Mastering Fixed Point Iteration and Cobweb Diagrams

In A-Level Mathematics, numerical methods provide powerful tools for solving equations that cannot be solved using standard algebraic techniques. Fixed point iteration is a fundamental method where we rearrange an equation into the form $x = g(x)$ to find an approximate root. Understanding this process is vital for your exams, as it tests both your algebraic manipulation and your ability to interpret numerical sequences.

Beyond the calculation, you will often be asked to visualise these iterations using cobweb or staircase diagrams. These graphical representations allow you to see whether a sequence converges to a root or diverges away from it. This article will guide you through the mechanics of the iterative formula and the geometry behind the diagrams.

Understanding the Iterative Formula

Fixed point iteration relies on the idea that if we have an equation $f(x) = 0$, we can rearrange it into the form $x = g(x)$. Once in this form, we can generate a sequence of approximations using the iterative formula:

$$x_{n+1} = g(x_n)$$

Starting with an initial value $x_0$, we calculate $x_1 = g(x_0)$, then $x_2 = g(x_1)$, and so on. If the sequence converges, the values will get closer and closer to the root $\alpha$, where $\alpha = g(\alpha)$.

Worked Example 1

Find the root of $x^3 - x - 4 = 0$ in the interval $(1, 2)$ using the iterative formula $x_{n+1} = \sqrt[3]{x_n + 4}$ with $x_0 = 1.5$.

1. $x_0 = 1.5$
2. $x_1 = \sqrt[3]{1.5 + 4} = \sqrt[3]{5.5} \approx 1.765$
3. $x_2 = \sqrt[3]{1.765 + 4} = \sqrt[3]{5.765} \approx 1.792$
4. $x_3 = \sqrt[3]{1.792 + 4} = \sqrt[3]{5.792} \approx 1.795$

The sequence is converging towards approximately $1.795$.

Visualising with Staircase and Cobweb Diagrams

To understand the behaviour of the iteration, we plot two graphs on the same axes: $y = x$ and $y = g(x)$. The root occurs where these two lines intersect.

• Staircase Diagram: This occurs when the gradient of $g(x)$ near the root is positive. The path moves in a "staircase" shape, stepping towards the intersection point.
• Cobweb Diagram: This occurs when the gradient of $g(x)$ near the root is negative. The path spirals or "weaves" around the intersection point, resembling a cobweb.

To construct these, start at $x_0$ on the x-axis. Move vertically to the curve $y = g(x)$, then horizontally to the line $y = x$. This new point on the line $y = x$ gives you the next $x$-value, $x_1$. Repeat this process to trace the path.

Convergence and Divergence

Not every rearrangement $x = g(x)$ will lead to a root. The success of the method depends on the gradient of $g(x)$ at the root. Specifically, if $|g'(x)| < 1$ at the root, the iteration will converge. If $|g'(x)| > 1$, the iteration will diverge, meaning the values will move further away from the root.

Worked Example 2

Consider $x = 2 - x^2$. If we start with $x_0 = 0.5$, we calculate:

1. $x_1 = 2 - (0.5)^2 = 1.75$
2. $x_2 = 2 - (1.75)^2 = -1.0625$
3. $x_3 = 2 - (-1.0625)^2 = 0.871$

In this case, the values oscillate. By sketching the cobweb diagram, you would see the path spiralling around the intersection point, illustrating the oscillatory convergence or divergence depending on the specific function's gradient.

Common Mistakes

1. Incorrect Rearrangement: Students often rearrange $f(x) = 0$ into $x = g(x)$ incorrectly. Always check that your rearrangement is algebraically equivalent to the original equation.
2. Ignoring the Initial Value: The choice of $x_0$ matters. If you are given an interval, ensure your $x_0$ is within or near that range to guarantee convergence.
3. Confusing the Graphs: Remember that the iteration moves between the curve $y = g(x)$ and the line $y = x$. Mixing these up will result in an incorrect diagram.
4. Rounding Errors: Always keep the full value in your calculator memory during iterations. Rounding too early can lead to significant errors in your final answer.

Frequently Asked Questions

What is the difference between a staircase and a cobweb diagram? A staircase diagram occurs when the function $g(x)$ has a positive gradient near the root, while a cobweb diagram occurs when $g(x)$ has a negative gradient.

How do I know if my iteration will converge? The iteration converges if the absolute value of the derivative $|g'(x)|$ is less than 1 at the root.

Can I use any rearrangement for fixed point iteration? No. Some rearrangements will diverge. If your sequence values are getting larger or oscillating wildly, try a different algebraic rearrangement of the original equation.

Conclusion

Fixed point iteration is a powerful numerical method that bridges the gap between algebra and geometry. By mastering the iterative formula and learning to sketch staircase and cobweb diagrams, you will be well-prepared for any numerical methods question on your A-Level Maths paper. For more practice and to see these concepts in motion, head over to MathInstructor AI to generate a free, narrated animated lesson on this topic.