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## 数据科学代写|数据分析代写Data Analysis代考|Strongly Convex Case

Recall from (2.19) that the smooth function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is strongly convex with modulus $m$ if there is a scalar $m>0$ such that
$$f(z) \geq f(x)+\nabla f(x)^T(z \quad x)+\frac{m}{2}|z \quad x|^2 .$$
Strong convexity asserts that $f$ can be lower bounded by quadratic functions. These functions change from point to point, but only in the linear term. It also tells us that the curvature of the function is bounded away from zero. Note that if $f$ is strongly convex and $L$-smooth, then $f$ is bounded above and below by simple quadratics (see (2.9) and (2.19)). This “sandwiching” effect enables us to prove the linear convergence of the steepest-descent method.

The simplest strongly convex function is the squared Euclidean norm $|x|^2$. Any convex function can be perturbed to form a strongly convex function by adding any small positive multiple of the squared Euclidean norm. In fact, if $f$ is any $L$-smooth function, then
$$f_\mu(x)=f(x)+\mu|x|^2$$
is strongly convex for $\mu$ large enough. (Exercise: Prove this!)
As another canonical example, note that a quadratic function $f(x)=$ $\frac{1}{2} x^T Q x$ is strongly convex if and only if the smallest eigenvalue of $Q$ is strictly positive. We saw in Theorem 2.8 that a strongly convex $f$ has a unique minimizer, which we denote by $x^*$.

Strongly convex functions are, in essence, the “easiest” functions to optimize by first-order methods. First, the norm of the gradient provides useful information about how far away we are from optimality. Suppose we minimize both sides of the inequality (3.9) with respect to $z$. The minimizer on the lefthand side is clearly attained at $z=x^$, while on the right-hand side, it is attained at $x-\nabla f(x) / m$. By plugging these optimal values into (3.9), we obtain \begin{aligned} f\left(x^\right) & \geq f(x) \quad \nabla f(x)^T\left(\frac{1}{m} \nabla f(x)\right)+\frac{m}{2}\left|\frac{1}{m} \nabla f(x)\right|^2 \ & =f(x) \quad \frac{1}{2 m}|\nabla f(x)|^2 . \end{aligned}
By rearrangement, we obtain
$$|\nabla f(x)|^2 \geq 2 m\left[f(x) \quad f\left(x^\right)\right]$$ If $|\nabla f(x)|<\delta$, we have $$f(x) \quad f\left(x^\right) \leq \frac{|\nabla f(x)|^2}{2 m} \leq \frac{\delta^2}{2 m} .$$

## 数据科学代写|数据分析代写Data Analysis代考|Comparison between Rates

It is straightforward to convert these convergence expressions into complexities using the techniques of Appendix A.2. We have, from (3.7), that an iteration $k$ will be found such that $\left|\nabla f\left(x^k\right)\right| \leq \epsilon$ for some $k \leq T$, where
$$T \geq \frac{2 L\left(f\left(x^0\right) \quad f^\right)}{\epsilon^2}$$ For the general convex case, we have from (3.8) that $f\left(x^k\right) \quad f^ \leq \epsilon$ when
$$k \geq \frac{L\left|x^0 \quad x^\right|^2}{2 \epsilon}$$ For the strongly convex case, we have from (3.15) that $f\left(x^k\right)-f^ \leq \epsilon$ for all $k$ satisfying
$$k \geq \frac{L}{m} \log \left(\left(f\left(x^0\right) \quad f^*\right) / \epsilon\right)$$

Note that in all three cases, we can get bounds in terms of the initial distance to optimality $\left|x^0 \quad x^\right|$ rather than the initial optimality gap $f\left(x^0\right) \quad f^$ by using the inequality
$$f\left(x^0\right) \quad f^* \leq \frac{L}{2}\left|x^0 \quad x^*\right|^2 .$$
The linear rate (3.17) depends only logarithmically on $\epsilon$, whereas the sublinear rates depend on $1 / \epsilon$ or $1 / \epsilon^2$. When $\epsilon$ is small (for example, $\epsilon=$ $10^{-6}$ ), the linear rate would appear to be dramatically faster, and, indeed, this is usually the case. The only exception would be when $m$ is extremely small, so that $m / L$ is of the same order as $\epsilon$. The problem is extremely ill conditioned in this case, and there is little difference between the linear rate (3.17) and the sublinear rate (3.16).

All of these bounds depend on knowledge of $L$. What happens when we do not know $L$ ? Even when we do know it, is the steplength $\alpha_k \equiv 1 / L$ good in practice? We have reason to suspect not, since the inequality (3.5) on which it is based uses the conservative global upper bound $L$ on curvature. (A sharper bound could be obtained in terms of the curvature in the neighborhood of the current iterate $x^k$.) In the remainder of this chapter, we expand our view to more general choices of search directions and steplengths.

## 数据科学代写|数据分析代写Data Analysis代考|Strongly Convex Case

$$f(z) \geq f(x)+\nabla f(x)^T(z \quad x)+\frac{m}{2}|z \quad x|^2$$

$$f_\mu(x)=f(x)+\mu|x|^2$$

## 数据科学代写|数据分析代写Data Analysis代考|Comparison between Rates

$$k \geq \frac{L}{m} \log \left(\left(f\left(x^0\right) \quad f^\right) / \epsilon\right)$$ 请注意，在所有这三种情况下，我们都可以根据与最优性的初始距离获得界限 缺少 \left 或额外的 \right 而不是最初的最优性差距缺少上标或下标参数 通 过使用不等式 $$f\left(x^0\right) \quad f^ \leq \frac{L}{2}\left|x^0 \quad x^*\right|^2$$

## 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。

Posted on Categories:Data Analysis, 数据分析, 数据科学代写

## avatest™帮您通过考试

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## 数据科学代写|数据分析代写Data Analysis代考|Characterizing Minima of Smooth Functions

The results of Section 2.2 give us the tools needed to characterize solutions of the unconstrained optimization problem
$$\min _{x \in \mathbb{R}^n} f(x)$$
where $f$ is a smooth function.
We start with necessary conditions, which give properties of the derivatives of $f$ that are satisfied when $x^*$ is a local solution. We have the following result.

Theorem 2.4 (Necessary Conditions for Smooth Unconstrained Optimization)
(a) Suppose that $f$ is continuously differentiable. If $x^$ is a local minimizer of (2.11), then $\nabla f\left(x^\right)=0$.
(b) Suppose that $f$ is twice continuously differentiable. If $x^$ is a local minimizer of (2.11), then $\nabla f\left(x^\right)=0$ and $\nabla^2 f\left(x^\right)$ is positive semidefinite. Proof We start by proving (a). Suppose for contradiction that $\nabla f\left(x^\right) \neq 0$, and consider a step $\alpha \nabla f\left(x^\right)$ away from $x^$, where $\alpha$ is a small positive number. By setting $p=\alpha \nabla f\left(x^\right)$ in formula (2.3) from Theorem 2.1, we have $$f\left(x^ \quad \alpha \nabla f\left(x^\right)\right)=f\left(x^\right) \quad \alpha \nabla f\left(x^* \quad \gamma \alpha \nabla f\left(x^\right)\right)^T \nabla f\left(x^\right),$$
for some $\gamma \in(0,1)$. Since $\nabla f$ is continuous, we have that
$$\nabla f\left(x^* \quad \gamma \alpha \nabla f\left(x^\right)\right)^T \nabla f\left(x^\right) \geq \frac{1}{2}\left|\nabla f\left(x^\right)\right|^2,$$ for all $\alpha$ sufficiently small, and any $\gamma \in(0,1)$. Thus, by substituting into (2.12), we have that $$f\left(x^ \quad \alpha \nabla f\left(x^\right)\right)=f\left(x^\right) \quad \frac{1}{2} \alpha\left|\nabla f\left(x^\right)\right|^2\right),$$
for all positive and sufficiently small $\alpha$. No matter how we choose the neighborhood $\mathcal{N}$ in the definition of local minimizer, it will contain points of the form $x^-\alpha \nabla f\left(x^\right)$ for sufficiently small $\alpha$. Thus, it is impossible to choose a neighborhood $\mathcal{N}$ of $x^$ such that $f(x) \geq f\left(x^\right)$ for all $x \in \mathcal{N}$, so $x^*$ is not a local minimizer.

## 数据科学代写|数据分析代写Data Analysis代考|Convex Sets and Functions

Convex functions take a central role in optimization precisely because these are the instances for which it is easy to verify optimality and for which such optima are guaranteed to be discoverable within a reasonable amount of computation.

A convex set $\Omega \subset \mathbb{R}^n$ has the property that
$$x, y \in \Omega \Rightarrow(1-\alpha) x+\alpha y \in \Omega \text { for all } \alpha \in[0,1] .$$
For all pairs of points $(x, y)$ contained in $\Omega$, the line segment between $x$ and $y$ is also contained in $\Omega$. The convex sets that we consider in this book are usually closed.
The defining property of a convex function is the following inequality: $f((1-\alpha) x+\alpha y) \leq(1-\alpha) f(x)+\alpha f(y), \quad$ for all $x, y \in \mathbb{R}^n$, all $\alpha \in[0,1]$.
$(2.15)$
The line segment connecting $(x, f(x))$ and $(y, f(y))$ lies entirely above the graph of the function $f$. In other words, the epigraph of $f$, defined as
$$\text { epi } f:=\left{(x, t) \in \mathbb{R}^n \times \mathbb{R} \mid t \geq f(x)\right} \text {, }$$
is a convex set. We sometimes call a function satisfying (2.15) as weakly convex function, to distinguish it from the special class called strongly convex functions, defined in Section 2.5 .

The concepts of “minimizer” and “solution” for the case of convex objective function and constraint set become more elementary in the convex case than in the general case of Section 2.1. In particular, the distinction between “local” and “global” solutions goes away.

## 数据科学代写|数据分析代写Data Analysis代考|Characterizing Minima of Smooth Functions

2.2 节的结果为我们提供了表征无约束优化问题解所需的工具
$$\min _{x \in \mathbb{R}^n} f(x)$$

## 数据科学代写|数据分析代写Data Analysis代考|Convex Sets and Functions

$$x, y \in \Omega \Rightarrow(1-\alpha) x+\alpha y \in \Omega \text { for all } \alpha \in[0,1] .$$

$(2.15)$

\left 缺少或无法识别的分隔符

## 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。

Posted on Categories:Data Analysis, 数据分析, 数据科学代写

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## 数据科学代写|数据分析代写Data Analysis代考|Matrix Factorization Problems

There are a variety of data analysis problems that require estimating a low-rank matrix from some sparse collection of data. Such problems can be formulated as natural extension of least squares to problems in which the data $a_j$ are naturally represented as matrices rather than vectors.

Changing notation slightly, we suppose that each $A_j$ is an $n \times p$ matrix, and we seek another $n \times p$ matrix $X$ that solves
$$\min X \frac{1}{2 m} \sum{j=1}^m\left(\left\langle A_j, X\right\rangle \quad y_j\right)^2,$$
where $\langle A, B\rangle:=\operatorname{trace}\left(A^T B\right)$. Here we can think of the $A_j$ as “probing” the unknown matrix $X$. Commonly considered types of observations are random linear combinations (where the elements of $A_j$ are selected i.i.d. from some distribution) or single element observations (in which each $A_j$ has 1 in a single location and zeros elsewhere). A regularized version of (1.6), leading to solutions $X$ that are low rank, is
$$\min X \frac{1}{2 m} \sum{j=1}^m\left(\left\langle A_j, X\right\rangle \quad y_j\right)^2+\lambda|X|_,$$ where $|X|_$ is the nuclear norm, which is the sum of singular values of $X$ (Recht et al., 2010). The nuclear norm plays a role analogous to the $\ell_1$ norm in (1.5), where as the $\ell_1$ norm favors sparse vectors, the nuclear norm favors lowrank matrices. Although the nuclear norm is a somewhat complex nonsmooth function, it is at least convex so that the formulation (1.7) is also convex. This formulation can be shown to yield a statistically valid solution when the true $X$ is low rank and the observation matrices $A_j$ satisfy a “restricted isometry property,” commonly satisfied by random matrices but not by matrices with just one nonzero element. The formulation is also valid in a different context, in which the true $X$ is incoherent (roughly speaking, it does not have a few elements that are much larger than the others), and the observations $A_j$ are of single elements (Candès and Recht, 2009).

## 数据科学代写|数据分析代写Data Analysis代考|Support Vector Machines

Classification via support vector machines (SVM) is a classical optimization problem in machine learning, tracing its origins to the 1960s. Given the input data $\left(a_j, y_j\right)$ with $a_j \in \mathbb{R}^n$ and $y_j \in{1,1}$, SVM seeks a vector $x \in \mathbb{R}^n$ and a scalar $\beta \in \mathbb{R}$ such that
$$\begin{array}{lll} a_j^T x & \beta \geq 1 & \text { when } y_j=+1, \ a_j^T x & \beta \leq 1 & \text { when } y_j=1 . \end{array}$$
Any pair $(x, \beta)$ that satisfies these conditions defines a separating hyperplane in $\mathbb{R}^n$, that separates the “positive” cases $\left{a_j \mid y_j=+1\right}$ from the “negative” cases $\left{a_j \mid y_j=-1\right}$. Among all separating hyperplanes, the one that minimizes $|x|^2$ is the one that maximizes the margin between the two classes that is, the hyperplane whose distance to the nearest point $a_j$ of either class is greatest.

We can formulate the problem of finding a separating hyperplane as an optimization problem by defining an objective with the summation form (1.2):
$$H(x, \beta)=\frac{1}{m} \sum_{j=1}^m \max \left(1-y_j\left(a_j^T x-\beta\right), 0\right) .$$
Note that the $j$ th term in this summation is zero if the conditions (1.9) are satisfied, and it is positive otherwise. Even if no pair $(x, \beta)$ exists for which $H(x, \beta)=0$, a value $(x, \beta)$ that minimizes (1.2) will be the one that comes as close as possible to satisfying (1.9) in some sense. A term $\lambda|x|_2^2$ (for some parameter $\lambda>0$ ) is often added to (1.10), yielding the following regularized version:
$$H(x, \beta)=\frac{1}{m} \sum_{j=1}^m \max \left(1 \quad y_j\left(a_j^T x \quad \beta\right), 0\right)+\frac{1}{2} \lambda|x|_2^2 .$$

## 数据科学代写|数据分析代写Data Analysis代考|Matrix Factorization Problems

$$\min X \frac{1}{2 m} \sum j=1^m\left(\left\langle A_j, X\right\rangle \quad y_j\right)^2$$

$$\min X \frac{1}{2 m} \sum j=1^m\left(\left\langle A_j, X\right\rangle \quad y_j\right)^2+\lambda|X|$$

## 数据科学代写|数据分析代写Data Analysis代考|Support Vector Machines

$$a_j^T x \quad \beta \geq 1 \quad \text { when } y_j=+1, a_j^T x \quad \beta \leq 1 \quad \text { when } y_j=1 .$$

$\backslash$ left 缺少或无法识别的分隔符 来自“负面”案例
$\backslash$ left 缺少或无法识别的分隔符 . 在所有分离超平面中，最小化的超平面 $|x|^2$ 是最 大化两个类之间的边距的超平面，即到最近点的距离的超平面 $a_j$ 任何一类都是最伟大的。

$$H(x, \beta)=\frac{1}{m} \sum_{j=1}^m \max \left(1-y_j\left(a_j^T x-\beta\right), 0\right) .$$

## 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。

Posted on Categories:Advanced Data Analysis, 数据科学代写, 统计代写, 统计代考, 高级数据分析

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 数据科学代写|数据分析代写Data Analysis代考|Art Shop web application

The art gallery application, with an inventory of artworks, was created with both MySQL(SQL), MongoDB(NoSQL) and a blockchain-based structure. With the art gallery application, the gallery owner can add and delete new artworks to their inventory and update the specified information of the works.

Relational database management system MySQL 8.0.21 and PhpMyAdmin 5.0.3 were used to implement the $\mathrm{SQL}$ database structure created with the tables in the diagram shown in Figure 3.2. Apache 2.4.41 is used for server management, and PHP $7.4$ is used for applications. As for servers, DigitalOcean’s servers with 2 GB memory/1 CPU were used with Ubuntu $20.04$ operating system.

For the NoSQL version of the Art Shop application, Mongo Atlas and MongoDB version $4.4 .6$ were used. On the server side, Ubuntu $20.04$ operating system and DigitalOcean’s 2 GB memory/1 CPU system were used. Strapi $3.6 .5$ was used as the content management system.

Three main collections were created on the Strapi in MongoDB: customers, artworks and transactions.

For the blockchain-supported version of the Art Shop web application, the Ubuntu $20.04$ operating system was used in DigitalOcean’s 2 GB memory, 1 CPU droplet. Starport $0.16$ was used to create and manage the blockchain. Go $1.16$ was installed to run Starport. Starport’s frontend application works with “Vue.js”. “Node.js” and “npm” were also installed to run these packages on the server.

## 数据科学代写|数据分析代写Data Analysis代考|Analysis

In the Art Shop application, the shop owner has three main options when they want to add pieces of art to the inventory:
1) adding data directly to the database with the command interface;
2) adding data using the graphical interface of the database management systems;
3) adding data using the specially developed application user interface.
The SQL- and NoSQL-enhanced custom commands can be used to add inputs in bulk with a json file or comma-separated values.

Adding new inputs is one of the tasks that can compare the performance of the systems. The scenario of a 150-line comma-separated values (CSV) file, seen in Figure 3.3, as sample data, and the art shop owner adding their inventory to the web application in one go, was implemented using the command interface. The SQL and NoSQL systems task was performed without any problems. It was completed in the times that can be seen in Figures $3.4$ and 3.5. The SQL system performed the task of adding bulk data faster than the NoSQL system. While the “LOAD DATA LOCAL INFILE” command is used directly on the server for $\mathrm{SQL}$, the “mongoimport” command, which provides connection to MongoDB Atlas servers via a local computer, is used for the NoSQL structure.

Multiple data were used in the $\mathrm{SQL}$ and NoSQL systems in order for the performance test to be meaningful during the addition of the entries, but since the Starport infrastructure used in the blockchain-supported system – as in all popular algorithms – actually uses Tendermint’s consensus algorithm, called BFT POS, the inputs must be added one by one (Ferdous et al. 2020). In order for a node to add more than one entry, the messages must be added to the Merkle-tree structure with unique hashes and proven one at a time.

## 数据科学代写|数据分析代写Data Analysis代考|Analysis

1）使用命令界面直接将数据添加到数据库；
2) 使用数据库管理系统的图形界面添加数据；
3) 使用专门开发的应用程序用户界面添加数据。
SQL 和 NoSQL 增强的自定义命令可用于通过 json 文件或逗号分隔值批量添加输入。

## 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。