversion, etc.) to be performed very rapidly, but only once the HSS representation of the matrix has been constructed. How to rapidly compute this representation in the first place is much less well understood. The present paper demonstrates that if an $N\times N$ matrix can be applied to a vector in $O(N)$ time, and if individual entries of the matrix can be computed rapidly, then provided that an HSS representation of the matrix exists, it can be constructed in $O(N\,k^{2})$ operations, where $k$ is an upper bound for the numerical rank of the off-diagonal blocks. The point is that when legacy codes (based on, e.g., the fast multipole method) can be used for the fast matrix-vector multiply, the proposed algorithm can be used to obtain the HSS representation of the matrix, and then well-established techniques for HSS matrices can be used to invert or factor the matrix.

Distributed non-equilibrium Green’s function algorithms for the simulation of nanoelectronic devices with scattering

Through the non-equilibrium Green’s function (NEGF) formalism, quantum-scale device simulation can be performed with the inclusion of electron-phonon scattering. However, the simulation of realistically sized devices under the NEGF formalism typically requires prohibitive amounts of memory and computation time. Two of the most demanding computational problems for NEGF simulation involve mathematical operations with structured matrices called semiseparable matrices. In this work, we present parallel approaches for these computational problems which allow for efficient distribution of both memory and computation based upon the underlying device structure. This is critical when simulating realistically sized devices due to the aforementioned computational burdens. First, we consider determining a distributed compact representation for the retarded Green’s function matrix GR. This compact representation is exact and allows for any entry in the matrix to be generated through the inherent semiseparable structure. The second parallel operation allows for the computation of electron density and current characteristics for the device. Specifically, matrix products between the distributed representation for the semiseparable matrix GR and the self-energy scattering terms in Σ< produce the less-than Green’s function G<. As an illustration of the computational efficiency of our approach, we stably generate the mobility for nanowires with cross-sectional sizes as large as 4.5 nm, assuming an atomistic model with scattering.

Distributed Control in Multiple Dimensions: A Structure Preserving Computational Technique

We consider the problem of analysis and control of discretely distributed systems in multidimensional arrays. For spatially invariant systems in one spatial dimension, we build an efficient arithmetic that preserves the rational Laurent operator structure, leading to fast iterative methods of solving Lyapunov and Riccati equations and block diagonalizations, and thus arbitrarily non-conservative stability analysis and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> distributed controller synthesis. These one-dimensional results are then used to build an efficient arithmetic and controller synthesis procedure in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensions by induction. The extension of these techniques from Laurent operators with rational symbols to sequentially semi-separable matrices yields a procedure for linear computational complexity analysis and controller synthesis for finite extent heterogeneous multidimensional systems with boundary conditions. The procedures are demonstrated on two computational examples.

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.

On some classes of structured matrices with algebraic trigonometric eigenvalues

Diagonal plus semiseparable matrices are constructed, the eigenvalues of which are algebraic numbers expressed by simple closed trigonometric formulas.

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.

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.

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.

Using semiseparable matrices to compute the SVD of a general matrix product/quotient

In this work we reduce the computation of the singular values of a general product/quotient of matrices to the computation of the singular values of an upper triangular semiseparable matrix. Compared to the reduction into a bidiagonal matrix the reduction into semiseparable form exhibits a nested subspace iteration. Hence, when there are large gaps between the singular values, these gaps manifest themselves already during the reduction algorithm in contrast to the bidiagonal case.

Orthogonal Rational Functions with real coefficients and semiseparable matrices

When one wants to use Orthogonal Rational Functions (ORFs) in system identification or control theory, it is important to be able to avoid complex calculations. In this paper we study ORFs whose numerator and denominator polynomial have real coefficients. These ORFs with real coefficients (RORFs) appear when the poles and the interpolation points appear in complex conjugate pairs, which is a natural condition. Further we deduce that there is a strong connection between RORFs and semiseparable matrices.

A multiple shift [formula omitted]-step for structured rank matrices

Eigenvalue computations for structured rank matrices are the subject of many investigations nowadays. There exist methods for transforming matrices into structured rank form, Q R -algorithms for semiseparable and semiseparable plus diagonal form, methods for reducing structured rank matrices efficiently to Hessenberg form and so forth. Eigenvalue computations for the symmetric case, involving semiseparable and semiseparable plus diagonal matrices have been thoroughly explored. A first attempt for computing the eigenvalues of nonsymmetric matrices via intermediate Hessenberg-like matrices (i.e. a matrix having all subblocks in the lower triangular part of rank at most one) was restricted to the single shift strategy. Unfortunately this leads in general to the use of complex shifts switching thereby from real to complex operations. This paper will explain a general multishift implementation for Hessenberg-like matrices (semiseparable matrices are a special case and hence also admit this approach). Besides a general multishift Q R -step, this will also admit restriction to real computations when computing the eigenvalues of arbitrary real matrices. Details on the implementation are provided as well as numerical experiments proving the viability of the presented approach.

A Superfast Algorithm for Toeplitz Systems of Linear Equations

In this paper we develop a new superfast solver for Toeplitz systems of linear equations. To solve Toeplitz systems many people use displacement equation methods. With displacement structures, Toeplitz matrices can be transformed into Cauchy-like matrices using the FFT or other trigonometric transformations. These Cauchy-like matrices have a special property, that is, their off-diagonal blocks have small numerical ranks. This low-rank property plays a central role in our superfast Toeplitz solver. It enables us to quickly approximate the Cauchy-like matrices by structured matrices called sequentially semiseparable (SSS) matrices. The major work of the constructions of these SSS forms can be done in precomputations (independent of the Toeplitz matrix entries). These SSS representations are compact because of the low-rank property. The SSS Cauchy-like systems can be solved in linear time with linear storage. Excluding precomputations the main operations are the FFT and SSS system solve, which are both very efficient. Our new Toeplitz solver is stable in practice. Numerical examples are presented to illustrate the efficiency and the practical stability.

Efficient solution of nonlinear elliptic problems using hierarchical matrices with Broyden updates

The numerical solution of nonlinear problems is usually connected with Newton’s method. Due to its computational cost, variants (so-called inexact and quasi–Newton methods) have been developed in which the arising inverse of the Jacobian is replaced by an approximation. In this article we present a new approach which is based on Broyden updates. This method does not require to store the update history since the updates are explicitly added to the matrix. In addition to updating the inverse we introduce a method which constructs updates of the LU decomposition. To this end, we present an algorithm for the efficient multiplication of hierarchical and semi-separable matrices. Since an approximate LU decomposition of finite element stiffness matrices can be efficiently computed in the set of hierarchical matrices, the complexity of the proposed method scales almost linearly. Numerical examples demonstrate the effectiveness of this new approach.

A QZ-method based on semiseparable matrices

This manuscript focusses on an alternative method for computing the eigenvalues of a pencil of two matrices, based on semiseparable matrices. An effective reduction of a matrix pair to lower semiseparable, upper triangular form will be presented as well as a QZ-iteration for this matrix pair. Important to remark is that this reduction procedure also inherits a kind of nested subspace iteration as was the case when solving the standard eigenvalue problem with semiseparable matrices. It will also be shown, that the QZ-iteration for a semiseparable-triangular matrix pair is closely related to the QZ-iteration for a Hessenberg-triangular matrix pair.

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 $$T$$ satisfies property A, one can easily construct a diagonal matrix $$\hat{D}$$ such that $$\hat{D}T\hat{D}$$ has the lowest condition number over all matrices $$DTD$$ , for any choice of diagonal matrix $$D$$ . 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.

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 n 3 term in the computational complexity. Finally we illustrate that special choices of the diagonal create a specific convergence behavior.

A bibliography on semiseparable matrices*

Currently there is a growing interest in semiseparable matrices and generalized semiseparable matrices. To gain an appreciation of the historical evolution of this concept, we present in this paper an extensive list of publications related to the field of semiseparable matrices. It is interesting to see that semiseparable matrices were investigated in different fields, e.g., integral equations, statistics, vibrational analysis, independently of each other. Also notable is the fact that leading statisticians at that time used semiseparable matrices without knowing their inverses to be tridiagonal. During this historical evolution the definition of semiseparable matrices has always been a difficult point leading to misunderstandings; sometimes they were defined as the inverses of irreducible tridiagonal matrices leading to generator representable matrices, while in other cases they were defined as matrices having low rank blocks below the diagonal. In this overview we present a list of interesting results which contributed to the evolution of the concept of semiseparable matrices as we now know them.

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.

A note on the representation and definition of semiseparable matrices

AbstractIn this paper the definition of semiseparable matrices is investigated. Properties of the frequently used definition and the corresponding representation by generators are deduced. Corresponding to the class of tridiagonal matrices another definition of semiseparable matrices is introduced preserving the nice properties dual to the class of tridiagonal matrices. Several theorems and properties are included showing the viability of this alternative definition.Because of the alternative definition, the standard representation of semiseparable matrices is not satisfying anymore. The concept of a representation is explicitly formulated and a new kind of representation corresponding to the alternative definition is given. It is proved that this representation keeps all the interesting properties of the generator representation. Copyright © 2005 John Wiley &amp; Sons, Ltd.