使用Matlab进行线性代数与优化实践

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"《Applied Linear Algebra and Optimization Using MATLAB》是一本关于使用MATLAB进行线性代数和优化的教材或参考书。书中涵盖了线性代数的基础知识，并结合MATLAB软件进行实践应用。购买或使用本书即表示同意授权条款，但不授予对书中内容的所有权。复制或传播书中的任何文本、代码、模拟图像等需获得出版商或其他内容所有者的许可。" 本文将深入探讨线性代数和优化这两个关键主题，以及它们在MATLAB环境中的应用。线性代数是现代数学、工程和科学领域的基础，它研究向量、矩阵、线性方程组、特征值和特征向量等概念。通过MATLAB，我们可以高效地执行这些计算，这对于理解和解决实际问题至关重要。首先，线性代数的基础部分可能包括以下内容： 1. 向量和标量：理解向量的基本操作，如加法、减法和标量乘法，以及向量的点积和叉积。 2. 矩阵：学习矩阵的定义、性质（如行列式、逆矩阵、转置和秩），以及矩阵运算（如乘法、求解线性方程组）。 3. 线性空间与线性映射：了解向量空间的概念，包括基、维数、子空间和线性独立性。 4. 特征值和特征向量：理解如何找到一个矩阵的特征值和特征向量，以及它们在系统稳定性分析中的作用。 5. 解析几何：通过坐标系和方程描述二维和三维空间中的几何形状。接下来，MATLAB在实现线性代数中的应用包括： 1. 矩阵函数：使用MATLAB内置函数，如`inv()`（求逆）、`det()`（求行列式）和`eig()`（求特征值和特征向量）进行计算。 2. 线性方程组求解：`linsolve()`或`\`操作符用于解线性方程组。 3. 矩阵运算：例如，使用`*`进行矩阵乘法，`.^`进行元素级幂运算。 4. 图形可视化：MATLAB的绘图功能可帮助用户直观理解向量和矩阵空间的关系。优化是寻找变量最佳值的过程，常用于工程设计、数据分析和决策制定。在MATLAB中，优化工具箱提供了一系列算法： 1. 无约束优化：`fminunc()`函数用于最小化未约束的函数。 2. 约束优化：`fmincon()`处理有约束的优化问题。 3. 线性规划：`linprog()`适用于线性目标函数和线性约束的问题。 4. 非线性规划：`fmincon()`也可处理非线性约束问题。 5. 数值微分和梯度：`fnder()`和`fgrad()`用于计算函数的导数和梯度，这对优化算法至关重要。此外，书中可能还会涵盖线性代数和优化在实际问题中的应用，如图像处理、机器学习模型训练、控制系统设计等。读者不仅可以学习到理论知识，还能通过MATLAB编程练习提升解决实际问题的能力。然而，出版商和软件开发者对于书中内容、代码和相关产品的使用具有严格的许可限制，未经许可，不得上传或在网络中分享。这意味着要合法使用这些资源，必须遵循相应的授权协议，尊重知识产权。《Applied Linear Algebra and Optimization Using MATLAB》是学习和提升这两方面技能的宝贵资源，结合实践操作，将有助于读者在学术和专业领域取得进步。

nonlinear programming, queuing systems, etc. In this book we emphasize only

linear and nonlinear programming problems.

A wide range of applications appears throughout the book. They have been

chosen and written to give the student a sense of the broad range of applicability of

linear algebra and optimization theory. These applications range from theoretical

applications such as the use of linear algebra in differential equations, difference

equations, and least squares analysis.

When dealing with linear algebra or optimization theory, we often need a

computer. We believe that computers can improve the conceptional understanding

of mathematics, not just enable the completion of complicated calculations. We

have chosen MATLAB as our standard package because it is a widely used

software for working with matrices. The surge of popularity in MATLAB is related

to the increasing popularity of UNIX and computer graphics. To what extent

numerical computations will be programmed in MATLAB in the future is uncertain.

A short introduction to MATLAB is given in Appendix C, and the programs in the

text serve as further examples.

The topics are discussed in a simplified manner with a number of examples

illustrating the different concepts and applications. Most of the sections contain a

fairly large number of exercises, some of which relate to real–life problems.

Chapter 1 covers the basic concepts of matrices and determinants and describes the

basic computational methods used to solve nonhomogeneous linear equations.

Direct methods, including Cramer's rule, the Gaussian elimination method and its

variants, the Gauss–Jordan method, and LU decomposition methods, are discussed.

It also covers the conditioning of linear systems. Many ill–conditioned problems

are discussed. The chapter closes with the many interesting applications of linear

systems. In Chapter 2, we discuss iterative methods, including the Jacobi method,

the Gauss–Seidel method, the SOR iterative method, the conjugate gradient method,

and the residual corrector method. Chapter 3 covers the selected methods of

computing matrix eigenvalues. The approach discussed here should help students

understand the relationship of eigenvalues to the roots of characteristic equations.

We define eigenvalues and eigenvectors and study several examples. We discuss

the diagonalization of matrices and the computation of powers of diagonalizable

matrices. Some interesting applications of the eigenvalues and eigenvectors of a

matrix are also discussed at the end of the chapter. In Chapter 4, various numerical

methods are discussed for the eigenvalues of matrices. Among them are the power

iterative methods, the Jacobi method, Given's method, the Householder method, the

QR iteration method, the LR method, and the singular value decomposition method.

Chapter 5 describes the approximation of functions. In this chapter we also

describe curve fitting of experimental data based on least squares methods. We

discuss linear, nonlinear, plane, and trigonometric function least squares

approximations. We use QR decomposition and singular value decomposition for

the solution of the least squares problem. In Chapter 6, we describe standard linear

programming formulations. The subject of linear programming, in general, involves

the development algorithms and methodologies in optimization. The field,

developed by George Dantzig and his associates in 1947, is now widely used in

industry and has its foundation in linear algebra. In keeping with the intent of this

book, this chapter presents the mathematical formulations of basic linear

programming problems. In Chapter 7, we describe nonlinear programming

formulations. We discuss many numerical methods for solving unconstrained and

constrained problems. In the beginning of the chapter some of the basic

mathematical concepts useful in developing optimization theory are presented. For

unconstrained optimization problems we discuss the golden–section search method

and quadratic interpolation method, which depend on the initial guesses that bracket

the single optimum, and Newton's method, which is based on the idea from calculus

that the minimum or maximum can be found by solving f

(x) = 0. For the functions of

several variables, we use the steepest descent method and Newton's method. For

handling nonlinear optimization problems with constraints, we discuss the

generalized reduced–gradient method, Lagrange multipliers, and KT conditions. At

the end of the chapter, we also discuss quadratic programming problems and the

separable programming problems.

In each chapter, we discuss several examples to guide students step–by–step

through the most complex topics. Since the only real way to learn mathematics is to

use it, there is a list of exercises provided at the end of each chapter. These

exercises range from very easy to quite difficult. This book is completely self–

contained, with all the necessary mathematical background given in it. Finally, this

book provides balanced convergence of the theory, application, and numerical

computation of all the topics discussed.

Appendix A covers different kinds of errors that are preparatory subjects for

numerical computations. To explain the sources of these errors, there is a brief

discussion of Taylor's series and how numbers are computed and saved in

computers. Appendix B consists of a brief introduction to vectors in space and a

review of complex numbers and how to do linear algebra with them. It is also

devoted to general inner product spaces and to how different notations and

processes generalize. In Appendix C, we discuss the basic commands for the

software package MATLAB. In Appendix D, we give answers to selected odd–

numbered exercises.

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使用Matlab进行线性代数与优化实践

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