Nonsymmetric Algebraic Riccati Equations Research Articles

Fast verified computation for solutions of algebraic Riccati equations arising in transport theory

SummaryA fast algorithm for enclosing the solution of the nonsymmetric algebraic Riccati equation arising in transport theory is proposed. The equation has a special structure, which is taken into account to reduce the complexity. By exploiting the structure, the enclosing process involves only quadratic complexity under a reasonable assumption. The algorithm moreover verifies the uniqueness and minimal positiveness of the enclosed solution. Numerical results show the efficiency of the algorithm.

Convergence Rates of a Class of Predictor-Corrector Iterations for the Nonsymmetric Algebraic Riccati Equation Arising in Transport Theory

AbstractIn this paper, we analyse the convergence rates of several different predictor-corrector iterations for computing the minimal positive solution of the nonsymmetric algebraic Riccati equation arising in transport theory. We have shown theoretically that the new predictor-corrector iteration given in [Numer. Linear Algebra Appl., 21 (2014), pp. 761–780] will converge no faster than the simple predictor-corrector iteration and the nonlinear block Jacobi predictor-corrector iteration. Moreover the last two have the same asymptotic convergence rate with the nonlinear block Gauss-Seidel iteration given in [SIAM J. Sci. Comput., 30 (2008), pp. 804–818]. Preliminary numerical experiments have been reported for the validation of the developed comparison theory.

Projected nonsymmetric algebraic Riccati equations and refining estimates of invariant and deflating subspaces

We consider the numerical solution of the projected nonsymmetric algebraic Riccati equations or their associated Sylvester equations via Newton’s method, arising in the refinement of estimates of invariant (or deflating subspaces) for a large and sparse real matrix A (or pencil A−λB). The engine of the method is the inversion of the matrix P2P2⊤A−γIn or Pl2Pl2⊤(A−γB), for some orthonormal P2 or Pl2 from Rn×(n−m), making use of the structures in A or A−λB and the Sherman–Morrison–Woodbury formula. Our algorithms are efficient, under appropriate assumptions, as shown in our error analysis and illustrated by numerical examples.

Numerical Study on Nonsymmetric Algebraic Riccati Equations

In this paper, we consider the nonsymmetric algebraic Riccati equation whose four coefficient matrices form an M-matrix K. When K is a regular M-matrix, the Riccati equation is known to have a minimal nonnegative solution. We present two new numerical methods which can be applied directly in the case where K is a regular M-matrix. Furthermore, we find that the alternately linearized implicit iteration method is also feasible. In addition, we study the monotone convergence property of the proposed methods. Numerical experiments show that the above three numerical iteration methods are feasible and effective for solving the nonsymmetric algebraic Riccati equation.

Characterization of solutions of non-symmetric algebraic Riccati equations

We give a new characterization of the solution set of non-symmetric algebraic Riccati equations involving real matrices. Our characterization involves the use of invariant subspaces of the coefficient matrices. We also give a poset structure on the solutions of ARE and explore some properties of this poset (partially ordered set).

A NEW VERSION OF THE SMITH METHOD FOR SOLVING SYLVESTER EQUATION AND DISCRETE-TIME SYLVESTER EQUATION

Recently, Xue etc. (37) discussed the Smith method for solving Sylvester equation AX + XB = C, where one of the matrices A and B is at least a nonsingular M-matrix and the other is an (singular or nonsingular) M-matrix. Furthermore, in order to nd the minimal non-negative solution of a certain class of non-symmetric algebraic Riccati equations, Gao and Bai (17) considered a doubling iteration scheme to inexactly solve the Sylvester equations. This paper discusses the iterative error of the standard Smith method used in (17) and presents the prior estimations of the accurate solution X for the Sylvester equation. Furthermore, we give a new version of the Smith method for solving discrete-time Sylvester equation or Stein equation AXB + X = C, while the new version of the Smith method can also be used to solve Sylvester equation AX + XB = C, where both A and B are positive denite. We also study the convergence rate of the new Smith method. At last, numerical examples are given to illustrate the eectiveness of our methods.

A modified SDA method for nonsymmetric algebraic Riccati \\ equations

Structure-preserving doubling algorithm (SDA) is one of the best methods for solving nonsymmetric algebraic Riccati equation associated with an $M$-matrix ($M$-NARE). In this paper, we modify the SDA-ss method proposed by Bini, Meini and Poloni so that it is more efficient for solving some types of $M$-NARE. Theoretical analysis and numerical experiments are given to show that our modification can improve SDA-ss in some cases.

Numerical methods for a quadratic matrix equation with a nonsingular M-matrix

We consider numerical solution of a quadratic matrix equation associated with a nonsingular M-matrix (M-QME), which arises in study of noisy Wiener–Hopf problems for Markov chain. We first transform the M-QME to a nonsymmetric algebraic Riccati equation (NARE) of special form, and then solve this special NARE by fixed-point iteration. Theoretical analysis and numerical experiments show that our method is effective and efficient.

Low-rank Newton-ADI methods for large nonsymmetric algebraic Riccati equations

The numerical treatment of large-scale, nonsymmetric algebraic Riccati equations (NAREs) by a low-rank variant of Newton׳s method is considered. We discuss a method to compute approximations to the solution of the NARE in a factorized form of low rank. The occurring large-scale Sylvester equations are dealt with using the factored alternating direction implicit iteration (fADI). Several performance enhancing strategies available for the factored ADI as well as the related Newton-ADI for symmetric algebraic Riccati equations are generalized to this combination. This includes the efficient computation of the norm of the residual matrix, adapted shift parameter strategies for fADI, and an acceleration of Newton׳s scheme by means of a Galerkin projection. Numerical experiments illustrate the capabilities of the proposed method to solve high-dimensional NAREs.

A new linearized implicit iteration method for nonsymmetric algebraic Riccati equations

For the nonsymmetric algebraic Riccati equation, we establish a new linearized implicit iteration method (LI) for computing its minimal nonnegative solution. And a modified linearized implicit iteration method (MLI) is obtained through Shamanskii technique. Under suitable conditions, we prove the monotone convergence of the LI and MLI iteration methods. Numerical experiments show that the LI and MLI iteration methods are feasible and effective. Moreover, the MLI iteration method outperforms the alternately linearized implicit iteration method (in: Bai et al., Numer. Linear Algebr. Appl. 13:655–674, 2006).

Condition number and backward errors of nonsymmetric algebraic Riccati equation

The normwise condition number and two kinds of backward errors are considered when nonsymmetric algebraic Riccati equation (NARE) has the minimal nonnegative solution. Based on the techniques for the symmetric case, we apply the condition number theory developed by Rice to define condition number for NARE. The explicit expression is derived in a uniform manner. Meanwhile, two kinds of backward errors are defined and evaluated by the explicit formulas. Numerical experiments are listed to illustrate and compare the practical performance.

Performance enhancement of doubling algorithms for a class of complex nonsymmetric algebraic Riccati equations

A new class of complex nonsymmetric algebraic Riccati equations has been studied by Liu & Xue (2012. Complex nonsymmetric algebraic Riccati equations arising in Markov modulated fluid flows. SIAM J. Matrix Anal. Appl., 33, 569–596), which is related to the M-matrix algebraic Riccati equations. Doubling algorithms, with properly chosen parameters, are used there for equations in this new class. It is pointed out that the number of iterations for the doubling algorithms may be relatively large in some situations. In this paper, we show that the performance of the doubling algorithms can often be improved significantly if a proper preprocessing procedure is used on the given Riccati equation. There are some difficult cases for which the preprocessing procedure does not help much by itself. We then propose new strategies for choosing parameters for doubling algorithms after using the preprocessing procedure. Numerical experiments show that our preprocessing procedure and the new parameter strategies are very effective.

Some predictor-corrector-type iterative schemes for solving nonsymmetric algebraic Riccati equations arising in transport theory

It is as well known that nonsymmetric algebraic Riccati equations arising in transport theory can be translated to vector equations. In this paper, we propose six predictor–corrector-type iterative schemes to solve the vector equations. And we give the convergence of these schemes. Unlike the previous work, we prove that all of them converge to the minimal positive solution of the vector equations by the initial vector (e,e), where e = (1,1, ⋯ ,1)T. Moreover, we prove that all the sequences generated by the iterative schemes are strictly and monotonically increasing and bounded above. In addition, some numerical results are also reported in the paper, which confirm the good theoretical properties of our approach. Copyright © 2014 John Wiley & Sons, Ltd.

Refining estimates of invariant and deflating subspaces for large and sparse matrices and pencils

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.

On applications of Schauderʼs fixed-point theorem for the solution of the non-symmetric algebraic Riccati equation

In this article, we give sufficient conditions for the existence of contractive solutions for the Non-Symmetric Algebraic Riccati Equation. The conditions are given in terms of the matrix of the coefficients – only. We also provide sufficient conditions for unique contractive solution (which is a unique minimal-norm solution). The results are generalized to contractive solutions with respect to generalized unit matrix balls.

A modified structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equations from transport theory

We consider the nonsymmetric algebraic Riccati equation arising in transport theory, where the n×n coefficient matrices A,B,C and E involved in the equation are rank structured. After a balancing strategy the matrix X̃T is the minimal positive solution of the dual algebraic Riccati equation, we can simplify the structure-preserving doubling algorithm (SDA) to this special equation and give a modified SDA, which has less computational cost at each iteration step. Also, we use numerical experiments to show the effectiveness of our new methods.

On the numerical solution of a structured nonsymmetric algebraic Riccati equation

A class of nonsymmetric algebraic Riccati equation, where one of the two linear coefficients is block diagonal, is studied. These equations arise in the modeling of an adaptive MMAP[K]/PH[K]/1 queue. Some theoretical results are proved, and two new algorithms are introduced that exploit the diagonal structure of the linear coefficient.

Low memory and low complexity iterative schemes for a nonsymmetric algebraic Riccati equation arising from transport theory

We reconsider Newton’s method and two fixed-point methods for finding the minimal positive solution of a nonsymmetric algebraic Riccati equation arising from transport theory. We rewrite the subproblem of the Newton and fixed-point iterative schemes into an equivalent form with some special structure. By the use of the particular structure of the subproblem, we then present low memory and low complexity versions of these iterative methods with a factored alternating-direction-implicit iteration. Some properties of eigenvalues for iterative coefficient matrices in solving the subproblem are derived and the convergence of the proposed methods is established. Numerical experiments show that the new iterative schemes are highly efficient to obtain the minimal positive solution. The proposed low memory and low complexity Newton’s method is particularly efficient for solving large scale Riccati equation arising from transport theory.

Solving Large-Scale Nonsymmetric Algebraic Riccati Equations by Doubling

We consider the solution of the large-scale nonsymmetric algebraic Riccati equation XCX − XD − AX + B =0 , withM ≡ (D, −C; −B, A) ∈ R(n1+n2)×(n1+n2) being a nonsingular M-matrix. In addition, A and D are sparselike, with the products A −1 u, A −� u, D −1 v ,a ndD −� v computable in O(n )c omplexity (withn =m ax{n1 ,n 2}), for some vectors u and v ,a ndB, C are low ranked. The structure-preserving doubling algorithms (SDA) by Guo, Lin, and Xu (Numer. Math., 103 (2006), pp. 392-412) is adapted, with the appropriate applications of the Sherman-Morrison- Woodbury formula and the sparse-plus-low-rank representations of various iterates. The resulting large-scale doubling algorithm has an O(n) computational complexity and memory requirement per iteration and converges essentially quadratically. A detailed error analysis, on the effects of truncation of iterates with an explicit forward error bound for the approximate solution from the SDA, and some numerical results will be presented.

The shift techniques for a nonsymmetric algebraic Riccati equation

In this paper, we want to analyze a special instance of a nonsymmetric algebraic matrix Riccati equation arising from transport theory. Traditional approaches for finding its minimal nonnegative solution are based on fixed point iterations and the speed of the convergence is linear. Recently, iterative methods such as Newton method and the structure-preserving doubling algorithm with quadratic convergence are designed for improving the speed of convergence. But, in some case, the speed of convergence will significantly decrease so that linear convergence becomes sublinear convergence and quadratic convergence becomes linear convergence. Our contribution in this work is to provide a thorough analysis to show that after the shift techniques, the speed of linear or quadratic convergence is preserved. Finally, we apply the shift procedures to the discussion of the simple iteration algorithm, improve its speed of convergence, and reduce its total elapsed CPU time.