Numerical Methods in MATLAB-Fourth Edition

本书提供了用Matlab进行数值计算的丰富资料，内容可读性、知识性和实用性都非常强。

This method can solve various partial differential equations and represents the latest numerical solution techniques. It is based on MATLAB programming, making it easier to understand and implement. By utilizing MATLAB, complex mathematical models become more accessible and the process of solving PDEs is streamlined for better clarity and efficiency.

工程数值方法与MATLAB应用是一本由外国作者撰写的关于数值计算方法的书籍，对工程领域的学习和实践具有重要的参考价值。

您想要轻松获取科学计算的最新方法吗？这本大幅扩展的第三版《Numerical Recipes》拥有比以往更广泛的覆盖范围，许多新的、扩展的和更新的章节，以及两个全新的章节。采用面向科学应用特别适合的面向对象风格的C++代码，现在采用彩色印刷，便于阅读。《Numerical Recipes》由四位来自学术界和工业界的领先科学家共同撰写，从基本数学和计算机科学开始，逐步演进到完整的工作例程。整本书采用了让早期版本如此受欢迎的非正式、易于阅读的风格。新材料的亮点包括：关于分类和推断的新章节，高斯混合模型，HMMs，层次聚类和SVMs；关于计算几何的新章节，涵盖KD树，四叉树和八叉树，Delaunay三角剖分，以及线段、多边形、三角形和球体的算法。

本程序介绍了应用最为广泛的椭圆型、双曲型、抛物型偏微分方程的数值解法，并详细编程实现了每种方程的多种常见数值解法。附件中使用MATLAB编程来实现这些算法。

该文档介绍了各种谱分析方法，并对其进行了MATLAB仿真、比较。内容涵盖了谱分析的理论基础、常用方法，如傅里叶变换、短时傅里叶变换、小波变换等。每种方法都配有详细的MATLAB仿真步骤，并对比了各方法在不同应用场景中的效果。此外，文档还深入探讨了谱分析方法在信号处理和特征提取中的实际应用场景，使读者可以直观理解各种方法的优缺点。

3.2 Genetic Operators (1) Crossover Operator The crossover operator randomly pairs individuals from the parent population for crossover operations, generating ( m ) offspring individuals to form the next generation. Two types of crossover are employed: single-point crossover and two-point crossover. Given two individuals for crossover ( P = {p_1, p_2, p_3, \dots, p_n} ) and ( Q = {q_1, q_2, q_3, \dots, q_n} ), a random crossover point ( b_1 ) is chosen from the range [1, n] for single-point crossover. The elements before ( b_1 ) in ( P ) are copied to offspring individual ( \text{new Individual1} ), while the remaining elements are copied from ( Q ). Similarly, a second offspring ( \text{new Individual2} ) is generated by swapping the roles of ( P ) and ( Q ). In two-point crossover, two random crossover points ( b_1 ) and ( b_2 ) are chosen, and the elements between ( b_1 ) and ( b_2 ) in ( P ) are copied to the offspring, with the remaining elements taken from ( Q ). (2) Mutation Operator After the crossover operation, two mutation operators are applied to the offspring individuals. The first is rotation mutation, where a random position ( \text{bit} ) is chosen, and with probability ( p_m1 ), the portion of the individual after ( \text{bit} ) is rotated. The second is position mutation, with a smaller probability ( p_m2 ), two integers ( \text{bit1} ) and ( \text{bit2} ) are randomly chosen from the range [1, n], and the corresponding parts of the individual are swapped. (3) Selection Operator The fitness of the mutated offspring individuals is evaluated using the lowest level line method. The parent and offspring individuals are ranked by their fitness in descending order, and the top ( m ) individuals are selected as the next generation's parents. 3.3 Termination Criteria The steps in sections 3.2(1), 3.2(2), and 3.2(3) are repeated until the fitness of the best solution meets the required threshold or the pre-defined number of generations is reached. At this point, the optimal solution is output. 4. Case Study To test the performance of the algorithm, two cases from literature [3] are solved. In Case 1, a large rectangle of size ( 15 \times 40 ) is divided into 25 smaller rectangles. Based on the lowest level line method, the corresponding coding sequence is ( \text{Opt} = {1, -9, 11, -15, 17, -24, -25, -10, -14, -22, -23, -2, -3, -5, 18, 7, -8, -12, 19, -20, 21, 6, 13, 4} ). The width is set at 40, and height considerations follow suit for the genetic algorithm implementation.

In this article, we explore the optimization of PID parameters using BAS (Beetle Antennae Search) and SOA (Swarm Optimization Algorithm) methods in MATLAB. By leveraging MATLAB's built-in BAS optimization and SOA optimization functions, users can enhance PID controller performance effectively. Key Methods BAS Optimization: The BAS algorithm simulates beetle behavior to locate optimal solutions efficiently, minimizing error in PID control. SOA Optimization: The SOA algorithm, inspired by swarm intelligence, is another powerful method to refine PID parameters, enabling improved control accuracy. Steps to Implement Setup MATLAB: Open MATLAB and access the BAS and SOA programs, adjusting parameters as needed for optimal PID performance. Run Simulink Models: Simulate the systems using provided Simulink diagrams for BAS and SOA to observe and compare optimization results. The use of BAS and SOA provides flexible, efficient paths to tuning PID controllers, beneficial across various applications requiring precise control mechanisms.

目前的脚本近似于原始和相关联的对偶（伴随）布拉修斯方程，如Kuehl等人[~11/2020]在关于“连续伴随补充到布拉修斯方程”中的研究所述。数值边值问题使用射击方法近似，其中要解决的初值问题采用4阶Runge-Kutta方法(RK4)。