On algebraic Riccati equations associated with M-matrices

We consider the algebraic Riccati equation for which the four coefficient matrices form an M-matrix K. When K is a nonsingular M-matrix or an irreducible singular M-matrix, the Riccati equation is known to have a minimal nonnegative solution and several efficient methods are available to find this solution. In this paper we are mainly interested in the case where K is a reducible singular M-matrix. Under a regularity assumption on the M-matrix K, we show that the Riccati equation still has a minimal nonnegative solution. We also study the properties of this particular solution and explain how the solution can be found by existing methods.     

Perturbation of systems of linear algebraic equations

A classical perturbation result for nonsingular systems of linear algebraic equations is extended to general consistent systems under any norm. An optimal perturbation result is also obtained for general linear least squares problems under a Euclidean norm.     

Numerical solution to systems of delay integrodifferential algebraic equations

The numerical solution of the initial value problem for a system of delay integrodifferential algebraic equations is examined in the framework of the parametric continuation method. Necessary and sufficient conditions are obtained for transforming this problem to the best argument, which is the arc length along the integral curve of the problem. The efficiency of the transformation is demonstrated using test examples.     

An iterative method for solving systems of linear algebraic equations

An iterative solution process for systems of linear algebraic equations is proposed. It converges starting from any initial approximation and theoretically does not require preliminary transformation of the input data.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 64–68, 1985.     

Sensitivity analysis of the algebraic Riccati equations

Summary. In this paper, some sharp perturbation bounds for the Hermitian positive semi-definite solution to an algebraic Riccati equation are developed. A further analysis for these bounds is done. This analysis shows that there is, presumably, some intrinsic relation between the sensitivity of the solution to the algebraic Riccati equation and the distance of the spectrum of the closed-loop matrix from the imaginary axis. Received December 16, 1994     

A subspace shift technique for nonsymmetric algebraic Riccati equations associated with an M‐matrix

The worst situation in computing the minimal nonnegative solution of a nonsymmetric algebraic Riccati equation associated with an M‐matrix occurs when the corresponding linearizing matrix has two very small eigenvalues, one with positive and one with negative real part. When both eigenvalues are exactly zero, the problem is called critical or null recurrent. Although in this case the problem is ill‐conditioned and the convergence of the algorithms based on matrix iterations is slow, there exist some techniques to remove the singularity and transform the problem to a well‐behaved one. Ill‐conditioning and slow convergence appear also in close‐to‐critical problems, but when none of the eigenvalues is exactly zero, the techniques used for the critical case cannot be applied. In this paper, we introduce a new method to accelerate the convergence properties of the iterations also in close‐to‐critical cases, by working on the invariant subspace associated with the problematic eigenvalues as a whole. We present numerical experiments that confirm the efficiency of the new method.Copyright © 2012 John Wiley & Sons, Ltd.     

Explicit preconditioning of systems of linear algebraic equations with dense matrices

For the acceleration of the convergence of the iterative methods for solving systems of linear algebraic equations with dense matrices, one suggests the use of sparse explicit preconditioners, based on the minimization of quadratic functionals and admitting adaptive refinement. One gives the results of test computations for exterior potential flow problems.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 154, pp. 90–100, 1986.The author considers it a pleasant duty to express here gratitude to A. T. Berland and V. A. Galaev for placing at my disposal a program for the generation of a test problem and to A. Yu. Eremin for assistance in carrying out the numerical experiments.     

An Arnoldi based algorithm for large algebraic Riccati equations

In the present work, we present a numerical method for the computation of approximate solutions for large continuous-time algebraic Riccati equations. The proposed method is a method of projection onto a matrix Krylov subspace. We use a matrix Arnoldi process to construct an orthonormal basis. We give some theoretical results and numerical experiments for large problems.     

The reproducing kernel algorithm for handling differential algebraic systems of ordinary differential equations

The aim of the present analysis is to implement a relatively recent computational method, reproducing kernel Hilbert space, for obtaining the solutions of differential algebraic systems for ordinary differential equations. The reproducing kernel Hilbert space is constructed in which the initial conditions of the systems are satisfied. While, two smooth kernel functions are used throughout the evolution of the algorithm in order to obtain the required grid points. An efficient construction is given to obtain the numerical solutions for the systems together with an existence proof of the exact solutions based upon the reproducing kernel theory. In this approach, computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to such systems. Copyright © 2016 John Wiley & Sons, Ltd.     

Controllability and stabilization for linear systems of algebraic and differential equations

In this paper, we study several reachability and controllability definitions for linear, autonomous systems of degenerate ordinary differential equations. We try to identify the most interesting among them; we prove the connections with stabilizability and full stabilizability properties of the systems, which are also studied. In particular, we show the connection of the stabilizability property with the existence of special solutions of a suitable version of the Riccati equation.This paper was written under the auspices of GNAFA-CNR (Gruppo Nazionale per l'Analisi Funzionale e le sue Applicazioni, Consiglio Nazionale delle Ricerche), Rome Italy.     

An efficient multistep iteration scheme for systems of nonlinear algebraic equations associated with integral equations

The aim of this research is to present a new iterative procedure in approximating nonlinear system of algebraic equations with applications in integral equations as well as partial differential equations (PDEs). The presented scheme consists of several steps to reach a high rate of convergence and also an improved index of efficiency. The theoretical parts are furnished, and several computational tests mainly arising from practical problems are given to manifest its applicability.     

Stability conditions for systems of linear nonautonomous delay differential equations

Systems of linear nonautonomous delay differential equations are considered which are of the form y′_i(t) = ∑_{k = 1}ⁿ ∝₀^T b_ik(t, s) y_k(t − s) dη_ik(s) − c_i(t) y_i(t), where I = 1,…, n. Sufficient conditions are derived for both the asymptotic stability and the instability of the zero solution. The main result is found by a monotone technique using elementary methods only. Moreover, additional criteria are obtained by using the method of Lyapunov functionals.     

Analysis and modificaton of Newton's method for algebraic Riccati equations

When Newton's method is applied to find the maximal symmetric solution of an algebraic Riccati equation, convergence can be guaranteed under moderate conditions. In particular, the initial guess need not be close to the solution. The convergence is quadratic if the Fréchet derivative is invertible at the solution. In this paper we examine the behaviour of the Newton iteration when the derivative is not invertible at the solution. We find that a simple modification can improve the performance of the Newton iteration dramatically.

     

Monotone convergence of Newton-like methods for M-matrix algebraic Riccati equations

For the algebraic Riccati equation whose four coefficient matrices form a nonsingular M-matrix or an irreducible singular M-matrix K, the minimal nonnegative solution can be found by Newton’s method and the doubling algorithm. When the two diagonal blocks of the matrix K have both large and small diagonal entries, the doubling algorithm often requires many more iterations than Newton’s method. In those cases, Newton’s method may be more efficient than the doubling algorithm. This has motivated us to study Newton-like methods that have higher-order convergence and are not much more expensive each iteration. We find that the Chebyshev method of order three and a two-step modified Chebyshev method of order four can be more efficient than Newton’s method. For the Riccati equation, these two Newton-like methods are actually special cases of the Newton–Shamanskii method. We show that, starting with zero initial guess or some other suitable initial guess, the sequence generated by the Newton–Shamanskii method converges monotonically to the minimal nonnegative solution.We also explain that the Newton-like methods can be used to great advantage when solving some Riccati equations involving a parameter.     

On the solution of systems of linear algebraic equations with symbolic elements

We introduce some general concepts and propositions relating to symbolic computation on a computer. We analyze the suitability of the known numerical methods for solving systems of linear algebraic equations with symbolic elements. We propose efficient algorithms for solving dense systems and certain sparse systems by branching continued fractions.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 29–35.