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# 数学代写|优化理论代写Optimization Theory代考|MATH6231 Numerical Methods and Their Classification

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## 数学代写|优化理论代写Optimization Theory代考|Numerical Methods and Their Classification

Numerical methods are the methods of the approximate or accurate solving problems of absolute or applied mathematics that are based on the constructing a finite consequence of actions upon a finite aggregate sequence of numbers [282].

There are many different criterion for classifying numerical methods. For example, according to the criteria of presentation of input data and the results of the solution that are analytical and numerical according to functional and some kind of qualitative characteristics, by the method of discretization, etc.

In this case, the classification of numerical methods should be performed in accordance with the mentioned above systematization of classes of problems. In other words, we put a method or a multitude of methods that are capable of (in a certain sense) solving problems of a given class or its subclass for each class of problems or its subclass.

Give a possible classification of numerical methods for some of the mentioned above classes of problems.

Numerical methods of solving problems of statistical processing of experimental data Numerical methods that cover a given class of problems are based, on the one hand, on methods of computational mathematics, and on the other hand, it is based on the computation of some functions with the necessary accuracy $[287,235]$.

Among the methods of computational and applied mathematics that are used, we will list the following ones:

• Numerical integration and differentiation
• Solving the systems of linear equations (SLE)
• Proportional and root-mean-square approximation
• Tabulation (according to A. G. Vitushkin) of different classes of functions
• The smallest squares
• Sectioning
• Rapid orthogonal transformations
• Solving linear integral equations of the first order
• Moments and so on

## 数学代写|优化理论代写Optimization Theory代考|Numerical methods of approximation of functions

Numerical methods of approximation of functions The numerical methods of solving problems of the approximation of functions are divided into subclasses primarily using the metric: proportional or root-mean-square, possibly with some weight. Further classification of methods is performed for:

• The properties and structure of a function that is approximated – periodic, non-periodic, convex, smooth, and so on;
• The presence of constrains that should satisfy the approaching function (in given sets of points to take the given values or other, more complex, constrains);
• Classes of functions that constrain- polynomials, splines, wavelets, etc.
• Dependence of the function that approximates, from the given parameters – linear or nonlinear;
• Ways of determining the parameters – iterative, line, combined;
• Other features.
A more detailed classification of methods can be found in the works on the approximation of functions. For example, we give a sufficiently detailed classification of the methods of the Chebyshev proportional approximation in [59].

## 数学代写|优化理论代写Optimization Theory代考|Numerical Methods and Their Classification

• 数值积分与微分
• 求解线性方程组 (SLE)
• 比例和均方根近似
• 不同类别函数的制表（根据 AG Vitushkin）
• 最小的正方形
• 切片
• 快速正交变换
• 求解一阶线性积分方程
• 精彩瞬间等

## 数学代写|优化理论代写Optimization Theory代考|Numerical methods of approximation of functions

• 近似函数的属性和结构——周期性、非周期性、凸函数、平滑函数等；
• 应满足逼近函数的约束的存在（在给定的点集中采用给定值或其他更复杂的约束）；
• 约束多项式、样条、小波等的函数类。
• 从给定参数逼近的函数的依赖性——线性或非线性；
• 确定参数的方式——迭代、直线、组合；
• 其他特性。
可以在有关函数逼近的著作中找到更详细的方法分类。例如，我们在 [59] 中对切比雪夫比例逼近的方法进行了足够详细的分类。

$$\operatorname{minimize} c^{\mathrm{T}} x \quad \text { subject to } A x \geq b, \quad x \geq 0$$

$$A \bar{x} \geq b, \bar{x} \geq 0 \quad \text { (primal feasibility) } \bar{y}^{\mathrm{T}} A \leq c^{\mathrm{T}}, \bar{y} \geq 0 \quad \text { (dual feasibility) } \bar{y}^{\mathrm{T}}(A \bar{x}-b)=0 \quad \text { (complementary slackness) }$$

$$\bar{y}_i>0 \Longrightarrow(A \bar{x}-b)_i=0 \quad \text { and } \quad \bar{x}_j>0 \Longrightarrow\left(\bar{y}^{\mathrm{T}} A-c\right)_j=0 .$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。