Minimal Nonnegative Solutions Research Articles

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.

Stochastic modeling on M/M/1/N inventory system with queue-dependent service rate and retrial facility

This paper investigates the queue-dependent service rates(QDSR) in the stochastic queueing-inventory system(SQIS). This SQIS consists a single server service channel, $ S $ number of inventories, and a finite queue. An arriving customer gets the service immediately if the server is free and there exists a positive stock in the SQIS. When the server is busy, they have to wait in the finite queue. Suppose they find that the waiting hall is full, either they leave the system or enter into an infinite orbit under the Bernoulli trial. The service rate of any arrival is dependent on the number of customers in the queue at present. The orbital customer can compete for the service only through joining into the waiting hall based on the classical retrial policy. Whenever the number of inventories in SQIS is reached a certain limit $ s $, the replenishment of $ Q( = S-s) $ items is placed. Due to the structure of rate matrix, the stability analysis, minimal non-negative solutions of the quadratic systems are derived through the Neuts matrix-geometric approximation(MGA). Further, the waiting time distribution(WTD) of arrival and necessary system characteristics are derived. Finally, adequate numerical examples are presented to highlight the proposed SQIS.

Elementwise Minimal Nonnegative Solutions for a Class of Nonlinear Matrix Equations

Elementwise Minimal Nonnegative Solutions for a Class of Nonlinear Matrix Equations

On algebraic Riccati equations associated with regular singular M-matrices

We further study algebraic Riccati equations associated with regular singular M-matrices. An M-matrix M is said to be regular if Mv≥0 for some v>0, so every irreducible singular M-matrix is a regular singular M-matrix. We prove a property about the product of the minimal nonnegative solution of such an algebraic Riccati equation and the minimal nonnegative solution of its dual equation. In the critical case, we show that the alternating-directional doubling algorithm (which includes the structure-preserving double algorithm as a special case) has linear convergence with rate 1/2. The results enhance our understanding of the behaviour of doubling algorithms for finding the minimal nonnegative solutions.

Existence and localization of positive solutions for a fractional boundary value problem at resonance

We investigate the existence of positive solutions for a fractional boundary value problem at resonance. By means of a fixed point theorem of increasing operators, the minimal and maximal nonnegative solutions for the problem are obtained.

Alternating-directional Doubling Algorithm for M-Matrix Algebraic Riccati Equations

A new doubling algorithm—the alternating-directional doubling algorithm (ADDA)—is developed for computing the unique minimal nonnegative solution of an $M$-matrix algebraic Riccati equation (MARE). It is argued by both theoretical analysis and numerical experiments that ADDA is always faster than two existing doubling algorithms: SDA of Guo, Lin, and Xu (Numer. Math., 103 (2006), pp. 393-412) and SDA-ss of Bini, Meini, and Poloni (Numer. Math., 116 (2010), pp. 553-578) for the same purpose. Also demonstrated is that all three methods are capable of delivering minimal nonnegative solutions with entrywise relative accuracies as warranted by the defining coefficient matrices of a MARE. The three doubling algorithms, differing only in their initial setups, correspond to three special cases of the general bilinear (also called Mobius) transformation. It is explained that ADDA is the best among all possible doubling algorithms resulted from all bilinear transformations.

Existence of large solutions of a class of quasilinear elliptic equations with singular boundary

We study a class of quasilinear elliptic equations with boundary blow-up. Based on super and sub-solution arguments and certain comparison principle, we prove the existence of non-negative solutions of (1.1). Moreover, we show the existence of maximal and minimal non-negative solutions of (1.1).

Alternately linearized implicit iteration methods for the minimal nonnegative solutions of the nonsymmetric algebraic Riccati equations

AbstractFor the non‐symmetric algebraic Riccati equations, we establish a class of alternately linearized implicit (ALI) iteration methods for computing its minimal non‐negative solutions by technical combination of alternate splitting and successive approximating of the algebraic Riccati operators. These methods include one iteration parameter, and suitable choices of this parameter may result in fast convergent iteration methods. Under suitable conditions, we prove the monotone convergence and estimate the asymptotic convergence factor of the ALI iteration matrix sequences. Numerical experiments show that the ALI iteration methods are feasible and effective, and can outperform the Newton iteration method and the fixed‐point iteration methods. Besides, we further generalize the known fixed‐point iterations, obtaining an extensive class of relaxed splitting iteration methods for solving the non‐symmetric algebraic Riccati equations. Copyright © 2006 John Wiley & Sons, Ltd.

A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation

In this paper, we propose a structure-preserving doubling algorithm (SDA) for the computation of the minimal nonnegative solution to the nonsymmetric algebraic Riccati equation (NARE), based on the techniques developed for the symmetric cases. This method allows the simultaneous approximation to the minimal nonnegative solutions of the NARE and its dual equation, requiring only the solutions to two linear systems and several matrix multiplications per iteration. Similar to Newton's method and the fixed-point iteration methods for solving NAREs, we also establish global convergence for SDA under suitable conditions, using only elementary matrix theory. We show that sequences of matrices generated by SDA are monotonically increasing and quadratically convergent to the minimal nonnegative solutions of the NARE and its dual equation. Numerical experiments show that the SDA algorithm is feasible and effective, and outperforms Newton's iteration and the fixed-point iteration methods.

On Matrix-Geometric Solution of Nested QBD Chains

In this paper, a generalization of the level dependent Quasi-Birth-and-Death (QBD) chains is presented. We analyze nested level dependent QBD chains and provide the complete characterization of their fundamental matrices in terms of minimal non-negative solutions of a number of matrix quadratic equations. Our results provide mixed matrix-geometric solution for the stationary distribution of nested QBD chains. Applications in overload control in communication networks are also discussed.

Minimal nonnegative solutions for nth order integro-differential equations in Banach spaces

In this paper, the cone theory and monotone iterative technique are used to investigate the minimal nonnegative solutions of initial value problems (IVPs) for nth order nonlinear integro-differential equations of mixed type on an infinite interval in a Banach space.

A class of iteration methods based on the moser formula for nonlinear equations in Markov chains

Many stochastic models in queueing, inventory, communications, and dam theories, etc., result in the problem of numerically determining the minimal nonnegative solutions for a class of nonlinear matrix equations. Various iterative methods have been proposed to determine the matrices of interest. We propose a new, efficient successive-substitution Moser method and a Newton-Moser method which use the Moser formula (which, originally, is just the Schulz method). These new methods avoid the inverses of the matrices, and thus considerable savings on the computational workloads may be achieved. Moreover, they are much more suitable for implementing on parallel multiprocessor systems. Under certain conditions, we establish monotone convergence of these new methods, and prove local linear convergence for the substitution Moser method and superlinear convergence for the Newton-Moser method.

Terminal value problems of impulsive integro‐differential equations in Banach spaces

This paper uses cone theory and the monotone iterative technique to investigate the existence of minimal nonnegative solutions of terminal value problems for first order nonlinear impulsive integro‐differential equations of mixed type in a Banach space.

Nonlinear Matrix Equations in Applied Probability—Solution Techniques and Open Problems

We draw attention to some important nonlinear equations commonly arising in applied probability, review some known techniques to compute their minimal nonnegative solutions, propose improved iterative techniques, and identify a number of open problems. Such equations arise ubiquitously in the numerical solution of many stochastic models in queueing, inventory, communications, and dam theories. The development of efficient algorithms and error analyses for these problems is, therefore, of substantial importance in enlarging the set of stochastic models that can be solved by algorithmic methods.

Errata: Fixed Point Theorems and Nonlinear Eigenvalue Problems in Ordered Banach Spaces

Previous article Full AccessErrata: Fixed Point Theorems and Nonlinear Eigenvalue Problems in Ordered Banach SpacesHerbert AmannHerbert Amannhttps://doi.org/10.1137/1019102PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout"Errata: Fixed Point Theorems and Nonlinear Eigenvalue Problems in Ordered Banach Spaces." SIAM Review, 19(4), p. vii Previous article FiguresRelatedReferencesCited ByDetails Three positive solutions of N -dimensional p -Laplacian with indefinite weightElectronic Journal of Qualitative Theory of Differential Equations, No. 19 | 1 January 2019 Cross Ref Global branch from the second eigenvalue for a semilinear Neumann problem in a ballJournal of Differential Equations, Vol. 249, No. 8 | 1 Oct 2010 Cross Ref A Priori Estimates and Continuation Methods for Positive Solutions of p-Laplace EquationsJournal of Differential Equations, Vol. 179, No. 1 | 1 Feb 2002 Cross Ref Uniform anti-maximum principle for polyharmonic boundary value problemsProceedings of the American Mathematical Society, Vol. 129, No. 2 | 28 August 2000 Cross Ref Competitive exclusion and coexistence for competitive systems on ordered Banach spacesTransactions of the American Mathematical Society, Vol. 348, No. 10 | 1 January 1996 Cross Ref Maximal and minimal nonnegative solutions of a semilinear Euler-Poisson-Darboux equationJournal of Mathematical Analysis and Applications, Vol. 90, No. 1 | 1 Nov 1982 Cross Ref Volume 19, Issue 4| 1977SIAM Reviewvi-vii History Submitted:13 July 1977Published online:17 February 2012 InformationCopyright © 1977 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1019102Article page range:pp. vii-viiISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics