Solving stable generalized Lyapunov equations with the matrix sign function

We investigate the numerical solution of the stable generalized Lyapunov equation via the sign function method. This approach has already been proposed to solve standard Lyapunov equations in several publications. The extension to the generalized case is straightforward. We consider some modifications and discuss how to solve generalized Lyapunov equations with semidefinite constant term for the Cholesky factor. The basic computational tools of the method are basic linear algebra operations that can be implemented efficiently on modern computer architectures and in particular on parallel computers. Hence, a considerable speed-up as compared to the Bartels–Stewart and Hammarling methods is to be expected. We compare the algorithms by performing a variety of numerical tests.     

Transformations between discrete-time and continuous-time algebraic Riccati equations

We introduce a transformation between the discrete-time and continuous-time algebraic Riccati equations. We show that under mild conditions the two algebraic Riccati equations can be transformed from one to another, and both algebraic Riccati equations share common Hermitian solutions. The transformation also sets up the relations about the properties, commonly in system and control setting, that are imposed in parallel to the coefficient matrices and Hermitian solutions of two algebraic Riccati equations. The transformation is simple and all the relations can be easily derived. We also introduce a generalized transformation that requires weaker conditions. The proposed transformations may provide a unified tool to develop the theories and numerical methods for the algebraic Riccati equations and the associated system and control problems.     

Numerical solution of Sylvester matrix equations with normal coefficients

Algorithms of the Bartels–Stewart type for the numerical solution of Sylvester matrix equations of modest size are modified for the case where the linear operators associated with these equations are normal. The superiority of the modified algorithms over the original ones is illustrated by numerical results.     

Dimensionality reduction and volume minimization—generalization of the determinant minimization criterion for reduced rank regression problems

In this article we propose a generalization of the determinant minimization criterion. The problem of minimizing the determinant of a matrix expression has implicit assumptions that the objective matrix is always nonsingular. In case of singular objective matrix the determinant would be zero and the minimization problem would be meaningless. To be able to handle all possible cases we generalize the determinant criterion to rank reduction and volume minimization of the objective matrix. The generalized minimization criterion is used to solve the following ordinary reduced rank regression problem:

min_rank(X)=kdet(B-XA)(B-XA)^T,     

Smooth factorizations of matrix valued functions and their derivatives

Summary In this paper we study the numerical factorization of matrix valued functions in order to apply them in the numerical solution of differential algebraic equations with time varying coefficients. The main difficulty is to obtain smoothness of the factors and a numerically accessible form of their derivatives. We show how this can be achieved without numerical differentiation if the derivative of the given matrix valued function is known. These results are then applied in the numerical solution of differential algebraic Riccati equations. For this a numerical algorithm is given and its properties are demonstrated by a numerical example.     

A numerical method for computing the Hamiltonian Schur form

We derive a new numerical method for computing the Hamiltonian Schur form of a Hamiltonian matrix that has no purely imaginary eigenvalues. We demonstrate the properties of the new method by showing its performance for the benchmark collection of continuous-time algebraic Riccati equations. Despite the fact that no complete error analysis for the method is yet available, the numerical results indicate that if no eigenvalues of are close to the imaginary axis then the method computes the exact Hamiltonian Schur form of a nearby Hamiltonian matrix and thus is numerically strongly backward stable. The new method is of complexity and hence it solves a long-standing open problem in numerical analysis. Volker Mehrmann was supported by Deutsche Forschungsgemeinschaft, Research Grant Me 790/11-3.     

On HSS and AHSS iteration methods for nonsymmetric positive definite Toeplitz systems

Two iteration methods are proposed to solve real nonsymmetric positive definite Toeplitz systems of linear equations. These methods are based on Hermitian and skew-Hermitian splitting (HSS) and accelerated Hermitian and skew-Hermitian splitting (AHSS). By constructing an orthogonal matrix and using a similarity transformation, the real Toeplitz linear system is transformed into a generalized saddle point problem. Then the structured HSS and the structured AHSS iteration methods are established by applying the HSS and the AHSS iteration methods to the generalized saddle point problem. We discuss efficient implementations and demonstrate that the structured HSS and the structured AHSS iteration methods have better behavior than the HSS iteration method in terms of both computational complexity and convergence speed. Moreover, the structured AHSS iteration method outperforms the HSS and the structured HSS iteration methods. The structured AHSS iteration method also converges unconditionally to the unique solution of the Toeplitz linear system. In addition, an upper bound for the contraction factor of the structured AHSS iteration method is derived. Numerical experiments are used to illustrate the effectiveness of the structured AHSS iteration method.     

On generalized symmetric SOR method for augmented systems

In this paper, we establish the generalized symmetric SOR method (GSSOR) for solving the large sparse augmented systems of linear equations, which is the extension of the SSOR iteration method. The convergence of the GSSOR method for augmented systems is studied. Numerical resume shows that this method is effective.     

Coupling dual reciprocity BEM and differential quadrature method for time-dependent diffusion problems

This paper presents the very first combined application of dual reciprocity BEM (DRBEM) and differential quadrature (DQ) method to time-dependent diffusion problems. In this study, the DRBEM is employed to discretize the spatial partial derivatives. The DQ method is then applied to analogize temporal derivatives. The resulting algebraic formulation is the known Lyapunov matrix equation, which can be very efficiently solved by the Bartels–Stewart algorithms. The mixed scheme combines strong geometry flexibility and boundary-only feature of the BEM and high accuracy and efficiency of the DQ method. Its superiority is demonstrated through the solution of some benchmark diffusion problems. The DQ method is shown to be numerically accurate, stable and computationally efficient in computing dynamic problems. In particular, the present study reveals that the DRBEM is also very efficient for transient diffusion problems with Dirichlet boundary conditions by coupling the DQ method in time discretization.     

ADI preconditioned Krylov methods for large Lyapunov matrix equations

In the present paper, we propose preconditioned Krylov methods for solving large Lyapunov matrix equations AX+XA^T+BB^T=0. Such problems appear in control theory, model reduction, circuit simulation and others. Using the Alternating Direction Implicit (ADI) iteration method, we transform the original Lyapunov equation to an equivalent symmetric Stein equation depending on some ADI parameters. We then define the Smith and the low rank ADI preconditioners. To solve the obtained Stein matrix equation, we apply the global Arnoldi method and get low rank approximate solutions. We give some theoretical results and report numerical tests to show the effectiveness of the proposed approaches.     

Method of Lyapunov functions for differential equations with piecewise constant delay

We address differential equations with piecewise constant argument of generalized type [5], [6], [7] and [8] and investigate their stability with the second Lyapunov method. Despite the fact that these equations include delay, stability conditions are merely given in terms of Lyapunov functions; that is, no functionals are used. Several examples, one of which considers the logistic equation, are discussed to illustrate the development of the theory. Some of the results were announced at the 14th International Congress on Computational and Applied Mathematics (ICCAM2009), Antalya, Turkey, in 2009.     

Time-harmonic solution for acousto-elastic interaction with controllability and spectral elements

The classical way of solving the time-harmonic linear acousto-elastic wave problem is to discretize the equations with finite elements or finite differences. This approach leads to large-scale indefinite complex-valued linear systems. For these kinds of systems, it is difficult to construct efficient iterative solution methods. That is why we use an alternative approach and solve the time-harmonic problem by controlling the solution of the corresponding time dependent wave equation.In this paper, we use an unsymmetric formulation, where fluid-structure interaction is modeled as a coupling between pressure and displacement. The coupled problem is discretized in space domain with spectral elements and in time domain with central finite differences. After discretization, exact controllability problem is reformulated as a least-squares problem, which is solved by the conjugate gradient method.     

Generalized biorthogonal bases and tridiagonalisation of matrices

Summary. In connection with the breakdown problem of the Lanczos algorithm a theory of generalized biorthogonal bases is developed. The connection between the generalized biorthogonal bases of Krylov chains and look-ahead Lanczos recursions is worked out in detail. It is shown how generalized biorthogonal bases with “antidiagonal blocks” can be constructed with small computational effort. Finally a special look-ahead Lanczos algorithm is derived which requires minimal computational effort and storage. Received September 21, 1995 / Revised version received August 12, 1996     

Convergence of a generalized MSSOR method for augmented systems

Recently, Wu et al. [S.-L. Wu, T.-Z. Huang, X.-L. Zhao, A modified SSOR iterative method for augmented systems, J. Comput. Appl. Math. 228 (1) (2009) 424-433] introduced a modified SSOR (MSSOR) method for augmented systems. In this paper, we establish a generalized MSSOR (GMSSOR) method for solving the large sparse augmented systems of linear equations, which is the extension of the MSSOR method. Furthermore, the convergence of the GMSSOR method for augmented systems is analyzed and numerical experiments are carried out, which show that the GMSSOR method with appropriate parameters has a faster convergence rate than the MSSOR method with optimal parameters.     

First order representations of Fliess models

Fuhrmann’s state-space construction (in its generalized form) is used to obtain a general theory of first order representations of Fliess models defined over an arbitrary noetherian commutative ring. The case of arbitrary linear delay differential equations is involved.     

几种广义非线性发展方程的新解

本文研究了构造了广义Kd V方程和广义KP-Burgers方程等几种广义非线性发展方程的新解的问题.利用三种辅助方程及其新解,获得了广义Kd V方程和广义KP-Burgers方程等几种广义非线性发展方程的新解.这些解由双曲余割函数、双曲正切函数、双曲正割函数、双曲余切函数和余割函数组成.     

Additive Schwarz algorithms for parabolic convection-diffusion equations

Summary In this paper, we consider the solution of linear systems of algebraic equations that arise from parabolic finite element problems. We introduce three additive Schwarz type domain decomposition methods for general, not necessarily selfadjoint, linear, second order, parabolic partial differential equations and also study the convergence rates of these algorithms. The resulting preconditioned linear system of equations is solved by the generalized minimal residual method. Numerical results are also reported.This work was supported in part by the National Science Foundation under Grant NSF-CCR-8903003 at the Courant Institute, New York University and in part by the National Science Foundation under contract number DCR-8521451 and ECS-8957475 at Yale University     

Accurate computation of the zeros of the generalized Bessel polynomials

Summary. A general method for approximating polynomial solutions of second-order linear homogeneous differential equations with polynomial coefficients is applied to the case of the families of differential equations defining the generalized Bessel polynomials, and an algorithm is derived for simultaneously finding their zeros. Then a comparison with several alternative algorithms is carried out. It shows that the computational problem of approximating the zeros of the generalized Bessel polynomials is not an easy matter at all and that the only algorithm able to give an accurate solution seems to be the one presented in this paper. Received July 25, 1997 / Revised version received May 19, 1999 / Published online June 8, 2000     

Study of a Generalized Nonlinear Euler-Poisson-Darboux System: Numerical and Bessel Based Solutions

In this paper a nonlinear Euler-Poisson-Darboux system is considered. In a first part, we proved the genericity of the hypergeometric functions in the development of exact solutions for such a systemin some special cases leading to Bessel type differential equations. Next, a finite difference scheme in two-dimensional case has been developed. The continuous system is transformed into an algebraic quasi linear discrete one leading to generalized Lyapunov-Sylvester operators. The discrete algebraic system is proved to be uniquely solvable, stable and convergent based on Lyapunov criterion of stability and Lax-Richtmyer equivalence theorem for the convergence. A numerical example has been provided at the end to illustrate the efficiency of the numerical scheme developed in section 3. The present method is thus proved to be more accurate than existing ones and lead to faster algorithms.     

On generalized successive overrelaxation methods for augmented linear systems

For the augmented system of linear equations, Golub, Wu and Yuan recently studied an SOR-like method (BIT 41(2001)71–85). By further accelerating it with another parameter, in this paper we present a generalized SOR (GSOR) method for the augmented linear system. We prove its convergence under suitable restrictions on the iteration parameters, and determine its optimal iteration parameters and the corresponding optimal convergence factor. Theoretical analyses show that the GSOR method has faster asymptotic convergence rate than the SOR-like method. Also numerical results show that the GSOR method is more effective than the SOR-like method when they are applied to solve the augmented linear system. This GSOR method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation, matrix relaxation and inexact iteration. Besides, we also demonstrate a complete version about the convergence theory of the SOR-like method. Subsidized by The Special Funds For Major State Basic Research Projects (No. G1999032803) and The National Natural Science Foundation (No. 10471146), P.R. China