A Schur method for solving algebraic Riccati equations

In this paper a new algorithm for solving algebraic Riccati equations (both continuous-time and discrete-time versions) is presented. The method studied is a variant of the classical eigenvector approach and uses instead an appropriate set of Schur vectors, thereby gaining substantial numerical advantages. Considerable discussion is devoted to a number of numerical issues. The method is apparently quite numerically stable and performs reliably on systems with dense matrices up to order 100 or so, storage being the main limiting factor.     

A Jacobi-like method for solving algebraic Riccati equations onparallel computers

An algorithm to solve continuous-time algebraic Riccati equations through the Hamiltonian Schur form is developed. It is an adaption for Hamiltonian matrices of an asymmetric Jacobi method of Eberlein (1987). It uses unitary symplectic similarity transformations and preserves the Hamiltonian structure of the matrix. Each iteration step needs only local information about the current matrix, thus admitting efficient parallel implementations on certain parallel architectures. Convergence performance of the algorithm is compared with the Hamiltonian-Jacobi algorithm of Byers (1990). The numerical experiments suggest that the method presented here converges considerably faster for non-Hermitian Hamiltonian matrices than Byers' Hamiltonian-Jacobi algorithm. Besides that, numerical experiments suggest that for the method presented here, the number of iterations needed for convergence can be predicted by a simple function of the matrix size     

A numerical method for solving systems of nonlinear equations

An iterative solution method for systems of nonlinear equations is proposed, making use of a nonlinear technique for the construction of the approximations. The method is efficient for solving quadratic equations.Translated from Kibernetika, No. 3, pp. 60–64, 69, May–June 1990.     

Numerical improvements for solving Riccati equations

In this paper, we discuss some ideas for improving the efficiency and accuracy of numerical methods for solving algebraic Riccati equations (AREs) based on invariant or deflating subspace methods. The focus is on AREs for which symmetric solutions exist, and our methods apply to both standard linear-quadratic-Gaussian (or H₂) AREs and to so-called H_∞-type AREs arising from either continuous-time or discrete-time models. The first technique is a new symmetric representation of a symmetric Riccati solution computed from an orthonormal basis of a certain invariant or deflating subspace. The symmetric representation does not require sign definiteness of the Riccati solution. The second technique relates to improving algorithm efficiency. Using a pencil-based approach, the solution of a Riccati equation can always be reformulated so that the deflating subspace whose basis is being sought corresponds to eigenvalues outside the unit circle. Thus, the natural tendency of the QZ algorithm to deflate these eigenvalues last, and hence, to appear in the upper left blocks of the appropriate pencils, then reduces the amount of reordering that must be done to a Schur form     

A new operational matrix for solving fractional-order differential equations

Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. For that reason we need a reliable and efficient technique for the solution of fractional differential equations. This paper deals with the numerical solution of a class of fractional differential equations. The fractional derivatives are described in the Caputo sense. Our main aim is to generalize the Legendre operational matrix to the fractional calculus. In this approach, a truncated Legendre series together with the Legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations. The main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem. The method is applied to solve two types of fractional differential equations, linear and nonlinear. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.     

A numerical method for solving parabolic equations with opposite orientations

The solution of parabolic control problems is characterized by a system of two equations parabolic with respect to opposite orientations. In this paper a fast iterative method for solving such problems is proposed.     

An exact line search method for solving generalized continuous-timealgebraic Riccati equations

We present a Newton-like method for solving algebraic Riccati equations that uses an exact line search to improve the sometimes erratic convergence behavior of Newton's method. It avoids the problem of a disastrously large first step and accelerates convergence when Newton steps are too small or too long. The additional work to perform the line search is small relative to the work needed to calculate the Newton step     

Comparison of numerical methods for solving Liapunov matrix equations†

A numerical comparison is made of most published methods for solving the linear matrix equations which arise when a quadratic form Liapunov function is applied to a constant linear system (continuous or discrete, real or complex). Generally, for the real equations direct methods are satisfactory for systems of order ten or less, whereas for larger order systems iterative methods (based upon expressing the solution in terms of an infinite series) are to be preferred. For the complex equations the most convenient numerical method uses an explicit representation for the solution in terms of the eigenvalues and vectors of the system matrix. If the system matrix is in companion form then algorithms taking account of this structure offer minor improvements.     

Weighted steepest descent method for solving matrix equations

This paper describes iterative methods for solving the general linear matrix equation including the well-known Lyapunov matrix equation, Sylvester matrix equation and some related matrix equations encountered in control system theory, as special cases. We develop the methods from the optimization point of view in the sense that the iterative algorithms are constructed to solve some optimization problems whose solutions are closely related to the unique solution to the linear matrix equation. Actually, two optimization problems are considered and, therefore, two iterative algorithms are proposed to solve the linear matrix equation. To solve the two optimization problems, the steepest descent method is adopted. By means of the so-called weighted inner product that is defined and studied in this paper, the convergence properties of the algorithms are analysed. It is shown that the algorithms converge at least linearly for arbitrary initial conditions. The proposed approaches are expected to be numerically reliable as only matrix manipulation is required. Numerical examples show the effectiveness of the proposed algorithms.     

The new numerical method for solving the system of two-dimensional Burgers’ equations

In this paper, the system of two-dimensional Burgers’ equations are solved by local discontinuous Galerkin (LDG) finite element method. The new method is based on the two-dimensional Hopf–Cole transformations, which transform the system of two-dimensional Burgers’ equations into a linear heat equation. Then the linear heat equation is solved by the LDG finite element method. The numerical solution of the heat equation is used to derive the numerical solutions of Burgers’ equations directly. Such a LDG method can also be used to find the numerical solution of the two-dimensional Burgers’ equation by rewriting Burgers’ equation as a system of the two-dimensional Burgers’ equations. Three numerical examples are used to demonstrate the efficiency and accuracy of the method.     

A new method for solving fuzzy linear differential equations

In this paper, a novel operator method is proposed for solving fuzzy linear differential equations under the assumption of strongly generalized differentiability. To this end, the equivalent integral form of the original problem is obtained then by using its lower and upper functions the solutions in the parametric forms are determined. The proposed method is illustrated with numerical examples.     

Comments on a numerical method for solving Boolean equations

A recent paper in this journal, by Abdel-Gawad, Atiya, and Darwish, presents a method of solving a system Boolean equations using the polynomial algebra invented by George Boole in 1854. The authors do not mention Boole, however, or the modern applications of this algebra. Their method entails reduction of the given system to a triangular system, which is solved by back-substitution. We show that the solutions of the triangular system include all those of the given system, but may include others that do not satisfy the given system.     

A new method for solving interval and fuzzy equations: Linear case

A new approach to the solution of interval and fuzzy equations based on the generalized procedure of interval extension called “interval extended zero” method is proposed. The central for the proposed approach is the treatment of the interval zero as an interval centered around zero. It is shown that such proposition is not of heuristic nature, but is the direct consequence of the interval subtraction operation. Some methodological problems concerned with this definition of interval zero are discussed. It is shown that the resulting solution of interval linear equations based on the proposed method may be naturally treated as a fuzzy number. An important advantage of a new method is that it substantially decreases the excess width effect. On the other hand, we show that it can be used as a reliable practical tool for solving linear interval and fuzzy equations as well as the systems of them. The features of the method are illustrated by the example of the solution of the well known Leontief input-output problem in the interval setting.     

Speeding up solving of differential matrix Riccati equations using GPGPU computing and MATLAB

In this work, we developed a parallel algorithm to speed up the resolution of differential matrix Riccati equations using a backward differentiation formula algorithm based on a fixed‐point method. The role and use of differential matrix Riccati equations is especially important in several applications such as optimal control, filtering, and estimation. In some cases, the problem could be large, and it is interesting to speed it up as much as possible. Recently, modern graphic processing units (GPUs) have been used as a way to improve performance. In this paper, we used an approach based on general‐purpose computing on graphics processing units. We used NVIDIA © GPUs with unified architecture. To do this, a special version of basic linear algebra subprograms for GPUs, called CUBLAS, and a package (three different packages were studied) to solve linear systems using GPUs have been used. Moreover, we developed a MATLAB © toolkit to use our implementation from MATLAB in such a way that if the user has a graphic card, the performance of the implementation is improved. If the user does not have such a card, the algorithm can also be run using the machine CPU. Experimental results on a NVIDIA Quadro FX 5800 are shown. Copyright © 2011 John Wiley & Sons, Ltd.     

应用Laguerre小波算子矩阵求二重数值积分

基于Laguerre小波函数及其对应的积分算子矩阵给出了一个求二重积分的数值方法。该方法通过对被积函数进行恰当的离散化,将二重数值积分问题转化为矩阵运算,从而易于求解、方便计算。该方法不仅适用于积分区域是矩形区域,也适用于二重变限积分的情况。数值算例验证了该方法的可行性及有效性。     

On the convergence rate of an iterative method for solving nonsymmetric algebraic Riccati equations

This paper is devoted to the convergence analysis of an iterative method for solving a nonsymmetric algebraic Riccati equation arising in transport theory. We give the convergence rate, and show that the iterative method converges linearly in one case and sublinearly in the other case.     

Some numerical considerations and Newton's method revisited forsolving algebraic Riccati equations

Analyzes some of the numerical aspects of solving the algebraic Riccati equation (ARE). This analysis applies to both the symmetric and unsymmetric cases. The author reconsiders the numerically relevant problems of balancing the ARE and the conditioning properties of the ARE and shows how these can be exploited by a solution algorithm. He proposes an estimator for the condition number of the Sylvester equation AX+XB=C based on iterative refinement. Also, he interprets Newton's method as a sequence of similarity transformations on the underlying system matrix. This closes the gap between so-called global and iterative methods for solving the ARE and also suggests an altogether revised implementation of Newton's method. One of the advantages of this revised implementation is that, in the case where Newton's method converges to a solution different from the desired solution, enough information emerges to allow a switch to the desired solution. The author examines the roundoff properties of the new algorithm and provides implementation considerations and numerical examples to highlight pros and cons