The Fixed Point Method for Solving Systems of Two Nonlinear Eqs.

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

We will now look at an extension to The Fixed Point Method for Approximating Roots. Suppose that a solution $(\alpha, \beta)$ exists to the system of two nonlinear equations:

(1)
\begin{align} \quad \left\{\begin{matrix} f(x, y) = 0 \\ g(x, y) = 0 \end{matrix}\right. \end{align}

We first rewrite this system of linear equations in the following form:

(2)
\begin{align} \left\{\begin{matrix} x = \phi (x, y) \\ y = \psi (x, y) \end{matrix}\right. \end{align}

Let $(x_0, y_0)$ be an initial solution to this system that is sufficiently close to $(\alpha, \beta)$. Then for $n = 0, 1, 2, ...$ define successive iterated approximations to the solution $(\alpha, \beta)$ with the formulas:

(3)
\begin{align} \quad x_{n+1} = \phi (x_n, y_n) \quad , \quad y_{n+1} = \psi (x_n, y_n) \end{align}

On the Convergence of The Fixed Point Method for Solving Systems of Two Nonlinear Equations we will provide criterion for which this method is guaranteed to converge to the solution $(\alpha, \beta)$ provided that the initial approximation $(x_0, y_0)$ meets certain requirements.