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## 数学代写|最优化作业代写optimization theory代考|Minimization of the Maximum Error

According to Chebyshev polynomial expansion, we know that $\phi_n$ presents the lowest maximum on the interval $[-1,1]$. Thus, it will be possible to minimize the error expressed under the form of a polynomial of degree $n$ on the interval $[-1,1]$ by equaling it to $\phi_n(x)$.
For example, we can minimize the error term of the interpolation polynomial
$$R_n(x)=\left[\prod_{i=0}^n\left(x-x_i\right)\right] \frac{f^{(n+1)}(\xi)}{(n+1) !}$$
The term $f^{(n+1)}(\xi)$ can be considered as constant and we can simply minimize the polynomial of degree $(n+1)$ which appears under the form of the product that we make equal to $\phi_{n+1}(x)$. Thus, the terms $\left(x-x_i\right)$ are simply the $(n+1)$ factors of $\phi_{n+1}(x)$; it results that the roots of $\phi_{n+1}(x)$ are the roots of the corresponding Chebyshev polynomial, that is
$$x_i=\cos \left[\frac{(2 i-1) \pi}{2 n+2}\right], \quad i=1, \ldots, n+1$$

The previous reasoning was made on the interval $x \in[-1,1]$; it can be generalized to any interval $z \in[a, b]$ by doing the change of variable
$$z=\frac{x(b-a)+(b+a)}{2}$$
The minimization thus performed of the maximum error constitutes the minimax principle.
Generalization:
In general, we desire to find a polynomial $P_n^(x)$ of degree $n$ which minimizes the maximum deviation with respect to a function $f(x)$ on the interval $[-1,1]$. If the function $f(x)$ can be expanded with respect to Chebyshev polynomials (Fourier series expansion) $$f(x)=\sum_{i=0}^{\infty} a_i T_i(x)$$ then the partial sum $$P_n(x)=\sum_{i=0}^n a_i T_i(x)$$ constitutes a good approximation of $P_n^(x)$, and $P_n(x)$ will nearly be minimax.

## 数学代写|最优化作业代写optimization theory代考|Runge Phenomenon

When the number $(n+1)$ of points used for the interpolation of a function $f(x)$ increases, we can expect the convergence of the Lagrange interpolation polynomial $P_n(x)$ to increase. Difficulties can be highlighted when the function is nearly constant (plane function) or linear on some part of the domain of $x$ and has a totally different behavior in the other parts of the domain.

First, examine the case of regularly spaced points. Consider on the interval $[-2,2]$ the function $y=\exp \left(-10 x^2\right)$ which tends rapidly to 0 outside a domain close to the origin. The polynomial very well approximates the function around the middle of the interpolation interval $[a, b]$ and thus there is convergence, but on the opposite a divergence occurs close to the endpoints of the interval (Figure 1.10) when the degree $n$ of the interpolation polynomial increases. This is Runge phenomenon (Hairer 1993).

When irregularly spaced points are used for interpolation according to the equation of Chebyshev polynomial roots (Figure 1.11), the convergence is neatly improved. With respect to Figure $1.10$, the same Lagrange interpolation polynomial was used, only the position of the interpolation points changed.

# 最优化代写

## 数学代写|最优化作业代写optimization theory代考|最小化最大错误

. Error

$$x_i=\cos \left[\frac{(2 i-1) \pi}{2 n+2}\right], \quad i=1, \ldots, n+1$$

$$z=\frac{x(b-a)+(b+a)}{2}$$
，它可以推广到任何区间$z \in[a, b]$。这样对最大误差进行的最小化构成了极大极小原则。一般来说，我们希望找到一个度为$n$的多项式$P_n^(x)$，它使在区间$[-1,1]$上对函数$f(x)$的最大偏差最小化。如果函数$f(x)$可以对切比雪夫多项式(傅立叶级数展开)$$f(x)=\sum_{i=0}^{\infty} a_i T_i(x)$$展开，那么部分和$$P_n(x)=\sum_{i=0}^n a_i T_i(x)$$构成了$P_n^(x)$的一个很好的近似，并且$P_n(x)$将接近极小极大

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## MATLAB代写

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