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# 数学代写|最优化作业代写optimization theory代考|ENGR62 Piecewise-Polynomial Interlineation

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## 数学代写|最优化作业代写optimization theory代考|Piecewise-Constant Splines

Functions approximation by algebraic and trigonometric interpolation polynomials has several disadvantages. To begin with, the interpolation of Lagrange’s polynomials on a constant of nodes disagrees with each continuous function with the increase of the number of interpolation nodes. Secondly, algebraic polynomials of high degree cannot be recommended for the approximation of the function on the line segments $[a, b]$ that are far away from the beginning of coordinates (in other words $|a|,|b| \gg 1)$.

To prove this statement, it is enough to give the following example. If the option of a function $g(x),|g(x)| \leq M, x \in\left[10^k, 10^k+1\right]$ approximation by a polynomial $P_n(x)$ of the degree $n$, then it is obvious that we will obtain small values $|g(t)|$ performing arithmetic operations with the numbers $10^{k s}, s=n, n-1, \ldots$, that have been already, for example, when $k \geq 10, n \geq 10$, required computation with a large number of discharges (in other words, on computing machinery with a large bit grid).
This concerns bit-grid computing with very small values of a variable $X$. For example, for an arithmetic device that meets the IEEE standard, for numbers $a$, that are smaller than eps $=2^{-52} \approx 2.220446 e-016$, the relation $1.0+a=1.0$ is performed.

Thirdly, the Gibbs phenomenon (emergence) arises when the threshold functions are approximated by the Fourier trigonometric sums at the change points of the first order. Essentially, increasing the number of the Fourier partial sum, the error of approximation does not decrease at the change point but leads to the value that depends on the jump at this point.

## 数学代写|最优化作业代写optimization theory代考|Polynomial Splines of Orders r, r $\geq 1$

Piecewise-constant splines are the most studied special case of piecewisepolynomial splines with the possible discontinuities at the nodal points. An interest in them is explained by the simplicity of their design and effective use (for example, in the constructing quadrature formulae of rectangles, trapezoids, parabolas, etc., the integral function is replaced by a spline of the zero, the first and the second order, respectively).

In approximating continuous and differential functions in building the appropriate formulae of interlineation and interflatation, the best approximation is obtained using splines of orders $m \in N$. The following algorithms are offered below for their constructing.
Definition 3.13 Let $g(x) \in C[a, b], \pi: a=x_0<x_1<\cdots<x_n=b$.
A function
$$s_{n, 1}(x)=g\left(x_{k-1}\right) \frac{x-x_k}{x_{k-1}-x_k}+g\left(x_k\right) \frac{x-x_{k-1}}{x_k-x_{k-1}}, x_{k-1} \leq x \leq x_k, k=\overline{1, n}$$
is called interpolation spline of the first degree (piecewise linear spline).
This spline has the following properties $\left(\Delta=\max {1 \leq k \leq n}\left(x_k-x{k-1}\right)\right): s_{n, 1}\left(x_k\right)=g(-$ $\left.x_k\right), k=\overline{0, n}$
$$\begin{gathered} g(x) \in P C^1[a, b] \Rightarrow\left|R_n(x)\right|_{\infty}=\sup {a \leq x \leq b}\left|g(x)-s{n, 1}(x)\right| \leq \frac{\Delta}{2}\left|g^{\prime}\right|_{L_{\infty}[a, b]}, \ g(x) \in P C^2[a, b] \Rightarrow\left|R_n(x)\right|_{\infty} \leq \frac{\Delta^2}{8}\left|g^{\prime \prime}\right|_{\infty} ;\left|R_n^{\prime}(x)\right|_{\infty} \leq \frac{\Delta}{2}\left|g^{\prime \prime}\right|_{\infty} . \end{gathered}$$

## 数学代写|最优化作业代写优化理论代考|多项式样条函数的阶r, r $\geq 1$

$$s_{n, 1}(x)=g\left(x_{k-1}\right) \frac{x-x_k}{x_{k-1}-x_k}+g\left(x_k\right) \frac{x-x_{k-1}}{x_k-x_{k-1}}, x_{k-1} \leq x \leq x_k, k=\overline{1, n}$$

## 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。