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# 数学代写|数值分析代写Numerical analysis代考|AMATH352 Newton’s Method

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## 数学代写|数值分析代写Numerical analysis代考|Newton’s Method

To derive an iteration method for solving (8.184), we generalize Newton’s method from Section 4.2. We begin by reviewing the use of a tangent plane approximation, from two-dimensional calculus.

Let $\left(x_0, y_0\right)$ be an initial guess for a solution $\alpha=(\xi, \eta)$ for the system (8.184). The graph of $z=f(x, y)$ is a surface in $x y z$-space, and we approximate it with a plane that is tangent to it at $\left(x_0, y_0, f\left(x_0, y_0\right)\right)$. The equation of this tangent plane is $z=p(x, y)$ with
$$p(x, y) \equiv f\left(x_0, y_0\right)+\left(x-x_0\right) f_x\left(x_0, y_0\right)+\left(y-y_0\right) f_y\left(x_0, y_0\right),$$
using the notation,
$$f_x(x, y)=\frac{\partial f(x, y)}{\partial x}, \quad f_y(x, y)=\frac{\partial f(x, y)}{\partial y},$$
the partial derivatives of $f$ with respect to $x$ and $y$, respectively. If $f\left(x_0, y_0\right)$ is sufficiently close to zero, then the zero curve of $p(x, y)$ will be an approximation of the zero curve of $f(x, y)$ for those points $(x, y)$ near to $\left(x_0, y_0\right)$. Because the graph of $z=p(x, y)$ is a plane, its zero curve is simply a straight line.

## 数学代写|数值分析代写Numerical analysis代考|The General Newton Method

To help motivate the form of Newton’s method for nonlinear systems of order larger than 2 , we first change the notation used for solving systems of order 2 . Replace (8.184) by
\begin{aligned} &F_1\left(x_1, x_2\right)=0 \ &F_2\left(x_1, x_2\right)=0 \end{aligned}
Introduce
$$\begin{gathered} x=\left[\begin{array}{l} x_1 \ x_2 \end{array}\right] \quad F(x)=\left[\begin{array}{l} F_1\left(x_1, x_2\right) \ F_2\left(x_1, x_2\right) \end{array}\right] \ F^{\prime}(x)=\left[\begin{array}{ll} \frac{\partial F_1}{\partial x_1} & \frac{\partial F_1}{\partial x_2} \ \frac{\partial F_2}{\partial x_1} & \frac{\partial F_2}{\partial x_2} \end{array}\right] \end{gathered}$$
$F^{\prime}(x)$ is the Frechet derivative of $F(x)$, and it is a generalization to higher dimensions of the ordinary derivative of a function of one variable. The system (8.193) can now be written as
$$F(x)=0$$
A solution of this equation will be denoted by $\alpha$.
Newton’s method (8.190) for solving $(8.184)$ becomes
\begin{aligned} F^{\prime}\left(x^{(k)}\right) \delta^{(k)} &=-F\left(x^{(k)}\right) \ x^{(k+1)} &=x^{(k)}+\delta^{(k)}, \quad k=0,1, \ldots \end{aligned}

## 数学代写数值分析代写Numerical analysis代考|Newton’s Method

$$p(x, y) \equiv f\left(x_0, y_0\right)+\left(x-x_0\right) f_x\left(x_0, y_0\right)+\left(y-y_0\right) f_y\left(x_0, y_0\right),$$

$$f_x(x, y)=\frac{\partial f(x, y)}{\partial x}, \quad f_y(x, y)=\frac{\partial f(x, y)}{\partial y}$$

## 数学代写|数值分析代写Numerical analysis代考|The General Newton Method

$$F_1\left(x_1, x_2\right)=0 \quad F_2\left(x_1, x_2\right)=0$$

$F^{\prime}(x)$ 是 Frechet 导数 $F(x)$ ，它是对一个变量的函数的普通导数的更高维度的推广。系统 (8.193) 现在可以写成
$$F(x)=0$$

$$F^{\prime}\left(x^{(k)}\right) \delta^{(k)}=-F\left(x^{(k)}\right) x^{(k+1)} \quad=x^{(k)}+\delta^{(k)}, \quad k=0,1, \ldots$$

## MATLAB代写

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