The Algorithm: Newton's Method for Solving Systems of Two Lin. Eqs.

# The Algorithm for Newton's Method for Solving Systems of Two Nonlinear Equations

We will now look at the algorithm for Newton's method for solving systems of two nonlinear equations.

Let $\left\{\begin{matrix} f(x, y) = 0\\ g(x, y) = 0 \end{matrix}\right.$ be a nonlinear system of equations and assume that a solution $(\alpha, \beta)$ exists. Obtain an initial approximation $(x_0, y_0)$ near $(\alpha, \beta)$. Set a maximum number of iterations, and prescribed accuracy $\epsilon$.

For each $k = 0, 1, 2, ...$ up to the maximum number of iterations prescribed:

Step 1: Solve the linear system below for $\begin{bmatrix} \delta_{x, k}\\ \delta_{y, k} \end{bmatrix}$:

(1)
\begin{align} \quad \begin{bmatrix} \frac{\partial}{\partial x} f(x_{k}, y_{k}) & \frac{\partial}{\partial y} f(x_{k}, y_{k})\\ \frac{\partial}{\partial x} f(x_{k}, y_{k}) & \frac{\partial}{\partial y} f(x_{k}, y_{k}) \end{bmatrix} \begin{bmatrix} \delta_{x, k}\\ \delta_{x, k} \end{bmatrix} = - \begin{bmatrix} f(x_k, y_k)\\ g(x_k, y_k) \end{bmatrix} \end{align}

Step 2: Obtain the successive approximations with the following formula:

(2)
\begin{align} \quad x_{k+1} = x_k + \delta_{x, k} \\ \quad y_{k+1} = y_k + \delta_{y, k} \end{align}

Step 3: Check the accuracy require below:

(3)
\begin{align} \quad \biggr \| \begin{bmatrix} x_{k+1}\\ y_{k+1} \end{bmatrix} \begin{bmatrix} x_k\\ y_k \end{bmatrix} \biggr \|_{\infty} = \max \{ \mid x_{k+1} - x_k \mid , \mid y_{k+1} - y_k \mid \} \: ≤ \: \epsilon \end{align}

If the accuracy is achieved, then stop the iterations. If the accuracy is not achieved, then continue to obtain successive iterations until the maximum number of iterations allowed is obtained. If the maximum number of iterations allowed is obtained and the accuracy is not achieved, then print out a failure message.