Search

We consider the Alternating Direction Implicit (ADI) method to compute the numerical solution of a continuous Sylvester equation A X + X B = C , based on the recently developed inexact ADI iteration, and we propose classical acceleration techniques to enhance its convergence rate. An extrapolated variant (EADI) and a block successive overrelaxation variant (block SOR-ADI) of the ADI iterative method are described. These relaxation approaches are similar to what is used in Gauss-Seidel and Jacobi methods for linear systems, and, to our knowledge, novel, especially the block SOR-ADI scheme. Convergence properties of these two relaxed variants are analyzed when the matrix A is positive definite and the matrix B is positive semi-definite (not necessarily Hermitian matrices), or conversely. Our numerical experiments suggest that these new schemes are computationally attractive. The convergence rate of the ADI method is usually increased, particularly with the block SOR-ADI variant. A comparison with the well-known Hermitian and skew-Hermitian splitting (HSS) method emphasizes the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

For a linear matrix function f in X ∈ R m × n we consider inhomogeneous linear matrix equations f (X) = E for E ≠ 0 that have or do not have solutions. For such systems we compute optimal norm constrained solutions iteratively using the Conjugate Gradient and Lanczos’ methods in combination with the More–Sorensen optimizer. We build codes for ten linear matrix equations, of Sylvester, Lyapunov, Stein and structured types and their T-versions, that differ only in two five times repeated equation specific code lines. Numerical experiments with linear matrix equations are performed that illustrate universality and efficiency of our method for dense and small data matrices, as well as for sparse and certain structured input matrices. Specifically we show how to adapt our universal method for sparse inputs and for structured data such as encountered when fusing image data sets via a Sylvester equation algorithm to obtain an image of higher resolution. [ABSTRACT FROM AUTHOR]

Dynamical systems that involve non-linearity of the dynamics is a major challenge encountered in learning these systems. Similarly, the lack of adequate models for phenomena that reflect the governing physics can be an obstacle to an appropriate analysis. Nonetheless, some numerically or experimentally measured data can be found. Based on this data, and using a data-driven method such as the Loewner framework, it is possible to manage this data to derive a high fidelity reduced dynamical system that mimics the behaviour of the original data. In this paper, we tackle the issue of large amount of data presented by samples of transfer functions in a frequency-domain. The main step in this framework consists in computing singular value decomposition (SVD) of the Loewner matrix which provides accurate reduced systems. However, the large amount of data prevents this decomposition from being computed properly. We exploit the fact that the Loewner and shifted Loewner matrices, the key tools of Loewner framework, satisfy certain large scale Sylvester matrix equations. Using an extended block Krylov subspace method, a good approximation in a factored form of the Loewner and shifted Loewner matrices can be obtained and a minimal computation cost of the SVD is ensured. This method facilitates the process of a large amount of data and guarantees a good quality of the inferred model at the end of the process. Accuracy and efficiency of our method are assessed in the final section. [ABSTRACT FROM AUTHOR]

Biclustering is a powerful data mining technique that allows simultaneously clustering rows (observations) and columns (features) in a matrix-format data set, which can provide results in a checkerboard-like pattern for visualization and exploratory analysis in a wide array of domains. Multiple biclustering algorithms have been developed in the past two decades, among which the convex biclustering can guarantee a global optimum by formulating in as a convex optimization problem. On the other hand, the application of biclustering has not progressed in parallel with the algorithm techniques. For example, biclustering for increasingly popular microbiome research data is under-applied possibly due to its compositional constraints for each sample. In this manuscript, we propose a new convex biclustering algorithm, called the bi-ADMM, under general setups based on the ADMM algorithm, which is free of extra smoothing steps to visualize informative biclusters required by existing convex biclustering algorithms. Furthermore, we tailor it to the algorithm named biC-ADMM specifically to tackle compositional constraints confronted in microbiome data. The key step of our methods is to utilize the Sylvester Equation to derive the ADMM algorithm, which is new to the clustering research. The effectiveness of the proposed methods is examined through a variety of numerical experiments and a microbiome data application. [ABSTRACT FROM AUTHOR]

The Sylvester equation plays an important role in many branches of mathematical physics. The goal of this paper is to show that a special case of the Sylvester equation can be related to the lattice B-type Kadomtsev–Petviashvili (BKP) system via the generalized Cauchy matrix method. We use the variables given in the Sylvester equation to define the function and several scalar functions that are closely related to the lattice BKP equation. After rederiving the lattice BKP equation, we make it clear that besides its multisoliton solutions, various other types of exact solutions also exist. Furthermore, Lax pairs for the lattice BKP equation are obtained in different ways. [ABSTRACT FROM AUTHOR]

This paper presents the numerical solution of the Black–Scholes partial differential equation (PDE) for the evaluation of European call and put options. The proposed method is based on the finite difference and Legendre wavelets aproximation scheme. We derive a matrix structure for the Legendre wavelets integral operator which has been widely used so far. Moreover, in order to use the payoff function, another operational matrix is derived. By the proposed combined method, the solving Black–Scholes PDE problem reduces to those of solving a Sylvester equation. The proposed algorithms show that in compared to literature methods, the proposed method is easy to be implemented and have high execution speed. Furthermore, we prove that the obtained Sylvester equation has a unique solution. In addition, the effect of the finite difference space step size to the computational accuracy is studied. For having suitable solution, the numerical solutions show that there is no need to select very small step size. Also only a small number of basis functions in the Legendre wavelets series is needed. The numerical results demonstrate efficiency and capability of the proposed method. [ABSTRACT FROM AUTHOR]

The Sylvester equation and its particular case, the Lyapunov equation, are widely used in image processing, control theory, stability analysis, signal processing, model reduction, and many more. We present basis-free solution to the Sylvester equation in Clifford (geometric) algebra of arbitrary dimension. The basis-free solutions involve only the operations of Clifford (geometric) product, summation, and the operations of conjugation. To obtain the results, we use the concepts of characteristic polynomial, determinant, adjugate, and inverse in Clifford algebras. For the first time, we give alternative formulas for the basis-free solution to the Sylvester equation in the case n = 4 , the proofs for the case n = 5 and the case of arbitrary dimension n. The results can be used in symbolic computation. [ABSTRACT FROM AUTHOR]

We investigate the factorized solution of generalized stable Sylvester equations such as those arising in model reduction, image restoration, and observer design. Our algorithms, based on the matrix sign function, take advantage of the current trend to integrate high performance graphics accelerators (also known as GPUs) in computer systems. As a result, our realisations provide a valuable tool to solve large-scale problems on a variety of platforms. [ABSTRACT FROM AUTHOR]

Recursive blocked algorithms have proven to be highly efficient at the numerical solution of the Sylvester matrix equation and its generalizations. In this work, we show that these algorithms extend in a seamless fashion to higher-dimensional variants of generalized Sylvester matrix equations, as they arise from the discretization of PDEs with separable coefficients or the approximation of certain models in macroeconomics. By combining recursions with a mechanism for merging dimensions, an efficient algorithm is derived that outperforms existing approaches based on Sylvester solvers. [ABSTRACT FROM AUTHOR]

This work is about a balanced truncation type order reduction method which is developed for stable and unstable large-scale continuous-time systems. In this method, a quantitative measure criterion for choosing the dominant eigenvalues helps in determining the steady-state and transient information of the dynamical system. These dominant eigenvalues are used to form a new substructure matrix that retains the dominant modes (or may desirable mode) of the original system. Retaining the dominant eigenvalues in the reduced mode assures stability and results in greater accuracy as the retained eigenvalues provides a physical link to the real system. In the quest to preserve the dominant eigenvalue of the real system, the proposed technique uses Sylvester equation for system transformation. Having obtained transformed model, the reduced model has been achieved by truncating the non-dominant eigenvalues using the singular perturbation approximation method. The efficiency and accuracy of the proposed method has been demonstrated by the benchmark test systems which were from the state-of-the-art models. [ABSTRACT FROM AUTHOR]

In this paper, a generalized matrix product is introduced and related properties are studied as well. Afterwards, we show how our approach can be applied to the so-called Sylvester and Lyaponov matrix equations for obtaining their related solutions in terms of the generalized matrix product. Numerical examples illustrating the theoretical study are also discussed. [ABSTRACT FROM AUTHOR]

Fractional partial differential equations have many applications in science and engineering. However, not only the analytical solution existed for a limited number of cases, but also the numerical methods are very complicated and difficult. The aim of this paper is to present an efficient wavelet operational method based on the second Chebyshev wavelet to solve the fractional partial differential equations. We derived the convergence of the two-dimensional second Chebyshev wavelet and give the second Chebyshev wavelet operational matrix of fractional integration. Then we present a computational method based on the above results for solving a class of fractional partial differential equations. The initial equations are transformed into a Sylvester equation. Some numerical examples are given to demonstrate the simplicity, clarity and powerfulness of the new method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a modernized and improved version of the Davison-Man method for the numerical resolution of Sylvester matrix equations. In the case of moderate size problems, we give some background facts about this iterative method, addressing the problem of stagnation and we propose an iterative refinement technique to improve its accuracy. Although not designed to solve large-scale Sylvester equations, we propose an approach combining the extended block Krylov algorithm and the Davison-Man method which gives interesting results in terms of accuracy and shows to be competitive with the classical Krylov subspaces methods. Numerical examples are given to illustrate the efficiency of this approach. [ABSTRACT FROM AUTHOR]

The conditions of regularization and the structure of generalized Green's operator for a re-gularized linear matrix differential-algebraic boundary-value problem are found. To solve the problem of regularization of a generalized matrix differential-algebraic boundary-value problem, the original conditions of solvability and the structure of the general solution of a matrix equation of the Sylvester type are used. [ABSTRACT FROM AUTHOR]

The model reduction problem for nonlinear systems and nonlinear time-delay systems based on the steady-state notion of moment is reviewed. We show how this nonlinear description of moment is used to pose and solve the model reduction problem by moment matching for nonlinear systems, to develop a notion of frequency response for nonlinear systems, and to solve model reduction problems in the presence of constraints on the reduced order model. Model reduction of nonlinear time-delay systems is then discussed. Finally, the problem of approximating the moment of nonlinear, possibly time-delay, systems from input/output data is briefly illustrated. [ABSTRACT FROM AUTHOR]

This paper investigates finite-time stability and its application for solving time-varying Sylvester equation by recurrent neural network. Firstly, a new finite-time stability criterion is given and a less conservative upper bound of the convergence time is also derived. Secondly, a sign-bi-power activation function with a linear term is presented for the recurrent neural network. The estimation of the upper bound of the convergence time is more less conservative. Thirdly, it is proposed a tunable activation function with three tunable positive parameters for the recurrent neural network. These parameters are not only helpful to reduce conservatism of the upper bound of the convergence time, accelerate convergence but also reduce sensitivity to additive noise. The effectiveness of our methods is shown by both theoretical analysis and numerical simulations. [ABSTRACT FROM AUTHOR]

The conditions of solvability and the structure of the generalized Green operator of a linear Noetherian matrix differential-algebraic boundary-value problem are found. The sufficient conditions for reducibility of the generalized matrix differential-algebraic equation to the traditional differential-algebraic equation with the unknown function in the form of a vector-column are obtained. In the solution of a generalized matrix differential-algebraic boundary-value problem, the original conditions of solvability and the structure of the general solution of a matrix Sylvester-type equation are used. [ABSTRACT FROM AUTHOR]

We set forth solvability conditions and construction of the generalized Green's operator for a Noetherian matrix linear differential-algebraic boundary value problem. Sufficient conditions of reducibility of a generalized matrix differential-algebraic operator to a conventional differential-algebraic equation with an unknown column vector are established. To solve a matrix differential-algebraic boundary value problem, we employ a special solvability conditions and the construction of a general solution to a matrix Sylvester-type equation. [ABSTRACT FROM AUTHOR]

This paper presents several new algorithms for the regularized reconstruction of a surface from its measured gradient field. By taking a matrix-algebraic approach, we establish general framework for the regularized reconstruction problem based on the Sylvester Matrix Equation. Specifically, Spectral Regularization via Generalized Fourier Series (e.g., Discrete Cosine Functions, Gram Polynomials, Haar Functions, etc.), Tikhonov regularization, Constrained Regularization by imposing boundary conditions, and regularization via Weighted least squares can all be solved expediently in the context of the Sylvester Equation framework. State-of-the-art solutions to this problem are based on sparse matrix methods, which are no better than $$\mathcal {O}\!\left( n^6\right) $$ algorithms for an $$m\times n$$ surface with $$m \sim n$$ . In contrast, the newly proposed methods are based on the global least squares cost function and are all $$\mathcal {O}\!\left( n^3\right) $$ algorithms. In fact, the new algorithms have the same computational complexity as an SVD of the same size. The new algorithms are several orders of magnitude faster than the state-of-the-art; we therefore present, for the first time, Monte-Carlo simulations demonstrating the statistical behaviour of the algorithms when subject to various forms of noise. We establish methods that yield the lower bound of their respective cost functions, and therefore represent the 'Gold-Standard' benchmark solutions for the various forms of noise. The new methods are the first algorithms for regularized reconstruction on the order of megapixels, which is essential to methods such as Photometric Stereo. [ABSTRACT FROM AUTHOR]

We present a nested splitting conjugate gradient iteration method for solving large sparse continuous Sylvester equation, in which both coefficient matrices are (non-Hermitian) positive semi-definite, and at least one of them is positive definite. This method is actually inner/outer iterations, which employs the Sylvester conjugate gradient method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergent and Hermitian positive definite splitting of the coefficient matrices. Convergence conditions of this method are studied and numerical experiments show the efficiency of this method. In addition, we show that the quasi-Hermitian splitting can induce accurate, robust and effective preconditioned Krylov subspace methods. [ABSTRACT FROM AUTHOR]

An analogue of the Davis-Kahan $${\sin 2\Theta}$$ theorem from (SIAM J Numer Anal 7:1-46, ) is proved under a general spectral separation condition. This extends the generic $${\sin 2\theta}$$ estimates recently shown by Albeverio and Motovilov in (Complex Anal Oper Theory 7:1389-1416, ). The result is applied to the subspace perturbation problem to obtain a bound on the arcsine of the norm of the difference of the spectral projections associated with isolated components of the spectrum of the unperturbed and perturbed operators, respectively. [ABSTRACT FROM AUTHOR]

We consider the refinement of estimates of invariant (or deflating) subspaces for a large and sparse real matrix (or pencil) in $$\mathbb {R}^{n \times n}$$ , through some (generalized) nonsymmetric algebraic Riccati equations or their associated (generalized) Sylvester equations via Newton's method. The crux of the method is the inversion of some well-conditioned unstructured matrices via the efficient and stable inversion of the associated structured but near-singular matrices. All computations have complexity proportional to $$n$$ , under appropriate assumptions, as illustrated by several numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the effective condition numbers for the generalized Sylvester equation ( AX − YB, DX − YE) = ( C, F), where A, D ∈ ℝ, B, E ∈ ℝ and C, F ∈ ℝ. We apply the small sample statistical method for the fast condition estimation of the generalized Sylvester equation, which requires O( m n + mn) flops, comparing with O( m + n) flops for the generalized Schur and generalized Hessenberg-Schur methods for solving the generalized Sylvester equation. Numerical examples illustrate the sharpness of our perturbation bounds. [ABSTRACT FROM AUTHOR]

Bartels-Stewart algorithm is an effective and widely used method with an O( n) time complexity for solving a static Sylvester equation. When applied to time-varying Sylvester equation, the computation burden increases intensively with the decrease of sampling period and cannot satisfy continuous realtime calculation requirements. Gradient-based recurrent neural network are able to solve the time-varying Sylvester equation in real time but there always exists an estimation error. In contrast, the recently proposed Zhang neural network has been proven to converge to the solution of the Sylvester equation ideally when time goes to infinity. However, this neural network with the suggested activation functions never converges to the desired value in finite time, which may limit its applications in realtime processing. To tackle this problem, a sign-bi-power activation function is proposed in this paper to accelerate Zhang neural network to finite-time convergence. The global convergence and finite-time convergence property are proven in theory. The upper bound of the convergence time is derived analytically. Simulations are performed to evaluate the performance of the neural network with the proposed activation function. In addition, the proposed strategy is applied to online calculating the pseudo-inverse of a matrix and nonlinear control of an inverted pendulum system. Both theoretical analysis and numerical simulations validate the effectiveness of proposed activation function. [ABSTRACT FROM AUTHOR]

Let A be a self-adjoint operator on a Hilbert space $$\mathfrak{H}$$. Assume that the spectrum of A consists of two disjoint components σ0 and σ1. Let V be a bounded operator on $$\mathfrak{H}$$, off-diagonal and J-self-adjoint with respect to the orthogonal decomposition $$\mathfrak{H} = \mathfrak{H}_{0}\, \oplus\, \mathfrak{H}_{1}$$ where $$\mathfrak{H}_{0}$$ and $$\mathfrak{H}_{1}$$ are the spectral subspaces of A associated with the spectral sets σ0 and σ1, respectively. We find (optimal) conditions on V guaranteeing that the perturbed operator L = A + V is similar to a self-adjoint operator. Moreover, we prove a number of (sharp) norm bounds on the variation of the spectral subspaces of A under the perturbation V. Some of the results obtained are reformulated in terms of the Krein space theory. As an example, the quantum harmonic oscillator under a $${\mathcal{PT}}$$-symmetric perturbation is discussed. [ABSTRACT FROM AUTHOR]

We present and analyze a new iterative scheme for large-scale solution of the well-known Sylvester equation. The proposed scheme is based on fixed point iteration approach and can make good use of the recently developed methods for solving block linear systems. It is shown mathematically that the iterative process converges under some assumptions on the coefficient matrices. Results on our numerical experiments with large-scale matrices are quite encouraging. In particular, the method compares favorably with the other block methods and a recently proposed method for Sylvester equation based on low-rank approximation of the right hand side matrix C. [ABSTRACT FROM AUTHOR]

We propose a family of kernels based on the Binet-Cauchy theorem, and its extension to Fredholm operators. Our derivation provides a unifying framework for all kernels on dynamical systems currently used in machine learning, including kernels derived from the behavioral framework, diffusion processes, marginalized kernels, kernels on graphs, and the kernels on sets arising from the subspace angle approach. In the case of linear time-invariant systems, we derive explicit formulae for computing the proposed Binet-Cauchy kernels by solving Sylvester equations, and relate the proposed kernels to existing kernels based on cepstrum coefficients and subspace angles. We show efficient methods for computing our kernels which make them viable for the practitioner. Besides their theoretical appeal, these kernels can be used efficiently in the comparison of video sequences of dynamic scenes that can be modeled as the output of a linear time-invariant dynamical system. One advantage of our kernels is that they take the initial conditions of the dynamical systems into account. As a first example, we use our kernels to compare video sequences of dynamic textures. As a second example, we apply our kernels to the problem of clustering short clips of a movie. Experimental evidence shows superior performance of our kernels. [ABSTRACT FROM AUTHOR]

In this paper we construct an approximation to the solution x of a linear system of equations Ax = b of tensor product structure as it typically arises for finite element and finite difference discretisations of partial differential operators on tensor grids. For a right-hand side b of tensor product structure we can prove that the solution x can be approximated by a sum of O (log(ε)2) tensor product vectors where ε is the relative approximation error. Numerical examples for systems of size 1024256 indicate that this method is suitable for high-dimensional problems. [ABSTRACT FROM AUTHOR]

We discuss convergence properties of the GMRES and FOM methods for solving large Sylvester equations of the form AX− XB= C. In particular we show the importance of the separation between the fields of values of A and B on the convergence behavior of GMRES. We also discuss the stagnation phenomenon in GMRES and its consequence on FOM. We generalize the issue of breakdown in the block-Arnoldi algorithm and explain its consequence on FOM and GMRES methods. Several numerical tests illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Solvability conditions are examined for the matrix equation $$X^{ - 1} X* = A$$ , which cannot be found in the well-known reference books on matrix theory. Methods for constructing solutions to this equation are indicated. [ABSTRACT FROM AUTHOR]

In the present paper, we propose block Krylov subspace methods for solving the Sylvester matrix equation AX− XB= C. We first consider the case when A is large and B is of small size. We use block Krylov subspace methods such as the block Arnoldi and the block Lanczos algorithms to compute approximations to the solution of the Sylvester matrix equation. When both matrices are large and the right-hand side matrix is of small rank, we will show how to extract low-rank approximations. We give some theoretical results such as perturbation results and bounds of the norm of the error. Numerical experiments will also be given to show the effectiveness of these block methods. [ABSTRACT FROM AUTHOR]

An effective and disturbance suppressible controller can be designed by assigning a left eigenstructure (eigenvalues/left eigenvectors) of a system. In this note, a novel left eigenstructure assignment scheme via Sylvester equation is proposed. The biorthogonality property between the right and left modal matrices of a system is utilized to develop the scheme. [ABSTRACT FROM AUTHOR]

We solve the operator equation AX-XB=C, where A and B are closed operators whose point spectra intersect. We obtain sufficient conditions for the existence of solutions and provide a way of constructing them. As a corollary, we obtain a result that gives us new insight on the matrix equations of the form AX=XB, where A and B share non-zero eigenvalues. Afterwards, we illustrate our results on Sturm-Liouville operators. [ABSTRACT FROM AUTHOR]