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# 数学代写|优化理论代写Optimization Theory代考|MATH6231 Integrals Computation from High-Oscillating Functions

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## 数学代写|优化理论代写Optimization Theory代考|Basic Approaches to Constructing the Accuracy Optimal and Close to Them Quadrature and Cubature Formulae of Integrals Computation from High-Oscillating Functions

Consider the problem of the computation of the integral that looks
\begin{aligned} &I_1(\omega)=\int_a^b f(x) e^{-i \omega x} d x \ &I_2(\omega)=\int_a^b f(x) \sin \omega x d x \ &I_3(\omega)=\int_a^b f(x) \cos \omega x d x \end{aligned}
assuming that $f(x) \in F(F)$ is a certain class of functions, and $\omega$ is a certain real number ( $\omega \mathrm{I} \geq 2 \pi(b-a)$ ).

Let the information about $f(x)$ be given by $N$ values at nodes $\left{x_i\right}_0^{N-1}$ from its definition domain: $\left{f_i\right}_0^{N-1}=\left{f\left(x_i\right)\right}_0^{N-1}, \varepsilon_i$ characterizes the accuracy of the problem $f\left(x_i\right)=f_i:\left|\tilde{f}_i-f_i\right| \leq \varepsilon_i, i=\overline{0, N-1}$.

## 数学代写|优化理论代写Optimization Theory代考|Theories of Computational Complexity

Despite the achievements in the application software of modern computers, today there are many problems for which it is impossible to obtain a solution with given accuracy at limited computing resources. This is all about the problems of mathematical modeling, crystallography, radio astronomy, control of fleeting processes, cryptanalysis, and problems of high dimension.

As a rule, the solution of the applied problems is reduced to the solving typical classes of problems of computational and applied mathematics. Thus, it is important to create methods for building high-speed efficient algorithms for calculating $\varepsilon$-solutions of problems that use minimal computer memory for software. This will improve applied mathematical software and provide an opportunity to solve problems with less computing resources and reduce losses from the uncertainty of conclusions based on approximate solutions.

The main attention in the chapter is given to the creation of the elements of the complexity theory. With the use of it, this would be possible to construct effective complexity algorithms for computation of $\varepsilon$-solutions problems of numerical mathematics with limited computing resources.

Important results in the theory of computing optimization on the computing machinery were obtained by M. S. Bakhvalov, P. S. Bondarenko, V. V. Voievodin, H. Vozhniakovsky, V. V. Ivanov, M. P. Korneichuk, I. M. Molchanov, S. M. Nikolski, A. Sard, I. V. Sergienko, S. L. Sobolev, J. Traub, and others. These results allow estimating $\varepsilon$.

Computational complexity is less investigated than other characteristics. The complexity of the problem in time essentially depends on the computing model (computer architecture). A question of problem classes narrowing, the ways of input data presentation, and the complete use of a priori information on the problem are relevant for computational complexity minimizing of algorithm complexity of $\varepsilon$ solution constructing.

## 数学代写|优化理论代写Optimization Theory代考|Basic Approaches to Constructing the Accuracy Optimal and Close to Them Quadrature and Cubature Formulae of Integrals Computation from High-Oscillating Functions

$$I_1(\omega)=\int_a^b f(x) e^{-i \omega x} d x \quad I_2(\omega)=\int_a^b f(x) \sin \omega x d x I_3(\omega)=\int_a^b f(x) \cos \omega x d x$$

\left 的分隔符缺失或无法识别 表征问题的准确性 $f\left(x_i\right)=f_i:\left|\tilde{f}_i-f_i\right| \leq \varepsilon_i, i=\overline{0, N-1}$.

## 数学代写|优化理论代写Optimization Theory代考|Theories of Computational Complexity

MS Bakhvalov、PS Bondarenko、VV Voievodin、H. Vozhniakovsky、VV Ivanov、MP Korneichuk、IM Molchanov、
SM Nikolski、A. Sard、IV Sergienko、SL Sobolev、J. 特劳布等人。这些结果允计估计 $\varepsilon$.

## 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。