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)
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.
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