I: Numerical methods: Solve equations approximately using simple iterative methods; be able to draw associated cobweb and staircase diagrams. Solve equations using the Newton-Raphson method and other recurrence relations of the form xn + 1 = g xn Understand how such methods can fail.

The idea

This topic covers approximate equation solving using iteration. You rearrange f(x)=0 into x=g(x) and use x_{n+1}=g(x_n). Cobweb and staircase diagrams show convergence or divergence: staircase when iterates approach a root monotonically, cobweb when they alternate. The Newton-Raphson method uses x_{n+1}=x_n - f(x_n)/f'(x_n) for faster convergence. You must understand failure: divergence when |g'(x)|>1, oscillation, or Newton-Raphson failing when f'(x_n)=0 or cycling. For example, solve x^3 - x - 1 = 0 using x_{n+1} = (x_n + 1)^{1/3} from x_0=1.5, and compare with Newton-Raphson. Diagrams help interpret convergence.

What earns the marks

• Rearranges an equation f(x)=0 into the iterative form x=g(x) correctly, ensuring the rearrangement is suitable for the starting value.
• Carries out iterations accurately, showing sufficient decimal places and continuing until a required degree of accuracy is reached.
• Sketches or describes a cobweb diagram when successive iterates alternate around the root, and a staircase diagram when they approach monotonically.
• Applies the Newton-Raphson formula x_{n+1}=x_n - f(x_n)/f'(x_n) correctly, including differentiation and substitution.
• Explains how an iterative method can fail, for example divergence when the derivative of g has magnitude greater than 1 near the root, or Newton-Raphson failure when f'(x)=0 or when iterates cycle.

Common mistakes and how to put them right

• Incorrect rearrangement leading to a divergent iteration.

  Put it right: Choose a rearrangement where |g'(x)|<1 near the root.

• Using degrees instead of radians when differentiating trigonometric functions in Newton-Raphson.

  Put it right: Use radians in calculus.

• Stopping iterations too early or rounding intermediate values excessively.

  Put it right: Keep full calculator precision and only round the final answer.

• Confusing cobweb and staircase diagrams.

  Put it right: Staircase occurs when iterates move steadily towards the root; cobweb occurs when they alternate sides.

Exam tips

• Show the iterative formula you are using and at least three iterations, even if the answer converges quickly, to demonstrate the method.
• When asked to justify a rearrangement, refer to the condition |g'(x)|<1 near the root or show that the iteration converges.
• For Newton-Raphson, write down f(x) and f'(x) clearly before substituting into the formula.
• If an iteration fails, explain why in terms of the gradient or the behaviour of the iterates, not just that it does not work.