# 线性代数网课代修|数值线性代数代写Numerical linear algebra代考|MATH5373

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• 数值分析
• 高等线性代数
• 矩阵论
• 优化理论
• 线性规划
• 逼近论

## 线性代数作业代写linear algebra代考|Polynomial Interpolation

Since there are $n+1$ interpolation conditions in (2.2) a natural choice for a function $g$ is a polynomial of degree $n$. As shown in most books on numerical methods such a $g$ is uniquely defined and there are good algorithms for computing it. Evidently, when $n=1, g$ is the straight line
$$g(x)=y_{1}+\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\left(x-x_{1}\right),$$
known as the linear interpolation polynomial.
Polynomial interpolation is an important technique which often gives good results, but the interpolant $g$ can have undesirable oscillations when $n$ is large. As an example, consider the function given by
$$f(x)=\arctan (10 x)+\pi / 2, \quad x \in[-1,1] .$$
The function $f$ and the polynomial $g$ of degree at most 13 satisfying (2.2) with $[a, b]=[-1,1]$ and $y_{i}=f\left(x_{i}\right), i=1, \ldots, 14$ is shown in Fig. 2.1. The interpolant has large oscillations near the end of the range. This is an example of the Runge phenomenon. Using larger $n$ will only make the oscillations bigger. ${ }^{1}$

## 线性代数作业代写linear algebra代考|The Buckling of a Beam

Consider a horizontal beam of length $L$ located between 0 and $L$ on the $x$-axis of the plane. We assume that the beam is fixed at $x=0$ and $x=L$ and that a force $F$ is applied at $(L, 0)$ in the direction towards the origin. This situation can be modeled by the boundary value problem
$$R y^{\prime \prime}(x)=-F y(x), \quad y(0)=y(L)=0,$$
where $y(x)$ is the vertical displacement of the beam at $x$, and $R$ is a constant defined by the rigidity of the beam. We can transform the problem to the unit interval $[0,1]$ by considering the function $u:[0,1] \rightarrow \mathbb{R}$ given by $u(t):=y(t L)$. Since $u^{\prime \prime}(t)=$ $L^{2} y^{\prime \prime}(t L)$, the problem $(2.24)$ then becomes
$$u^{\prime \prime}(t)=-K u(t), \quad u(0)=u(1)=0, \quad K:=\frac{F L^{?}}{R} .$$
Clearly $u=0$ is a solution, but we can have nonzero solutions corresponding to certain values of the $\mathrm{K}$ known as eigenvalues. The corresponding function $u$ is called an eigenfunction. If $F=0$ then $K=0$ and $u=0$ is the only solution, but if the force is increased it will reach a critical value where the beam will buckle and maybe break. This critical value corresponds to the smallest eigenvalue of (2.25). With $u(t)=\sin (\pi t)$ we find $u^{\prime \prime}(t)=-\pi^{2} u(t)$ and this $u$ is a solution if $K=\pi^{2}$. It can be shown that this is the smallest eigenvalue of (2.25) and solving for $F$ we find $F=\frac{\pi^{2} R}{L^{2}}$.

## 线性代数作业代写linear algebra代考|Polynomial Interpolation

$$g(x)=y_{1}+\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\left(x-x_{1}\right)$$

$$f(x)=\arctan (10 x)+\pi / 2, \quad x \in[-1,1] .$$

## 线性代数作业代写linear algebra代考|The Buckling of a Beam

$$R y^{\prime \prime}(x)=-F y(x), \quad y(0)=y(L)=0,$$

$$u^{\prime \prime}(t)=-K u(t), \quad u(0)=u(1)=0, \quad K:=\frac{F L^{?}}{R}$$

# 计量经济学代写

## 在这种情况下，如何学好线性代数？如何保证线性代数能获得高分呢？

1.1 mark on book

【重点的误解】划重点不是书上粗体，更不是每个定义，线代概念这么多，很多朋友强迫症似的把每个定义整整齐齐用荧光笔标出来，然后整本书都是重点，那期末怎么复习呀。我认为需要标出的重点为

A. 不懂，或是生涩，或是不熟悉的部分。这点很重要，有的定义浅显，但证明方法很奇怪。我会将晦涩的定义，证明方法标出。在看书时，所有例题将答案遮住，自己做，卡住了就说明不熟悉这个例题的方法，也标出。

B. 老师课上总结或强调的部分。这个没啥好讲的，跟着老师走就对了

C. 你自己做题过程中，发现模糊的知识点

1.2 take note

1.3 understand the relation between definitions