Cauchy-like Matrix Research Articles

A Parallel Structured Divide-and-Conquer Algorithm for Symmetric Tridiagonal Eigenvalue Problems

In this paper, a parallel structured divide-and-conquer (PSDC) eigensolver is proposed for symmetric tridiagonal matrices based on ScaLAPACK and a parallel structured matrix multiplication algorithm, called PSMMA. Computing the eigenvectors via matrix-matrix multiplications is the most computationally expensive part of the divide-and-conquer algorithm, and one of the matrices involved in such multiplications is a rank-structured Cauchy-like matrix. By exploiting this particular property, PSMMA constructs the local matrices by using generators of Cauchy-like matrices without any communication, and further reduces the computation costs by using a structured low-rank approximation algorithm. Thus, both the communication and computation costs are reduced. Experimental results show that both PSMMA and PSDC are highly scalable and scale to 4096 processes at least. PSDC has better scalability than PHDC that was proposed in [J. Comput. Appl. Math. 344 (2018) 512--520] and only scaled to 300 processes for the same matrices. Comparing with \texttt{PDSTEDC} in ScaLAPACK, PSDC is always faster and achieves $1.4$x--$1.6$x speedup for some matrices with few deflations. PSDC is also comparable with ELPA, with PSDC being faster than ELPA when using few processes and a little slower when using many processes.

Involutory-Multiple-Lightweight MDS Matrices based on Cauchy-type Matrices

<p style='text-indent:20px;'>One of the best methods for constructing maximum distance separable (<inline-formula><tex-math id="M1">\begin{document}$ \operatorname{MDS} $\end{document}</tex-math></inline-formula>) matrices is based on making use of Cauchy matrices. In this paper, by using some extensions of Cauchy matrices, we introduce several new forms of <inline-formula><tex-math id="M2">\begin{document}$ \operatorname{MDS} $\end{document}</tex-math></inline-formula> matrices over finite fields of characteristic 2. A known extension of a Cauchy matrix, called the Cauchy-like matrix, with application in coding theory was introduced in 1985. One of the main contributions of this paper is to apply Cauchy-like matrices to introduce <b><inline-formula><tex-math id="M3">\begin{document}$ 2n \times 2n $\end{document}</tex-math></inline-formula> involutory <inline-formula><tex-math id="M4">\begin{document}$ \operatorname{MDS} $\end{document}</tex-math></inline-formula> matrices</b> in the semi-Hadamard form which is a generalization of the previously known methods. We make use of Cauchy-like matrices to construct <b>multiple <inline-formula><tex-math id="M5">\begin{document}$ \operatorname{MDS} $\end{document}</tex-math></inline-formula> matrices</b> which can be used in the Feistel structures. We also introduce a new extension of Cauchy matrices to be referred to as <i>Cauchy-light matrices</i>. The introduced Cauchy-light matrices are applied to construct <inline-formula><tex-math id="M6">\begin{document}$ n \times n $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M7">\begin{document}$ \operatorname{MDS} $\end{document}</tex-math></inline-formula> matrices having at least <inline-formula><tex-math id="M8">\begin{document}$ 3n-3 $\end{document}</tex-math></inline-formula> entries equal to the unit element <inline-formula><tex-math id="M9">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>; such a matrix is called a <b>lightweight <inline-formula><tex-math id="M10">\begin{document}$ \operatorname{MDS} $\end{document}</tex-math></inline-formula> matrix</b> and can be used in the lightweight cryptography. A simple closed-form expression is given for the determinant of Cauchy-light matrices.</p>

Robust Finite-Time Control of Linear System with Non-Differentiable Time-Varying Delay

Practical systems such as hybrid power systems are currently implemented around the world. In order to get the system to work properly, the systems usually require their behavior to be maintained or state values to stay within a certain threshold. However, it is difficult to form a perfect mathematical model for describing behavior of the practical systems since there may be some information (uncertainties) that is not observed. Thus, in this article, we studied the stability of an uncertain linear system with a non-differentiable time-varying delay. We constructed Lyapunov-Krasovskii functionals (LKFs) containing several symmetric positive definite matrices to obtain robust finite-time stability (RFTS) and stabilization (RFTU) of the uncertain linear system. With the controller and uncertainties in the considered system, there exist nonlinear terms occurring in the formulation process. Past research handled these nonlinear terms as new variables but this led to some difficulty from a computation point of view. Instead, we applied a novel approach via Cauchy-like matrix inequalities to handle these difficulties. In the end, we present three numerical simulations to show the effectiveness of our proposed theory.

Block pivoting implementation of a symmetric Toeplitz solver

Toeplitz matrices are characterized by a special structure that can be exploited in order to obtain fast linear system solvers. These solvers are difficult to parallelize due to their low computational cost and their closely coupled data operations. We propose to transform the Toeplitz system matrix into a Cauchy-like matrix since the latter can be divided into two independent matrices of half the size of the system matrix and each one of these smaller arising matrices can be factorized efficiently in multicore computers. We use OpenMP and store data in memory by blocks in consecutive positions yielding a simple and efficient algorithm. In addition, by exploiting the fact that diagonal pivoting does not destroy the special structure of Cauchy-like matrices, we introduce a local diagonal pivoting technique which improves the accuracy of the solution and the stability of the algorithm.

Superfast and Stable Structured Solvers for Toeplitz Least Squares via Randomized Sampling

We present some superfast ($O((m+n)\log^{2}(m+n))$ complexity) and stable structured direct solvers for $m\times n$ Toeplitz least squares problems. Based on the displacement equation, a Toeplitz matrix $T$ is first transformed into a Cauchy-like matrix $\mathcal{C}$, which can be shown to have small off-diagonal numerical ranks when the diagonal blocks are rectangular. We generalize standard hierarchically semiseparable (HSS) matrix representations to rectangular ones, and construct a rectangular HSS approximation to $\mathcal{C}$ in nearly linear complexity with randomized sampling and fast multiplications of $\mathcal{C}$ with vectors. A new URV HSS factorization and a URV HSS solution are designed for the least squares solution. We also present two structured normal equation methods. Systematic error and stability analysis for our HSS methods is given, which is also useful for studying other HSS and rank structured methods. We derive the growth factors and the backward error bounds in the HSS factoriz...

Wiener–Hopf and spectral factorization of real polynomials by Newton’s method

We present a new method for factoring a real polynomial into the product of two polynomials which have their zeros inside and outside the unit circle, respectively. The approach is based on solving a nonlinear system by Newton’s method. The Jacobi matrix is a polynomial of a companion matrix and thus a Bezoutian times the inverse of a triangular Toeplitz matrix. After getting deeper understanding of the displacement structure of the Bezoutian, each Newton step can be reduced to a few applications of discrete Fourier or cosine transforms and the LU-decomposition of a conceived linear system with a Cauchy-like matrix. These structural features are employed to design a fast algorithm, which shows excellent numerical behavior. For example, in the case of the extremely ill-conditioned test polynomial (2zn+∑k=0n-1zk)(2+∑k=1nzk) of the degree 2n=20000, we obtain the factorization after 22 Newton steps with an error of 3.4·10-8 in the Euclidean norm, the execution time on a laptop being a couple of minutes. The local quadratic convergence of our Newton method is proved and explicit convergence balls are given on the basis of estimates for the Lipschitz constant which occurs in Kantorovich’s theorem. We also design and test two superfast versions of our method, which, for polynomials of degree 20000, typically need about half a minute to produce an initial coefficient vector and then perform the Newton iteration within 1s.

A Superfast Structured Solver for Toeplitz Linear Systems via Randomized Sampling

We propose a superfast solver for Toeplitz linear systems based on rank structured matrix methods and randomized sampling. The solver uses displacement equations to transform a Toeplitz matrix $T$ into a Cauchy-like matrix $\mathcal{C}$, which is known to have low-numerical-rank off-diagonal blocks. Thus, we design a fast scheme for constructing a hierarchically semiseparable (HSS) matrix approximation to $\mathcal{C}$, where the HSS generators have internal structures. Unlike classical HSS methods, our solver employs randomized sampling techniques together with fast Toeplitz matrix-vector multiplications, and thus converts the direct compression of the off-diagonal blocks of $\mathcal{C}$ into the compression of much smaller blocks. A strong rank-revealing QR factorization method is used to generate/preserve certain special structures, and also to ensure stability. A fast ULV HSS factorization scheme is provided to take advantage of the special structures. We also propose a precomputation procedure for the HSS construction so as to further improve the efficiency. The complexity of these methods is significantly lower than some similar Toeplitz solvers for large matrix size $n$. Detailed flop counts are given, with the aid of a rank relaxation technique. The total cost of our methods includes $O(n)$ flops for HSS operations and $O(n\log^{2} n)$ flops for matrix multiplications via FFTs, where $n$ is the order of $T$. Various numerical tests on classical examples, including ill-conditioned ones, demonstrate the efficiency, and also indicate that the methods are stable in practice. This work shows a practical way of using randomized sampling in the development of fast rank structured methods.

A fast solver for linear systems with displacement structure

We describe a fast solver for linear systems with reconstructable Cauchy-like structure, which requires O(rn^2) floating point operations and O(rn) memory locations, where n is the size of the matrix and r its displacement rank. The solver is based on the application of the generalized Schur algorithm to a suitable augmented matrix, under some assumptions on the knots of the Cauchy-like matrix. It includes various pivoting strategies, already discussed in the literature, and a new algorithm, which only requires reconstructability. We have developed a software package, written in Matlab and C-MEX, which provides a robust implementation of the above method. Our package also includes solvers for Toeplitz(+Hankel)-like and Vandermonde-like linear systems, as these structures can be reduced to Cauchy-like by fast and stable transforms. Numerical experiments demonstrate the effectiveness of the software.

A note on the O(n)-storage implementation of the GKO algorithm and its adaptation to Trummer-like matrices

We propose a new O(n)-space implementation of the GKO-Cauchy algorithm for the solution of linear systems with Cauchy-like matrix. Despite its slightly higher computational cost, this new algorithm makes a more efficient use of the processor cache memory. Thus, for matrices of size larger than about 500-1000, it outperforms the existing algorithms. We present an applicative case of Cauchy-like matrices with non-reconstructible main diagonal. In this special instance, the O(n) space algorithms can be adapted nicely to provide an efficient implementation of basic linear algebra operations in terms of the low displacement-rank generators.

Fast QR factorization of Cauchy-like matrices

In this paper, we present two fast numerical methods for computing the QR factorization of an n × n Cauchy-like matrix C, C = QR , with data points lying on the real axis or on the unit circle in the complex plane. It is shown that the rows of the Q -factor of C are the eigenvectors of a rank structured matrix partially determined by some prescribed spectral data. This property establishes a basic connection between the computation of Q and the solution of an inverse eigenvalue problem for a rank structured matrix. Exploiting the structure of this problem enables us to develop quadratic time, i.e., O ( n 2 ) , QR factorization algorithms.

A FAST FACTORIZATION ALGORITHM FOR A CONFLUENT CAUCHY MATRIX

This paper presents a fast factorization algorithm for confluent Cauchy-like matrices. The algorithm consists of two parts. First. a confluent Cauchy-like matrix is transformed into a Cauchy-like matrix available to pivot without changing its structure. Second. a fast partial pivoting factorization algorithm for the Cauchy-like matrix is presented. A new displacement structure cannot possibly generate all entries of a transformed matrix, which is called by 'partially reconstructible'. This paper also discusses how the proposed factorization algorithm can be generally applied to partially reconstructive matrices.

An efficient solution for Cauchy-likesystems of linear equations

Using the displacement approach and transformation of vectors, the complexity ofsolving strongly nonsingular Cauchy-like systems of linear equations C x ˜ = υ ˜ will be improved off a factor of log n from the known algorithms of the recursive triangular factorization, where F is a field and v¯ ∈ F n×1 is a vector and C ∈ F n×n is a strongly nonsingular Cauchy-like matrix. The efficient algorithms were presented by Morf, Bitmead and Anderson to solve strongly nonsingular Toeplitz-like equations of linear systems by using the recursive triangular factorization in 1980. Recently, this approach of the recursive triangular factorization was extended to solve Cauchy-like systems with the complexity of O(n log 3 n) operations by Pan et al. This is the best known complexity bound by using the direct approach of recursive triangular factorization in Cauchy-like cases. However, these algorithms are still slower than the well-known algorithms with the asymptotic bound of O(n log 2 n) operations, which have been proposed by the means of reducing Cauchy-like matrices into Toeplitz-like matrices. In our present paper, the algorithms of the recursive triangular factorization are modified so that the complexity bound of the direct recursive approach can be decreased to O(n log 2 n) operations. This matches the asymptotic bound without transforming to Toeplitz-like matrices. Our improvement of the direct recursive approach is by a factor of log n due to changing the original vectors expressed in the given Cauchy-like matrix into the special vectors, where the entries are unit roots. The applications of structured matrices include Nevanlinna Pick tangential interpolation problems.

Superfast algorithms for Cauchy-like matrix computations and extensions

An effective algorithm of [M. Morf, Ph.D. Thesis, Department of Electrical Engineering, Stanford University, Stanford, CA, 1974; in: Proceedings of the IEEE International Conference on ASSP, IEEE Computer Society Press, Silver Spring, MD, 1980, pp. 954–959; R.R. Bitmead and B.D.O. Anderson, Linear Algebra Appl. 34 (1980) 103–116] computes the solution x → =T −1 b → to a strongly nonsingular Toeplitz or Toeplitz-like linear system T x → = b → , a short displacement generator for the inverse T −1 of T, and det T. We extend this algorithm to the similar computations with n×n Cauchy and Cauchy-like matrices. Recursive triangular factorization of such a matrix can be computed by our algorithm at the cost of executing O(nr 2 log 3 n) arithmetic operations, where r is the scaling rank of the input Cauchy-like matrix C (r=1 if C is a Cauchy matrix). Consequently, the same cost bound applies to the computation of the determinant of C, a short scaling generator of C −1 , and the solution to a nonsingular linear system of n equations with such a matrix C. (Our algorithm does not use the reduction to Toeplitz-like computations.) We also relax the assumptions of strong nonsingularity and even nonsingularity of the input not only for the computations in the field of complex or real numbers, but even, where the algorithm runs in an arbitrary field. We achieve this by using randomization, and we also show a certain improvement of the respective algorithm by Kaltofen for Toeplitz-like computations in an arbitrary field. Our subject has close correlation to rational tangential (matrix) interpolation under passivity condition (e.g., to Nevanlinna–Pick tangential interpolation problems) and has further impact on the decoding of algebraic codes.

On a property of Cauchy-like matrices

We introduce in this Note a property connecting the generators of a Cauchy-like matrix to the generators of its Schur complements. This property (formulas (6)) yields to a recursive process that can be used to devise a fast and direct inversion algorithm for Cauchy-like matrices (Section 4). The approach considered here is substantially simpler than those developed in some recent works on the subject ([2], [4], [6], [7], [8]).

Pivoted Cauchy-Like Preconditioners for Regularized Solution of Ill-Posed Problems

Many ill-posed problems are solved using a discretization that results in a least squares problem or a linear system involving a Toeplitz matrix. The exact solution to such problems is often hopelessly contaminated by noise, since the discretized problem is quite ill conditioned, and noise components in the approximate null-space dominate the solution vector. Therefore we seek an approximate solution that does not have large components in these directions. We use a preconditioned conjugate gradient algorithm to compute such a regularized solution. A unitary change of coordinates transforms the Toeplitz matrix to a Cauchy-like matrix, and we choose our preconditioner to be a low rank Cauchy-like matrix determined in the course of Gu's fast modified complete pivoting algorithm. We show that if the kernel of the ill-posed problem is smooth, then this preconditioner has desirable properties: the largest singular values of the preconditioned matrix are clustered around one, the smallest singular values, corresponding to the lower subspace, remain small, and the upper and lower spaces are relatively unmixed. The preconditioned algorithm costs only O(n lg n) operations per iteration for a problem with n variables. The effectiveness of the preconditioner for filtering noise is demonstrated on three examples.