Minimal Nonnegative Solution Research Articles

Finding an NARE whose minimal nonnegative solution represents first passage quantities in the two-dimensional Brownian motion

Finding an NARE whose minimal nonnegative solution represents first passage quantities in the two-dimensional Brownian motion

On the Iterative Methods for the Solution of Three Types of Nonlinear Matrix Equations

In this paper, we investigate the iterative methods for the solution of different types of nonlinear matrix equations. More specifically, we consider iterative methods for the minimal nonnegative solution of a set of Riccati equations, a nonnegative solution of a quadratic matrix equation, and the maximal positive definite solution of the equation X+A∗X−1A=Q. We study the recent iterative methods for computing the solution to the above specific type of equations and propose more effective modifications of these iterative methods. In addition, we make comments and comparisons of the existing methods and show the effectiveness of our methods by illustration examples.

Computing the extremal nonnegative solutions of the [formula omitted]-tensor equation with a nonnegative right-side vector

We consider the tensor equation whose coefficient tensor is a nonsingular M-tensor and whose right-side vector is nonnegative. Such a tensor equation may have a large number of nonnegative solutions. It is already known that the tensor equation has a maximal nonnegative solution and a minimal nonnegative solution (called extremal solutions collectively). However, the existing proofs do not show how the extremal solutions can be computed. The existing numerical methods can find one of the nonnegative solutions, without knowing whether the computed solution is an extremal solution. In this paper, we present new proofs for the existence of extremal solutions. Our proofs are much shorter than existing ones and more importantly they give numerical methods that can compute the extremal solutions. Linear convergence of these numerical methods is also proved under mild assumptions. Some of our discussions also allow the coefficient tensor to be a Z-tensor or allow the right-side vector to have some negative elements.

Computation of the Stabilizing Solution of Nash-Riccati Equations

Consider an iterative modification of the linearized Newton method for computing the minimal nonnegative solution of a a nonsymmetric Nash-Riccati equation. The equation has arisen in linear quadratic games for positive systems. The Newton procedure for computing the minimal nonnegative solution is well known in the the literature. Our proposal is effective one because it employs small number of matrix multiplication at each iteration step and there is a variant to exploit the block structure of matrix coefficients of the Nash-Riccati equation. Moreover, in this reason, it is easy to extend the proposed iterative modification depending on the number of players of a given game model. We provide a numerical example where compare the results from experiment with the proposed iteration an the linearized Newton method.

Nonsymmetric Algebraic Riccati Equations under the Tensor Product

The nonsymmetric algebraic Riccati equation is proposed by using the tensor product. The existence of its minimal nonnegative solution is studied. The sufficient condition of the existence and the uniqueness of the minimal nonnegative solution is given by -tensor as well. The solution can be obtained by the fast Fourier transform which save computational cost of computing the required solution. Some numerical experiments are performed.

Relaxed fixed point iterations for matrix equations arising in Markov chain modeling

We present some accelerated variants of fixed point iterations for computing the minimal non-negative solution of the unilateral matrix equation associated with an M/G/1-type Markov chain. These variants derive from certain staircase regular splittings of the block Hessenberg M-matrix associated with the Markov chain. By exploiting the staircase profile, we introduce a two-step fixed point iteration. The iteration can be further accelerated by computing a weighted average between the approximations obtained at two consecutive steps. The convergence of the basic two-step fixed point iteration and of its relaxed modification is proved. Our theoretical analysis, along with several numerical experiments, shows that the proposed variants generally outperform the classical iterations.

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.

SOR-based alternately linearized implicit iteration method for nonsymmetric algebraic Riccati equations

<abstract><p>In this paper, we propose a class of successive over relaxation-based alternately linearized implicit iteration method for computing the minimal nonnegative solution of nonsymmetric algebraic Riccati equations. Under certain conditions, we prove the convergence of the iterative method. Finally, numerical examples are given to show the iterative method is efficient.</p></abstract>

Fast enclosure for the minimal nonnegative solution to the nonsymmetric T-Riccati equation

Fast enclosure for the minimal nonnegative solution to the nonsymmetric T-Riccati equation

Extremal solutions of strongly coupled nonlinear elliptic systems and L∞-boundedness

The paper concerns positive solutions for the Dirichlet problem{−Lu=ΛF(x,u)inΩ,u=0on∂Ω, where Ω is a smooth bounded domain in Rn, n≥2, u=(u1,…,um):Ω‾→Rm, m≥1, Lu=(L1u1,…,Lmum), where each Li denotes a uniformly elliptic linear operator of second order in nondivergence form in Ω, Λ=(λ1,…,λm)∈Rm, F=(f1,…,fm):Ω×Rm→Rm and ΛF(x,u)=(λ1f1(x,u),…,λmfm(x,u)). For a general class of maps F we prove that there exists a hypersurface Λ⁎ in R+m:=(0,∞)m such that tuples Λ∈R+m below Λ⁎ correspond to minimal positive strong solutions of the above system. Stability of these solutions is also discussed. Already for tuples above Λ⁎, there is no nonnegative strong solution. The shape of the hypersurface Λ⁎ depends on growth on u of the nonlinearity F in a sense to be specified. When Λ∈Λ⁎ and the coefficients of each operator Li are slightly smooth, the problem admits a unique minimal nonnegative weak solution, called extremal solution. Furthermore, when F depends only on u and all Li are Laplace operators, we investigate the L∞ regularity of this solution for any m≥1 in dimensions 2≤n≤9 for balls and n=2 and n=3 for convex domains.

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 family of fast fixed point iterations for M/G/1-type Markov chains

Abstract We consider the problem of computing the minimal non-negative solution $G$ of the nonlinear matrix equation $X=\sum _{i=-1}^\infty A_iX^{i+1}$ where $A_i$, for $i\geqslant -1$, are non-negative square matrices such that $\sum _{i=-1}^\infty A_i$ is stochastic. This equation is fundamental in the analysis of M/G/1-type Markov chains, since the matrix $G$ provides probabilistic measures of interest. A new family of fixed point iterations for the numerical computation of $G$, which includes the classical iterations, is introduced. A detailed convergence analysis proves that the iterations in the new class converge faster than the classical iterations. Numerical experiments confirm the effectiveness of our extension.

Improved SDA-ss algorithm for nonsymmetric algebraic Riccati equations

It is well known that nonsymmetric algebraic Riccati equation can be efficiently solved by using the structure-preserving doubling algorithm (SDA) with the shift-and-shrink transformation and the corresponding doubling algorithm is called the SDA-ss. In this paper, we propose an improved SDA-ss algorithm. Preliminary numerical experiments show that the algorithm is efficient to derive the minimal nonnegative solution of nonsymmetric algebraic Riccati equation with M-matrix.

Two-parameters numerical methods of the non-symmetric algebraic Riccati equation

In this paper, we focus on finding the minimal non-negative solution of a certain class of non-symmetric algebraic Riccati equations whose four coefficient matrices form a regular M-matrix. Firstly, we present two new numerical algorithms called modified inexact Newton method (MINewton) and modified structure-preserving doubling method (MSDA). Then we introduce two parameters in these algorithms, and establish the convergence theory by using Cayley transformation. Finally we give some numerical results to show the effectiveness of our derived methods compared with other iteration methods.

Solving Quadratic Matrix Equations Arising in Random Walks in the Quarter Plane

Quadratic matrix equations of the kind $A_1X^2+A_0X+A_{-1}=X$ are encountered in the analysis of Quasi--Birth-Death stochastic processes where the solution of interest is the minimal nonnegative solution $G$. In many queueing models, described by random walks in the quarter plane, the coefficients $A_1,A_0,A_{-1}$ are infinite tridiagonal matrices with an almost Toeplitz structure. Here, we analyze some fixed point iterations, including Newton's iteration, for the computation of $G$ and introduce effective algorithms and acceleration strategies which fully exploit the Toeplitz structure of the matrix coefficients and of the current approximation. Moreover, we provide a structured perturbation analysis for the solution $G$. The results of some numerical experiments which demonstrate the effectiveness of our approach are reported.

ITERATIVE COMPUTING THE MINIMAL SOLUTION OF THE COUPLED NONLINEAR MATRIX EQUATIONS IN TERMS OF NONNEGATIVE MATRICES

We investigate a set of nonlinear matrix equations with nonnegative matrix coefficients which has arisen in applied sciences. There are papers where the minimal nonnegative solution of the set of nonlinear matrix equations is computed applying the different procedures. The alternate linear implicit method and its modifications have intensively investigated because they have simple computational scheme. We construct a new decoupled modification of the alternate linear implicit procedure to compute the minimal nonnegative solution of the considered set of equations. The convergence properties of the proposed iteration are derived and a sufficient condition for convergence is derived. The performance of the proposed algorithm is illustrated on several numerical examples. On the basis of the experiments we derive conclusions for applicability of the computational schemes.

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.

Convergence of a modified Newton method for a matrix polynomial equation arising in stochastic problem

The Newton iteration is considered for a matrix polynomial equation which arises in stochastic problem. In this paper, it is shown that the elementwise minimal nonnegative solution of the matrix polynomial equation can be obtained using Newton's method if the equation satisfies the sufficient condition, and the convergence rate of the iteration is quadratic if the solution is simple. Moreover, it is shown that the convergence rate is at least linear if the solution is non-simple, but a modified Newton method whose iteration number is less than the pure Newton iteration number can be applied. Finally, numerical experiments are given to compare the effectiveness of the modified Newton method and the standard Newton method.