The Algorithm for The F.P. Method: Solving Systems of Two Nonlin. Eqs.

# The Algorithm for The Fixed Point Method for Solving Systems of Two Nonlinear Equations

We will now summarize The Fixed Point Method for Solving Systems of Two Nonlinear Equations in the following algorithm.

Let $\left\{\begin{matrix} f(x, y) = 0\\ g(x, y) = 0 \end{matrix}\right.$ be a system of two nonlinear equations and let $(\alpha, \beta)$ be a solution to this system. Rewrite this system in the form $\left\{\begin{matrix} x = \phi (x, y) \\ y = \psi (x, y) \end{matrix}\right.$. Prescribe a desired level of accuracy $\epsilon$ and a maximum number of iterations.

Step 1: Select an initial approximation $(x_0, y_0)$ to $(\alpha, \beta) \in D$ where $D$ is the box such that $D = [a, b] \times [c, d]$.

Step 2: For $n = 0, 1, 2, ...$ to the maximum number of iterations, compute the successive approximations to the actual solution:

(1)
\begin{align} \quad x_{n+1} = \phi (x_n, y_n) \\ \quad y_{n+1} = \psi (x_n, y_n) \end{align}

Step 3: For each $n$, check the error $\biggr \| \begin{bmatrix} x_n\\ y_n \end{bmatrix} - \begin{bmatrix} x_{n-1}\\ y_{n-1} \end{bmatrix} \biggr \|_1 = \mid x_n - x_{n-1} \mid + \mid y_n - y_{n-1} \mid < \epsilon$. If the accuracy is achieved at some step, then finish. If not, the continue to compute successive approximations until the accuracy is achieved. If the accuracy is not achieved after the maximum number of iterations prescribed, then stop and print that the method failed to obtain the desired accuracy in the maximum number of iterations prescribed.