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## 数学代写|最优化作业代写optimization theory代考|Interpolation by Cubic Hermite Polynomial

The interpolation of a function given by a cubic Hermite polynomial (also called cubic Hermite spline) lies on the principle that the interpolation function passes through the endpoints of the interval and possesses the same derivatives at these endpoints.

Consider a variable $t \in[0,1]$ such that the function $f(t)$ to approximate and its derivatives are given at the endpoints of the interval. Note
$$f(t=0)=f_0, \quad f(t=1)=f_1, \quad f^{\prime}(t=0)=f_0^{\prime}, \quad f^{\prime}(t=1)=f_1^{\prime}$$
This imposes four constraints for the interpolating function $\tilde{f}(t)$ which can thus be a polynomial of degree 3 , that is
$$\tilde{f}(t)=a_0+a_1 t+a_2 t^2+a_3 t^3$$
The respect of the constraints results in the following linear system:

Numerical Methods and Optimization
23
\begin{aligned} &a_0=f_0 \ &a_0+a_1+a_2+a_3=f_1 \ &a_1=f_0^{\prime} \ &a_1+2 a_2+3 a_3=f_1^{\prime} \end{aligned}
from which the values of the polynomial coefficients are drawn
\begin{aligned} &a_0=f_0 \ &a_1=f_0^{\prime} \ &a_2=-3 f_0+3 f_1-2 f_0^{\prime}-f_1^{\prime} \ &a_3=2 f_0-2 f_1+f_0^{\prime}+f_1^{\prime} \end{aligned}
It is possible to reorder the interpolating function under the form
$$\tilde{f}(t)=f_0\left(1-3 t^2+2 t^3\right)+f_1\left(3 t^2-2 t^3\right)+f_0^{\prime}\left(t-2 t^2+t^3\right)+f_0^{\prime}\left(-t^2+t^3\right)$$
which is noted as
$$\tilde{f}(t)=f_0 h_{00}(t)+f_1 h_{01}(t)+f_0^{\prime} h_{10}(t)+f_1^{\prime} h_{11}(t)$$
where the polynomials $h_{i j}(t)$ are Hermite basis polynomials defined by
\begin{aligned} &h_{00}(t)=1-3 t^2+2 t^3 \ &h_{01}(t)=3 t^2-2 t^3 \ &h_{10}(t)=t-2 t^2+t^3 \ &h_{11}(t)=-t^2+t^3 \end{aligned}
which are displayed in Figure 1.12. The interpolation here defined on $[0,1]$ of course can be extended to subintervals $\left[x_i, x_{i+1}\right]$ of any domain $[a, b]$. This type of interpolation by Hermite cubic polynomials is frequently used to simulate ordinary differential equations with boundary conditions (Section 6.6) or partial differential equations (Section 7.10.2).

## 数学代写|最优化作业代写optimization theory代考|Interpolation by Spline Functions

The spline functions ${ }^1$ are extremely appreciated for their quality of graphical interpolation (de Boor 1978). There exist several types of spline functions, B-splines, cubic splines, and exponential splines.

The spline functions are produced by associating to a partition $\left[a=x_1, x_2\right.$, $\ldots, x_{n-1}, b=x_n$ ] a set of piecewise polynomial functions, i.e. to each subinterval $\left[x_i, x_{i+1}\right]$ a different polynomial $P_i(x)$ is associated. Nevertheless, all polynomials will have the same degree. Moreover, it is possible to add a condition such that the interpolated functions $f$ coincide at right with the polynomials $P_i\left(x_i\right)=f\left(x_i\right)$ except at $b$.

In this section, only the cubic splines which are the most used are described. A spline function $S(x)$ is a real function defined on the partition $\left[a=x_1, x_2, \ldots, x_{n-1}, b=x_n\right]$ which possesses the following properties:

• $S$ is twice continuously differentiable on $[a, b]$.
• $S$ coincides on each subinterval $\left[x_i, x_{i+1}\right]$ of $[a, b]$ with a polynomial of degree 3 .
Thus, a spline function is composed of pieces of cubic polynomials connected together so that, on one side, the first derivatives and, on another side, the second derivatives coincide at the nodes.

Frequently, one of the following conditions is added to ensure the unicity of the spline function $S$ :
(a) $S^{\prime \prime}(a)=0, S^{\prime \prime}(b)=0$ (natural spline function).
(b) $S^{(k)}(a)=S^{(k)}(b)$ for $k=0,1,2 ; S$ is periodic.
(c) $S^{\prime}(a)=y_1^{\prime}, S^{\prime}(b)=y_n^{\prime}$ with $y_1^{\prime}$ and $y_n^{\prime}$ given.
An important property of spline functions is that the spline function minimizes the following norm:
$$|f|^2=\int_a^b\left|f^{\prime \prime}(x)\right|^2 d x$$
which is thus the integral of the square of the absolute value of the curvature on the considered interval. This integral represents the energy of the spline.

The spline function can be compared to the curve drawn by a designer who takes his French curve ruler to draw by eye the best curve passing through a set of points.
On each subinterval $\left[x_i, x_{i+1}\right]$, a function $S_i(x)$ is defined so that the function $S(x)$ can be considered as the family of functions $S_i(x)$ connected on the subintervals composing the interval $[a, b]$.

# 最优化代写

## 数学代写|最优化作业代写优化理论代考|插值立方Hermite多项式

$$f(t=0)=f_0, \quad f(t=1)=f_1, \quad f^{\prime}(t=0)=f_0^{\prime}, \quad f^{\prime}(t=1)=f_1^{\prime}$$

$$\tilde{f}(t)=a_0+a_1 t+a_2 t^2+a_3 t^3$$

23
\begin{aligned} &a_0=f_0 \ &a_0+a_1+a_2+a_3=f_1 \ &a_1=f_0^{\prime} \ &a_1+2 a_2+3 a_3=f_1^{\prime} \end{aligned}

\begin{aligned} &a_0=f_0 \ &a_1=f_0^{\prime} \ &a_2=-3 f_0+3 f_1-2 f_0^{\prime}-f_1^{\prime} \ &a_3=2 f_0-2 f_1+f_0^{\prime}+f_1^{\prime} \end{aligned}

$$\tilde{f}(t)=f_0\left(1-3 t^2+2 t^3\right)+f_1\left(3 t^2-2 t^3\right)+f_0^{\prime}\left(t-2 t^2+t^3\right)+f_0^{\prime}\left(-t^2+t^3\right)$$

$$\tilde{f}(t)=f_0 h_{00}(t)+f_1 h_{01}(t)+f_0^{\prime} h_{10}(t)+f_1^{\prime} h_{11}(t)$$

\begin{aligned} &h_{00}(t)=1-3 t^2+2 t^3 \ &h_{01}(t)=3 t^2-2 t^3 \ &h_{10}(t)=t-2 t^2+t^3 \ &h_{11}(t)=-t^2+t^3 \end{aligned}

## 数学代写|最优化作业代写optimization theory代考|插值的样条函数

$S$在$[a, b]$上是二次连续可微的。 $S$在$[a, b]$的子区间$\left[x_i, x_{i+1}\right]$上与3次多项式重合。

(a) $S^{\prime \prime}(a)=0, S^{\prime \prime}(b)=0$(自然样条函数).
(b) $S^{(k)}(a)=S^{(k)}(b)$对于$k=0,1,2 ; S$是周期性的。
(c) $S^{\prime}(a)=y_1^{\prime}, S^{\prime}(b)=y_n^{\prime}$, $y_1^{\prime}$和$y_n^{\prime}$给定。样条函数的一个重要性质是样条函数使以下范数最小化:
$$|f|^2=\int_a^b\left|f^{\prime \prime}(x)\right|^2 d x$$

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