M-matrix Algebraic Riccati Equation Research Articles

ALGORITHMS FOR THE RICCATI EQUATION WITH A NONSINGULAR M-MATRIX

We consider different algorithms with linear rate of convergence for computing the minimal nonnegative solution of M-matrix algebraic Riccati equation. The performance of the considered algorithms are illustrated on numerical examples.

Accurate numerical solution for structured [formula omitted]-matrix algebraic Riccati equations

This paper is concerned with a M-matrix algebraic Riccati equation (MARE) XDX−AX−XB+C=0 for which A is block-diagonal and its defining matrix W=B−D−CA is a nonsingular or irreducible singular M-matrix. Such an MARE can be decomposed into many coupled algebraic Riccati equations (AREs) that can be solved by the Jacobi- or Gauss–Seidel-like iteration updating scheme at the outer-loop while by a doubling algorithm in the inner loop for each coupled ARE, as first proposed by Meini (2013). The goals of this paper are two-fold. One is to resolve a critical technical detail in Meini’s algorithm that was not addressed. It is about whether each ARE in the inner loop has a minimal nonnegative solution. It is proved that the defining matrix of each coupled ARE during a doubling iteration is indeed a nonsingular or irreducible singular M-matrix and, as a result, they do have minimal nonnegative solutions and a doubling algorithm is an efficient way to compute them. The other goal is to design a highly accurate implementation of the doubling algorithm for the inner loop so that all entries of the minimal nonnegative solution to the original MARE are calculated with high entrywise relative accuracies, regardless of their magnitudes. This is made possible by a novel way of constructing triplet representations for the coupled ARE during doubling iterations. Numerical examples are presented to demonstrate that the resulting algorithm can indeed deliver an entrywise relatively accurate solution.

A novel linear iteration method for M-matrix algebraic Riccati equations

A novel linear iteration method for M-matrix algebraic Riccati equations

Accurate Numerical Solution for Shifted M-Matrix Algebraic Riccati Equations

An algebraic Riccati equation (are) is called a shifted M-matrix algebraic Riccati equation (mare) if it can be turned into an mare after its matrix variable is partially shifted by a diagonal matrix. Such an are can arise from computing the invariant density of a Markov modulated Brownian motion. Sufficient and necessary conditions for an are to be a shifted mare are obtained. Based on the conditions, a highly accurate implementation of the alternating directional doubling algorithm (adda) is established to compute the extremal solution of a shifted mare, as well as a quantity needed for computing the invariant density in the application, with high entrywise relative accuracy. Numerical examples are presented to demonstrate the theory and algorithms.

Highly accurate doubling algorithm for quadratic matrix equation from quasi-birth-and-death process

A highly accurate doubling algorithm to solve the most fundamental quadratic matrix equation in the quasi-birth-and-death (qbd) process is developed. It follows from the general framework of the doubling algorithm for the first standard form (SF1) but can be implemented to compute the minimal nonnegative solution with high entrywise relative accuracy for all entries, large or tiny. The algorithm is globally and quadratically convergent, except for qbd equations in the critical case where convergence is linear with the linear rate 1/2. Numerical examples are presented to demonstrate and confirm our claims. The development here parallels the recent work of Xue and Li (2017) [3] on the M-matrix algebraic Riccati equation.

Modified alternately linearized implicit iteration method for M-matrix algebraic Riccati equations

Research on the theories and efficient numerical methods of M-matrix algebraic Riccati equation (MARE) has become a hot topic in recent years. In this paper, we consider numerical solution of M-matrix algebraic Riccati equation and propose a modified alternately linearized implicit iteration method (MALI) for computing the minimal nonnegative solution of MARE. Convergence of the MALI method is proved by choosing proper parameters for the nonsingular M-matrix or irreducible singular M-matrix. Theoretical analysis and numerical experiments show that the MALI method is effective and efficient in some cases.

The double deflating technique for irreducible singular M-matrix algebraic Riccati equations in the critical case

ABSTRACTAs is known, Alternating-Directional Doubling Algorithm (ADDA) is quadratically convergent for computing the minimal nonnegative solution of an irreducible singular M-matrix algebraic Riccati equation (MARE) in the noncritical case or a nonsingular MARE, but ADDA is only linearly convergent in the critical case. The drawback can be overcome by deflating techniques for an irreducible singular MARE so that the speed of quadratic convergence is still preserved in the critical case and accelerated in the noncritical case. In this paper, we proposed an improved deflating technique to accelerate further the convergence speed – the double deflating technique for an irreducible singular MARE in the critical case. We proved that ADDA is quadratically convergent instead of linearly when it is applied to the deflated algebraic Riccati equation (ARE) obtained by a double deflating technique. We also showed that the double deflating technique is better than the deflating technique from the perspective of dimension of the deflated ARE. Numerical experiments are provided to illustrate that our double deflating technique is effective.

New Alternately Linearized Implicit Iteration for M-matrix Algebraic Riccati Equations

New Alternately Linearized Implicit Iteration for M-matrix Algebraic Riccati Equations

Highly accurate doubling algorithms for M-matrix algebraic Riccati equations

The doubling algorithms are very efficient iterative methods for computing the unique minimal nonnegative solution to an M-matrix algebraic Riccati equation (MARE). They are globally and quadratically convergent, except for MARE in the critical case where convergence is linear with the linear rate 1 / 2. However, the initialization phase and the doubling iteration kernel of any doubling algorithm involve inverting nonsingular M-matrices. In particular, for MARE in the critical case, the M-matrices in the doubling iteration kernel, although nonsingular, move towards singular M-matrices at convergence. These inversions are causes of concerns on entrywise relative accuracy of the eventually computed minimal nonnegative solution. Fortunately, a nonsingular M-matrix can be inverted by the GTH-like algorithm of Alfa et al. (Math Comp 71:217---236, 2002) to almost full entrywise relative accuracy, provided a triplet representation of the matrix is known. Recently, Nguyen and Poloni (Numer Math 130(4):763---792, 2015) discovered a way to construct triplet representations in a cancellation-free manner for all involved M-matrices in the doubling iteration kernel, for a special class of MAREs arising from Markov-modulated fluid queues. In this paper, we extend Nguyen's and Poloni's work to all MAREs by also devising a way to construct the triplet representations cancellation-free. Our construction, however, is not a straightforward extension of theirs. It is made possible by an introduction of novel recursively computable auxiliary nonnegative vectors. As the second contribution, we propose an entrywise relative residual for an approximate solution. The residual has an appealing feature of being able to reveal the entrywise relative accuracies of all entries, large and small, of the approximation. This is in marked contrast to the usual legacy normalized residual which reflects relative accuracies of large entries well but not so much those of very tiny entries. Numerical examples are presented to demonstrate and confirm our claims.

Componentwise accurate fluid queue computations using doubling algorithms

Markov-modulated fluid queues are popular stochastic processes frequently used for modelling real-life applications. An important performance measure to evaluate in these applications is their steady-state behaviour, which is determined by the stationary density. Computing it requires solving a (nonsymmetric) M-matrix algebraic Riccati equation, and indeed computing the stationary density is the most important application of this class of equations. Xue, Xu and Li [Numer. Math., 2012] provided a componentwise first-order perturbation analysis of this equation, proving that the solution can be computed to high relative accuracy even in the smallest entries, and suggested several algorithms for computing it. An important step in all proposed algorithms is using so-called triplet representations, which are special representations for M-matrices that allow for a high-accuracy variant of Gaussian elimination, the GTH-like algorithm. However, triplet representations for all the M-matrices needed in the algorithm were not found explicitly. This can lead to an accuracy loss that prevents the algorithms to converge in the componentwise sense. In this paper, we focus on the structured doubling algorithm, the most efficient among the proposed methods in Xue et al., and build upon their results, providing (i) explicit and cancellation-free expressions for the needed triplet representations, allowing the algorithm to be performed in a really cancellation-free fashion; (ii) an algorithm to evaluate the final part of the computation to obtain the stationary density; and (iii) a componentwise error analysis for the resulting algorithm, the first explicit one for this class of algorithms. We also present numerical results to illustrate the accuracy advantage of our method over standard (normwise-accurate) algorithms.

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.

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.

Accurate solutions of M-matrix algebraic Riccati equations

This paper is concerned with the relative perturbation theory and its entrywise relatively accurate numerical solutions of an M-matrix Algebraic Riccati Equations (MARE) $$XDX - AX - XB + C = 0 $$ by which we mean the following conformally partitioned matrix $$ \left( \begin{array}{ll}B\, \, -D\\-C\, \, A\end{array}\right)$$ is a nonsingular or an irreducible singular M-matrix. It is known that such an MARE has a unique minimal nonnegative solution $${\Phi}$$. It is proved that small relative perturbations to the entries of A, B, C, and D introduce small relative changes to the entries of the nonnegative solution $${\Phi}$$. Thus the smaller entries $${\Phi}$$ do not suffer bigger relative errors than its larger entries, unlike the existing perturbation theory for (general) Algebraic Riccati Equations. We then discuss some minor but crucial implementation changes to three existing numerical methods so that they can be used to compute $${\Phi}$$ as accurately as the input data deserve. Current study is based on a previous paper of the authors’ on M-matrix Sylvester equation for which D = 0.