Numerical methods for a quadratic matrix equation associated with an M-matrix

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Numerical methods for a quadratic matrix equation

associated with an M-matrix

Zubair Ahmed

指导教师姓名：

卢 琳 璋 教授

专 业 名 称：

计 算 数 学

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2016 年

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厦门大学博硕士论文摘要库

摘

摘

摘要

要

要

在科学与工程计算中常常遇到非线性矩阵方程, 尤其是二次矩阵方程. 本文研究 了一类特殊的二次矩阵方程 X2 _{− EX − F = 0, 其中 E, F ∈ R}n×n_{, E 是正对角矩阵,} 而 F 是 M-矩阵. 这种类型的二次矩阵方程在 Markov 链的含噪 Wiener-Hopf 问题中 有着一定的应用. 本文主要内容包括: 第一章, 简单介绍了二次矩阵方程的相关理论与算法, 并对相关的预备知识也作 了简要说明. 主要内容包括非负矩阵, M-矩阵与一般矩阵理论的基本知识. 第二章, 简单介绍了非对称代数 Riccati 方程的相关理论与数值算法. 第三章, 在相关文献的基础上利用新的变换将这种特殊的二次矩阵方程转化为 非对称代数 Riccati 方程, 并进而利用非对称代数 Riccati 方程的相关数值算法去求解. 理论分析和数值实验表明新的变换具有更好的性质. 第四章, 将上述结果推广到 F 是不可约奇异 M-矩阵的情形, 并给出了相应的理 论分析和数值实验. 第五章, 研究了这类二次矩阵方程的特殊情况, M-矩阵的平方根. 理论分析和数 值实验表明新的方法相比之前的已有方法具有一定的优势. 第六章, 总结和展望了本文中的方法和相关的课题. 关键词：二次矩阵方程; 非对称代数 Riccati 方程; 最小非负解; M-矩阵; 不动点迭代 法; Newton 迭代法; 保结构加倍算法; 矩阵平方根. – I –

厦门大学博硕士论文摘要库

Abstract

Nonlinear matrix equation often arises in areas of science and engineering computation. This thesis, discusses numerical solution for a quadratic matrix equation associated with an M-matrix (M-QME) i.e. X2 − EX − F = 0, where E, F ∈ Rn×n, E is a diagonal matrix and F is an M-matrix. This type of quadratic matrix equation arises in the study of noisy Wiener-Hopf problems for the Markov chains. We first transform the M-QME equation to a nonsymmetric algebraic Riccati equation (NARE) of special form and then considers some numerical methods on this special NARE to show how to compute the desired M-matrix solution, since the solution of practical interest is the M-matrix solution.

It is argued by both theoretical analysis and numerical experiments that the new formation used in some numerical methods are effective and efficient than the existing trans-formation in Chun-Hua Guo [IMA journal of Numerical Analysis. 2003].

Since square roots of a matrix play an important role in many applications of matrix theory. Therefore, we generalize the above numerical method for finding the square root of a nonsingular M-matrix. Theoretical analysis and numerical experiments show that the new method is more effective and efficient.

Key Words: Quadratic matrix equation; Nonsymmetric algebraic Riccati equations; Min-imal nonnegative solution; M-matrix; Fixed point iteration; Newton method; Alternating-directional doubling algorithm; Matrix square root

– III –

CONTENTS

Abstract (in Chinese)

. . . I

Abstract (in English)

. . . III

List of figures

. . . vi

Chapter 1

Introduction

. . . 01

1.1

Background

. . . 01

1.2

Notation

. . . 03

1.3

Kronecker Product

. . . 04

1.4

Frechet derivative

. . . 04

1.5

Regular and weak regular splitting of matrices

. . . 05

1.6

Irreducible matrix and Perron-Frobenius theorem

. . . 05

1.7

Some results on M- matrix

. . . 06

Chapter 2

Nonsymmetric algebraic Ricatti equation

. . . 08

2.1

Introduction to nonsymmetric algebraic Riccati equation

. . . 08

2.2

Existence of minimal nonnegative solution

. . . 09

2.3

Numerical methods

. . . 10

2.3.1

Fixed point iteration method

. . . 10

2.3.2

Newton iteration method

. . . 12

2.3.3

ALI iteration method

. . . 13

2.3.4

Doubling algorithm (SDA)

. . . 14

2.3.5

Alternating-directional doubling algorithm (ADDA)

. . . 17

Chapter 3

Numerical methods for a quadratic matrix equation

with a nonsingular M-matrix

. . . 19

3.1

Introduction

. . . 19

3.2

Theoretical analysis

. . . 20 – V –

CONTENTS

3.3

Numerical methods

. . . 22

3.3.1

Fixed point iteration method

. . . 22

3.3.2

Newton iteration method

. . . 25

3.4.3

Alternating-directional doubling algorithm (ADDA)

. . . 26

3.4

Numerical experiments

. . . 27

3.5

Conclusion remarks

. . . 29

Chapter 4

Numerical methods for a quadratic matrix equation

with an M-matrix: the irreducible singular case

. . . 34

4.1

Introduction

. . . 34

4.2

Theoretical analysis

. . . 35

4.3

Numerical methods

. . . 38

4.4.1

Fixed point iteration method

. . . 38

4.4.2

Newton iteration method

. . . 40

4.4.3

Alternating-directional doubling algorithm (ADDA)

. . . 40

4.4

Numerical experiments

. . . 41

4.5

Conclusion remarks

. . . 51

Chapter 5

New numerical method for square root of a nonsingular

M-matrix

. . . 52

5.1

Introduction

. . . 52

5.2

Some previous methods

. . . 52

5.3

New method for square root of a nonsingular M-matrix

. . . 54

5.4

Numerical experiments

. . . 56

5.5

Conclusion remarks

. . . 59

Chapter 6

Summary of whole thesis

. . . 60

6.1

Summary

. . . 60

References

. . . 61

Academic achievements

. . . 67

Acknowledgements

. . . 69 – VI –

厦门大学博硕士论文摘要库

List of figures

3.1 Computational results of Experiment 2 . . . 28

3.2 Computational results of Experiment 3 . . . 29

3.3 Computational results of Experiment 4 . . . 30

4.1 Computational results of Experiment 1 . . . 41

4.2 Computational results of Experiment 2 case(2) . . . 42

4.3 Computational results of Experiment 2 case(3) . . . 43

4.4 Computational results of Experiment 2 case(4) . . . 44

4.5 Computational results of Experiment 3 . . . 45

4.6 Computational results of Experiment 4 . . . 46

5.1 Computational results of Experiment 1 . . . 52

5.2 Computational results of Experiment 2 . . . 53

5.3 Computational results of Experiment 3 . . . 54

5.4 Computational results of Experiment 4 . . . 55

– VII –

Chapter 1

Introduction

1.1 Background

Quadratic matrix equations arise in a number of areas of engineering applications and scientific computing. The quadratic matrix equation is just one kind of many non-linear matrix equations that need further research. The well known quadratic matrix equation is the algebraic Riccati equation. Numerical methods and theory of algebraic Riccati equations are well developed, which occur in a variety of applications[26, 47, 52].

The quadratic matrix equation (QME)

X2+ EX + F = 0; E, F ∈ Rn×n (1.1)

occurs in many applications. Motivation for studying the quadratic equation comes either from the quadratic eigen value problems (see [24, 29, 35, 44, 61])

(λ2+ λE + F )x = 0, E, F ∈ Rn×n, (1.2)

or from the noisy Wiener-Hopf problems for markov chains [19, 42, 57, 58].

The answer to the question, whether the normal formula of the roots of the scalar quadratic can be generalized to (1.1), would be no unless it is specified for the special case, where E and F would not only commute but also E2−4F would has a square root. Obvious-ly, such solutions of (1.1) would be complicated and their characterization and computing would be interesting and challenging.

Newton’s method is applied by Higham and Kim [35] with and without exact line searches to solve QME (1.1). Besides this, Bai et al. [63] prove local convergence and developed a Newton method and successive approximation method by converting the QME into an equivalent fixed point iteration. More specifically, if E is diagonal and F is an matrix Guo[20] developed an iterative method and prove existence and uniqueness of M-matrix solution of QME (1.1),. He achieved this by transforming QME (1.1) into a special NARE, and proved that the monotonic convergence can be obtained by iterative methods.

This thesis mainly focuses on the numerical solution of a quadratic matrix equation of the form

X2− EX − F = 0; E, F ∈ Rn×n (1.3) – 1 –

1.1 Background

where E is a diagonal matrix and F is an M-matrix. The M-QME (1.3) has application in Wiener-Hopf problems in Markov chains, (see [20, 57]). It is proved in [20] that the M-QME (1.3) has a unique non-singular M-matrix solution, which is of practical importance. Some numerical methods have been developed for solving the M-QME by transforming it into an equivalent

NARE to solve (see [20, 25, 46]). This study, mainly focus on further development of the transformation methods.

To compute the M-matrix solution of the M-QME (1.3), Guo in [20] turned it into a special NARE to solve by the transformation

X = αI− Y

The motivational thoughts behind this are that many effective numerical methods for the NARE can be utilized to solve M-QME (1.3). This research uses a more general transfor-mation to expect to achieve better effectiveness.

Let A = (aij)∈ Cn×n, a matrix X ∈ Cn×nis called a square root of A if

X2 = A. (1.4)

The study of square root of a general (real or complex) matrix can be traced back to the early works of Sylvester, Cayley, Frobenius in the 19th century. These researchers were followed by the works of Cecioni and Kreis in the early 20th century. The square roots of a matrix play significant role in many applications of matrix theory, for example computation of the matrix logarithm, the boundary value problems, etc. For the background of the square root of matrix, we refer to [34].

The computation of square root of real matrix A has been studied for many years by several authors, see [9, 14, 32, 34, 45] and references therein. If A is real, it may or may not have a real square root. A sufficient condition for one to exist is that A has no real negative eigenvalue see [32, 37]. Besides then, any matrix, with no non-positive real eigen values, has a unique square root. The eigen value for this class of matrices lies in the open right plane and is at times named as the principal square root, which is of usual interest. A class of matrices with no non-positive real eigen values is a highly studied class of non-singular M-matrices. The methods for finding square root of a matrix can be divided into two category, direct methods and iterative methods. In the class of direct methods, for example Schur method (see [14]) and the iterative methods depends on later class (see [30, 31, 33, 56, 62]). This work considers the computation of square root of the non-singular M-matrix of the

– 2 –

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