An interactive calculator using Newton's method [1] to approximate the solutions of a system of two equations in two variables is presented. It approximates the solutions, if there is any, and gives a table of all values of iterations for educational purposes.

Newton's Method for Systems of Equations

Newton's method is numerical method used to find the roots of an equation by iterations starting from an approximate initial solution. When dealing with a system of equations, the method extends naturally by considering the Jacobian matrix and its determinant.

One Variable Newton's Method

Suppose that we need to solve the following equation
\[ f(x) = 0 \]
Taylor expansion of \( f(x+\Delta x) \) is given by
\[ f(x+\Delta x) \approx f(x) + \Delta x f'(x) \]
We now solve \( f(x+\Delta x) = 0 \) which gives
\[ f(x) + \Delta x f'(x) = 0 \]
which gives
\[ \Delta x \approx - \dfrac{f(x)}{f'(x)} \]
Suppose we know an approximate value \( x_n \) of the root of the equation, the approximate root \( x_{n+1} \) defined by
\[ \Delta x = x_{n+1} - x_n \]
is given by
\[ x_{n+1} \approx x_{n} - \dfrac{f(x)}{f'(x)} \]

System of Equations and the Jacobian Matrix

Consider a system of two equations in two variables \( x \) and \( y \):
\[
\begin{align*}
f(x, y) &= 0 \\
g(x, y) &= 0
\end{align*}
\]
The Jacobian matrix \( J \) of the system is given by:
\[
J = \begin{bmatrix}
\frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \\
\frac{\partial g}{\partial x} & \frac{\partial g}{\partial y}
\end{bmatrix}
\]

Newton's Method Update

The update formulas for Newton's method for a system of equations are given by:
\[
\begin{aligned}
\Delta x &= \frac{-f \cdot g_y + g \cdot f_y}{\text{D}} \\\\
\Delta y &= \frac{-g \cdot f_x + f \cdot g_x}{\text{D}}
\end{aligned}
\]
hence
\[
\begin{aligned}
x_{n+1} &\approx x_n + \frac{-f \cdot g_y + g \cdot f_y}{\text{D}} \\\\
y_{n+1} &\approx y_n + \frac{-g \cdot f_x + f \cdot g_x}{\text{D}}
\end{aligned}
\]
Where \( f \) and \( g \) are the functions evaluated at the current \( (x_n, y_n) \) values.
\( f_x, f_y, g_x, g_y \) are the partial derivatives of \( f \) and \( g \) with respect to \( x \) and \( y \), respectively.
\(\text{D} = f_x \cdot g_y - f_y \cdot g_x\) is the determinant of the Jacobian matrix.

Iterative Process

Newton's method iteratively updates the variables \( x \) and \( y \) using the above formulas until a stopping criterion is met. The common stopping criteria include:
Convergence tolerance: Stop when the difference between consecutive iterations is below a certain threshold.
Maximum iterations: Stop after a maximum number of iterations is reached.
1. Initialization: Start with initial guesses for \( x \) and \( y \):
One way to obtain close initial guesses for the solution to the system is to graph \( f(x,y) \) and \( g(x,y) \), approximate their points of interscetion use them as initial guesses.
2. Evaluate Functions and Derivatives: Compute \( f(x, y) \), \( g(x, y) \), and their partial derivatives at the current \( (x, y) \) values.
3. Calculate Determinant: Compute the determinant of the Jacobian matrix.
4. Update Variables: Use the Newton's method update formulas to compute \( \Delta x \) and \( \Delta y \).
5. Iterate: Update \( x \) and \( y \) using \( \Delta x \) and \( \Delta y \), and repeat until convergence or maximum iterations are reached.
6. The tolerance \( \epsilon \) is used to test the absolute value of \( f(x,y) \) and \( g(x,y) \) as follows:
When \( |f(x,y)| \lt \epsilon \) and \( |g(x,y)| \lt \epsilon \), the iteration process stops.
7. The calculator approximate one solution at the time.
Newton's method provides a robust and efficient way to approximate the solutions of a system of equations in two variables, given that the initial guesses are close enough to the actual solutions and the functions are differentiable in the neighborhood of the solutions.

Calculator

Results

Table including the iteration values of \( x, y, f(x,y) \) and \( g(x,y) \).

Iteration

\( x \)

\( y \)

\( f(x, y) \)

\( g(x, y) \)

More References and links

University Calculus - Early Transcendental - Joel Hass, Maurice D. Weir, George B. Thomas, Jr., Christopher Heil - ISBN-13 : 978-0134995540
Calculus - Gilbert Strang - MIT - ISBN-13 : 978-0961408824
Calculus - Early Transcendental - James Stewart - ISBN-13: 978-0-495-01166-8