### 数学代写|优化算法代写optimization algorithms代考|Elements of the Computing Theory

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|优化算法代写optimization algorithms代考|Elements of the Computing Theory

Since every year the complexity of scientific and applied problems is increasing, there is an uprise of the need of a large amount of process of input data with the use of computing machinery for the numerical computation, and optimization of computations by all rational methods and means is a topical and important question. As a rule, the solution of the applied problems reduces to solving the typical problems of computational mathematics. Computational mathematics is a science about the methods of numerical solving problems of applied mathematics using modern computing machinery. The subject of computational mathematics is numerical methods (NM) or, that is the same thing, the multitude of computational algorithms (CA) and the question of their substantiation: the convergence and speed of NM convergence, their stability and error, the optimality on different criterion, the implementation time on the computing machinery, the necessary memory of computing machinery, etc.

The purpose of this chapter is to explain the main concepts and some results of computational mathematics, which are repeatedly used in the monograph, and at the same time, they have an independent value. The presentation of this information, in our opinion, is necessary for a confident possession of the given material.
In the first chapter:

• Certain supporting data in the theory of computing is given for determining the accuracy of the approximate solving problem, the quality of the used algorithm, and the comparative study of various algorithms according to some criteria.
• The principal features of the CA are determined (accuracy, execution time, necessary memory of the computing machinery), and the methods of their estimate are provided.
• There is a definition of optimal (inaccuracy and high speed), asymptotically optimal, and optimal in the order algorithms.
• The questions of identifying and clarifying a priori information on the problem are considered in cases when a priori information is not incorrectly set in the order to improve the quality of the obtained approximate solutions of a problem.

## 数学代写|优化算法代写optimization algorithms代考|Theory of Error Computations

One of the main directions of development of modern computational mathematics is the improvement of the theory of error computations, the research of comparative study of CA, and the optimal algorithm development in the solving problems of computational and applied mathematics.

The solution of the majority of the problems with modern computing tools is based on the computational experiment (CE), which seamlessly unites the mathematical model (MM), the computational algorithm (CA), the computations on the computing machinery, and the experiment. CE is necessary for obtaining an adequate quantitative description of the studied phenomenon or process itself with the necessary accuracy for the least possible machine time. The process of observations and comparing them with the matched model of MM is conducted on the computing machinery to check the validity of MM. The implementation of MM is performed with the use of numerical methods (NM), which approximate the input model and make it suitable for practical use. The errors of this approximation, as well as the rounding errors in the implementation of NM on the computing machinery and the errors of measurement or observation of implementation of the studied process, must be considered in determining errors of MM or the adequacy of MM and in the process itself. The errors of input MM must be considered in the process of definition of the requirements for the accuracy of the solution of various problems that are solved within this model.

In this monograph, the most important computational aspects of the determination of the error of MM and constructing its optimal implementation are analyzed. The multifaceted approach is based on the analysis of the three main characteristics of the corresponding NM-accuracy, implementation time, and necessary memory on computing machinery. According to these characteristics, the comparative study and the optimization of the corresponding NM are performed.

Main Characteristics of CA Let the problem $P(I)$ is solved by the algorithm $A(X)$ on the computing machinery $C(Y)$, where $I, X, Y$ are finite multitudes (vectors) of parameters from which essentially $P, A, C$ are, respectively, dependent on. The components of the vector $I$ may include data on a priori qualities of solving a problem, for example, constant that constrains the absolute values of the order of the derivatives from the given functions, data on the accuracy of the input values, etc. A number of iterations of the algorithm, the degree of approximation, the step of the grid, etc., can be the vector $X$ components. Vector $Y$ can contain the number of cell digits of the computing machinery memory, the total volume of its RAM, the run time of the computing machinery, the quantity of the computing machinery

processors, the characteristics of the used operational systems and compiler programs, etc.

In the practice of computational solving problems on the computing machinery, the following are the characteristics of problems, algorithms, and computing machinery: $E(I, X, Y)$ is a global error of solution $E$ of the problem $P$ on the computing machinery $C$ with algorithm $A, T(I, X, Y)$ is the time that is required to obtain a solution of the problem, $M(I, X, Y)$ is a required computing machinery memory, and fef is a coefficient of technical and economic efficiency. Explain the characteristics of $T, M$, and fef..

Total time $T$ – The period of time from setting the problem $P(I)$ to its solution of CA $A(X)$ on computing machinery $C(Y)$ can be estimated as follows:
$$T=T_{1}+T_{2}+T_{3}+T_{4}$$
where $T_{1}$ is the time to set the problem and develop or select CA $A$ and computing machinery $C ; T_{2}$ is the programming time, transmission motion, and debugging $\mathrm{CA}$ $A ; T_{3}$ is the implementation time of $\mathrm{CA}, A$ on the computing machinery $C$; and $T_{4}$ is the time of interpretation of the obtained solution and its comprehension. Practically when estimating $T_{3}$, only the essential operators are often considered to the number and time of the operation performing of the computing machinery. In further detail, the given characteristic Tis described below and in the works $[2,85,97,114]$.

## 数学代写|优化算法代写optimization algorithms代考|Problem Statement of Optimization of Computation

Optimization of computations lies in the optimization of one of the introduced characteristics (in a general way on $I, X, Y$ ) in accordance with certain constraints on others $[118,111]$.

The first primary problem is to minimize time $T(I, X, Y)$ while upholding the real (Re) constraints on $M, E$, and $f e f$ :
$$\begin{gathered} T(I, X, Y)=\min {I, X, Y} \ M(I, X, Y) \leq M{\mathrm{Re}}, E(I, X, Y) \leq E_{\mathrm{Re}}, f e f(I, X, Y) \leq f e f_{\mathrm{Re}} . \end{gathered}$$

The second primal problem is to minimize the global error $E(I, X, Y)$ with constraints on $M, T$, and $f e f$.
$$\begin{gathered} E(I, X, Y)=\min {l, X, Y} \ M(I, X, Y) \leq M{\mathrm{Re},} \quad T(I, X, Y) \leq T_{\mathrm{Re}}, \quad f e f(I, X, Y) \leq f e f_{\mathrm{Re}} \end{gathered}$$
The third primal problem is to maximize a coefficient of the technical and economic effectiveness $f e f(I, X, Y)$ :
$$\begin{gathered} f e f(I, X, Y)=\max {I, X, Y} \ M(I, X, Y) \leq M{\mathrm{Re}}, \quad T(I, X, Y) \leq T_{\mathrm{Re}}, \quad E(I, X, Y) \leq E_{\mathrm{Re}} \end{gathered}$$
We make some remarks on the mentioned statements of the primal problems. Let the computing machinery $C(Y)$ be fixed. Then $T, M, E$, and fef depend only on $I, X$. It is convenient to consider $I$ to be a random value and consider the probabilistic characteristics of the values $T, M, E$, and $f e f$, which will also be the characteristics of CA $A$ and will depend only on $X$. We designate each of the characteristics $T, M, E$, and fef through $H(I, X)$ and designate the frequency distribution accordingly to $H, I$ through $P(H), P(I)$. The essential characteristics of $\mathrm{CA} A(X)$ is the mathematical expectation $M_{H}(X)$ and the dispersion $D_{H}(X)$ :
\begin{aligned} &M_{H}(X)=\int_{G} H P(I) d I=\int_{-\infty}^{\infty} H P(H) d H \ &D_{H}(X)=\int_{G}\left(H-M_{H}\right)^{2} P(I) d I=\int_{-\infty}^{\infty}\left(H-M_{H}\right)^{2} P(H) d H \end{aligned}
where $G$ is a domain of possible values of $I$.

## 数学代写|优化算法代写optimization algorithms代考|Elements of the Computing Theory

• 给出了计算理论中的一定支持数据，用于确定近似求解问题的准确性、所用算法的质量，以及按一定标准对各种算法进行比较研究。
• 确定了 CA 的主要特征（准确性、执行时间、计算机的必要内存），并提供了它们的估计方法。
• 在顺序算法中有最优（不准确和高速）、渐近最优和最优的定义。
• 在没有错误设置先验信息的情况下，考虑识别和澄清有关问题的先验信息的问题，以提高获得的问题近似解的质量。

## 数学代写|优化算法代写optimization algorithms代考|Theory of Error Computations

CA的主要特点让问题磷(一世)由算法解决一种(X)在计算机器上C(是)， 在哪里一世,X,是是有限数量的参数（向量），它们本质上是磷,一种,C分别依赖于。向量的组成部分一世可能包括有关解决问题的先验质量的数据，例如，约束给定函数的导数阶的绝对值的常数、有关输入值准确性的数据等。 算法的多次迭代，逼近的程度，网格的步长等，可以是向量X组件。向量是可以包含计算机内存的单元位数，其RAM的总容量，计算机的运行时间，计算机的数量

## 数学代写|优化算法代写optimization algorithms代考|Problem Statement of Optimization of Computation

F和F(一世,X,是)=最大限度一世,X,是 米(一世,X,是)≤米R和,吨(一世,X,是)≤吨R和,和(一世,X,是)≤和R和

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## MATLAB代写

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