A modified Newton method for solving non-symmetric algebraic Riccati equations arising in transport theory

Liang Bao ^{The non-symmetric algebraic Riccati equation arising in transporttheory can be rewritten as a vector equation and the minimalpositive solution of the non-symmetric algebraic Riccati equationcan be obtained by solving the vector equation. In this paper,we apply the modified Newton method to solve the vector equation.Some convergence results are presented. Numerical tests showthat the modified Newton method is feasible and effective, andoutperforms the Newton method.}     

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

A 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.     

The King-Werner method for solving nonsymmetric algebraic Riccati equation

For the nonsymmetric algebraic Riccati equation arising from transport theory, we concern about solving its minimal positive solution. In [1], Lu transferred the equation into a vector form and pointed out that the minimal positive solution of the matrix equation could be obtained via computing that of the vector equation. In this paper, we use the King-Werner method to solve the minimal positive solution of the vector equation and give the convergence and error analysis of the method. Numerical tests show that the King-Werner method is feasible to determine the minimal positive solution of the vector equation.     

Convergence analysis of the Newton–Shamanskii method for a nonsymmetric algebraic Riccati equation

In this paper, we consider the nonsymmetric algebraic Riccati equation arising in transport theory. An important feature of this equation is that its minimal positive solution can be obtained via computing the minimal positive solution of a vector equation. We apply the Newton–Shamanskii method to solve the vector equation. Convergence analysis shows that the sequence of vectors generated by the Newton–Shamanskii method is monotonically increasing and converges to the minimal positive solution of the vector equation. Numerical experiments show that the Newton–Shamanskii method is feasible and effective, and outperforms the Newton method. Copyright © 2008 John Wiley & Sons, Ltd.     

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

For 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 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.     

Convergence rates of some iterative methods for nonsymmetric algebraic Riccati equations arising in transport theory

We determine and compare the convergence rates of various fixed-point iterations for finding the minimal positive solution of a class of nonsymmetric algebraic Riccati equations arising in transport theory.     

求解非对称代数Riccati 方程几个新的预估-校正法

来源于输运理论的非对称代数Riccati 方程可等价地转化成向量方程组来求解. 本文提出了求解该向量方程组的几个预估-校正迭代格式,证明了这些迭代格式所产生的序列是严格单调递增且有上界,并收敛于向量方程 组的最小正解. 最后,给出了一些数值实验,实验结果表明,本文所提出的算法是有效的.     

On inexact Newton methods based on doubling iteration scheme for non‐symmetric algebraic Riccati equations

Newton iteration method can be used to find the minimal non‐negative solution of a certain class of non‐symmetric algebraic Riccati equations. However, a serious bottleneck exists in efficiency and storage for the implementation of the Newton iteration method, which comes from the use of some direct methods in exactly solving the involved Sylvester equations. In this paper, instead of direct methods, we apply a fast doubling iteration scheme to inexactly solve the Sylvester equations. Hence, a class of inexact Newton iteration methods that uses the Newton iteration method as the outer iteration and the doubling iteration scheme as the inner iteration is obtained. The corresponding procedure is precisely described and two practical methods of monotone convergence are algorithmically presented. In addition, the convergence property of these new methods is studied and numerical results are given to show their feasibility and effectiveness for solving the non‐symmetric algebraic Riccati equations. Copyright © 2010 John Wiley & Sons, Ltd.     

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

As 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.     

A new two‐phase structure‐preserving doubling algorithm for critically singular M‐matrix algebraic Riccati equations

Among numerous iterative methods for solving the minimal nonnegative solution of an M‐matrix algebraic Riccati equation, the structure‐preserving doubling algorithm (SDA) stands out owing to its overall efficiency as well as accuracy. SDA is globally convergent and its convergence is quadratic, except for the critical case for which it converges linearly with the linear rate 1/2. In this paper, we first undertake a delineatory convergence analysis that reveals that the approximations by SDA can be decomposed into two components: the stable component that converges quadratically and the rank‐one component that converges linearly with the linear rate 1/2. Our analysis also shows that as soon as the stable component is fully converged, the rank‐one component can be accurately recovered. We then propose an efficient hybrid method, called the two‐phase SDA, for which the SDA iteration is stopped as soon as it is determined that the stable component is fully converged. Therefore, this two‐phase SDA saves those SDA iterative steps that previously have to have for the rank‐one component to be computed accurately, and thus essentially, it can be regarded as a quadratically convergent method. Numerical results confirm our analysis and demonstrate the efficiency of the new two‐phase SDA. Copyright © 2015 John Wiley & Sons, Ltd.     

Solving the nonnegative solution for a (shifted) nonsymmetric algebraic Riccati equation in the critical case

We are interested in computing the nonnegative solution of a nonsymmetric algebraic Riccati equation arising in transport theory. The coefficient matrices of this equation have two parameters c and α. There have been some iterative methods presented by Lu in [13] and Bai et al. in [2] to solve the minimal positive solution for or . While the equation has a unique nonnegative solution when c=1 and α=0, all the methods presented by Lu and Bai cannot be used to find the nonnegative solution. To cope with this problem, a shifted technique is used in this paper to transform the original Riccati equation into a new one so that all the methods can be effectively employed to solve the nonnegative solution. Numerical experiments are given to illustrate the results.     

Adaptive iterative regularization schemes for two‐dimensional integral‐algebraic systems

The main idea of this paper is to utilize the adaptive iterative schemes based on regularization techniques for moderately ill‐posed problems that are obtained by a system of linear two‐dimensional Volterra integral equations with a singular matrix in the leading part. These problems may arise in the modeling of certain heat conduction processes as well as in the dynamic simulation packages such as compressible flow through a plant piping network. Owing to the ill‐posed nature of the first kind Volterra equation that appears in the system, we will focus on the two families of regularization algorithms, ie, the Landweber and Lavrentiev type methods, where we treat both the exact and perturbed data. Our aim is to work directly with the original Volterra equations without any kind of reduction. Two fast iterative algorithms with reasonable computational complexity are developed. Numerical experiments on a few test problems are used to illustrate the validity and efficiency of the proposed iterative methods in comparison with the classical regularization methods.     

On the convergence of basic iterative methods for convection–diffusion equations

In this paper we analyze convergence of basic iterative Jacobi and Gauss–Seidel type methods for solving linear systems which result from finite element or finite volume discretization of convection–diffusion equations on unstructured meshes. In general the resulting stiffness matrices are neither M‐matrices nor satisfy a diagonal dominance criterion. We introduce two newmatrix classes and analyse the convergence of the Jacobi and Gauss–Seidel methods for matrices from these classes. A new convergence result for the Jacobi method is proved and negative results for the Gauss–Seidel method are obtained. For a few well‐known discretization methods it is shown that the resulting stiffness matrices fall into the new matrix classes. Copyright © 1999 John Wiley & Sons, Ltd.     

Some iterative algorithms for positive definite solution to nonlinear matrix equations

This paper is concerned with the unique positive definite solution to a system of nonlinear matrix equations $X-A^*\bar{Y}^{-1}A=I_n$ and $Y-B^*\bar{X}^{-1}B=I_n$, where $A,B\in\mathbb{C}^{n\times n}$ are given matrices. Based on the special structure of the system of nonlinear matrix equations, the system can be equivalently reformulated as $V-C^*\bar{V}^{-1}C=I_{2n}$. Moreover, by means of Sherman-Moorison-Woodbury formula, we derive the relationship between the solutions of $V-C^*\bar{V}^{-1}C =I_{2n}$ and the well studied standard nonlinear matrix equation $Z+D^*Z^{-1}D=Q$, where $D$, $Q$ are uniquely determined by $C$. Then, we present a structure-preserving doubling algorithm and two modified structure-preserving doubling algorithms to compute the positive definite solution of the system. Furthermore, cyclic reduction algorithm and two modified cyclic reduction algorithms for the positive definite solution of the system are proposed. Finally, some numerical examples are presented to illustrate the efficiency of the theoretical results and the behavior of the considered algorithms.     

New predictor‐corrector scheme for solving nonlinear differential equations with Caputo‐Fabrizio operator

In this paper, we develop a new, simple, and accurate scheme to obtain approximate solution for nonlinear differential equation in the sense of Caputo‐Fabrizio operator. To derive this new predictor‐corrector scheme, which suits on Caputo‐Fabrizio operator, firstly, we obtain the corresponding initial value problem for the differential equation in the Caputo‐Fabrizio sense. Hence, by fractional Euler method and fractional trapeziodal rule, we obtain the predictor formula as well as corrector formula. Error analysis for this new method is derived. To test the validity and simplicity of this method, some illustrative examples for nonlinear differential equations are solved.     

Numerical analysis for the quadratic matrix equations from a modification of fixed‐point type

In this paper, we study the quadratic matrix equations. To improve the application of iterative schemes, we use a transform of the quadratic matrix equation into an equivalent fixed‐point equation. Then, we consider an iterative process of Chebyshev‐type to solve this equation. We prove that this iterative scheme is more efficient than Newton's method. Moreover, we obtain a local convergence result for this iterative scheme. We finish showing, by an application to noisy Wiener‐Hopf problems, that the iterative process considered is computationally more efficient than Newton's method.     

An adaptive learning rate backpropagation‐type neural network for solving n × n systems on nonlinear algebraic equations

This paper presents an MLP‐type neural network with some fixed connections and a backpropagation‐type training algorithm that identifies the full set of solutions of a complete system of nonlinear algebraic equations with n equations and n unknowns. The proposed structure is based on a backpropagation‐type algorithm with bias units in output neurons layer. Its novelty and innovation with respect to similar structures is the use of the hyperbolic tangent output function associated with an interesting feature, the use of adaptive learning rate for the neurons of the second hidden layer, a feature that adds a high degree of flexibility and parameter tuning during the network training stage. The paper presents the theoretical aspects for this approach as well as a set of experimental results that justify the necessity of such an architecture and evaluate its performance. Copyright © 2015 John Wiley & Sons, Ltd.     

Nonstandard finite difference schemes for Michaelis–Menten type reaction‐diffusion equations

We compare and investigate the performance of the exact scheme of the Michaelis–Menten (M–M) ordinary differential equation with several new nonstandard finite difference (NSFD) schemes that we construct using Mickens' rules. Furthermore, the exact scheme of the M–M equation is used to design several dynamically consistent NSFD schemes for related reaction‐diffusion equations, advection‐reaction equations, and advection‐reaction‐diffusion equations. Numerical simulations that support the theory and demonstrate computationally the power of NSFD schemes are presented. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013