Mathematics · Year 13 · Numerical Methods · Summer Term

Iteration and Fixed Point Iteration

Learn to solve equations of the form f(x) = 0 by rearranging them into the form x = g(x) and using an iterative formula x(n+1) = g(x(n)) to find a root.

TL;DR:This topic introduces students to fixed-point iteration, a powerful numerical method for finding approximate solutions to equations that cannot be solved algebraically.

• National Curriculum Attainment Targets
• DfE Subject Content for Mathematics: I2 - Solve equations approximately using simple iterative methods, including recurrence relations of the form x(n+1) = g(x(n)).

About This Topic

Fixed-point iteration is a core component of the Numerical Methods section within the A-Level Mathematics curriculum. It provides students with an algorithmic approach to finding approximate roots for equations that are difficult or impossible to solve analytically, such as those involving a mix of polynomial, trigonometric, or exponential functions. This topic builds upon the foundational understanding of locating roots within an interval (e.g., via a change of sign) and introduces a more direct method for homing in on the solution.

The core concept involves rearranging an equation f(x) = 0 into the form x = g(x). The root of the original equation is then a 'fixed point' of the function g(x), graphically represented by the intersection of the curves y = g(x) and y = x. The iterative formula x(n+1) = g(x(n)) generates a sequence of values that, under certain conditions, will converge to this fixed point. A crucial part of this topic is the analysis of convergence, which depends on the gradient of g(x). Students must understand that for convergence, the condition |g'(x)| < 1 must be satisfied in the vicinity of the root. The graphical representations of this process, staircase and cobweb diagrams, are powerful visual tools that help students intuitively grasp the concepts of convergence and divergence.

Key Questions

1. Explain how rearranging an equation f(x) = 0 into the form x = g(x) relates to finding a fixed point.
2. Compare different rearrangements of the same equation and their effect on the convergence of the iterative process.
3. Analyse a graphical representation of an iterative process, such as a cobweb or staircase diagram, to determine if it converges or diverges.

Learning Objectives

• Rearrange an equation f(x) = 0 into an appropriate form x = g(x).
• Apply an iterative formula to calculate a sequence of approximations for a root.
• Determine the accuracy of a root by considering the convergence of the iterative sequence.
• Construct and interpret cobweb and staircase diagrams to show convergence or divergence.
• Demonstrate that a root lies within a given interval by showing a change of sign of f(x).

Key Vocabulary

Iteration	The process of repeating a calculation, where the output from one step becomes the input for the next.
Fixed Point	A value 'a' for which a function g(x) satisfies the equation g(a) = a. It is the x-coordinate of the intersection of y=g(x) and y=x.
Convergence	The process by which a sequence of approximations generated by an iterative formula approaches a specific, finite value (the root).
Divergence	The process by which a sequence of approximations moves progressively further away from the root.
Cobweb Diagram	A graphical depiction of an iteration that spirals inwards (convergence) or outwards (divergence), typically occurring when the gradient g'(x) is negative.
Staircase Diagram	A graphical depiction of an iteration that forms step-like movements towards (convergence) or away from (divergence) the fixed point, typically occurring when g'(x) is positive.

Watch Out for These Misconceptions

Common Misconception

Any rearrangement of f(x) = 0 into x = g(x) will successfully find the root.

What to Teach Instead

Convergence is not guaranteed. The iterative process only converges to a root 'a' if the gradient of g(x) is sufficiently shallow near the root, specifically, if |g'(a)| < 1. Different rearrangements lead to different g(x) functions with different gradients, so only some will work.

Common Misconception

The result of an iteration is the exact value of the root.

What to Teach Instead

Iteration provides a sequence of approximations that get progressively closer to the root. The process is stopped when a desired level of accuracy is achieved (e.g., the value is stable to 4 decimal places), but the result is still an approximation, not an exact value.

Common Misconception

A calculator's rounded answer can be used for the next step in the iteration.

What to Teach Instead

Using a rounded value as the input for the next step introduces rounding errors that can accumulate and affect the final accuracy. Students should always use the full, unrounded answer stored in the calculator's memory for the subsequent iteration.

Active Learning Ideas

Collaborative Problem-Solving

Rearrangement Race

In small groups, students are given an equation like x³ + x - 5 = 0 and must find as many valid rearrangements into the form x = g(x) as possible. They then test each rearrangement with the same starting value to see which ones converge, diverge, or oscillate, fostering a discussion on why some are better than others.

25 min·Small Groups

Collaborative Problem-Solving

Graphical Iteration Explorer

Using graphing software like GeoGebra or Desmos, students plot y=x and a given y=g(x). They then manually trace the iterative path (from x₀ to g(x₀) on the curve, then horizontally to y=x, then vertically to the curve again) to create their own cobweb or staircase diagram, visually reinforcing the process.

30 min·Pairs

Collaborative Problem-Solving

The Convergence Condition

Students are given several iterative formulae and the roots they converge to. They use their differentiation skills to calculate the value of |g'(x)| at the root for each formula and categorise the results based on whether the iteration converges or diverges, helping them to discover the condition |g'(x)| < 1.

20 min·Individual

Real-World Connections

• Calculating the yield to maturity for a bond in financial mathematics, which often involves solving complex polynomial equations.
• In physics and engineering, finding the equilibrium position of a system, such as the deflection of a loaded beam.
• Computer science algorithms for optimisation and machine learning, such as gradient descent, use iterative principles to find minimum values.
• Modelling population dynamics in ecology, where the population in one year is a function of the population in the previous year.
• In pharmacology, determining the steady-state concentration of a drug in the body after repeated doses.

Assessment Ideas

Quick Check

Use mini-whiteboards for a quick check. Give students an iterative formula and a starting value, and ask them to calculate and display the value of x₁, x₂, and x₃.

Quick Check

Set a multi-part exam-style question that requires students to first show a root exists in an interval, then use a given iterative formula to find the root to a specified accuracy, and finally sketch a diagram to illustrate why the iteration converges.

Quick Check

Provide students with a problem and a fully worked solution. Ask them to mark their own work, identifying where they made errors and writing a short sentence explaining what they need to remember for next time.

Frequently Asked Questions

Why do we need iterative methods if we can just solve equations on a calculator?

While modern calculators can solve many equations, understanding the underlying numerical methods is crucial in mathematics. These methods are the basis for the algorithms that calculators and computers use, and they are essential for solving problems in science and engineering where bespoke solutions are needed.

How do I choose a good starting value, x₀?

A good starting value should be close to the root. You can find an approximate location of the root by sketching the graph of f(x) to see where it crosses the x-axis, or by finding a narrow interval [a, b] where f(a) and f(b) have opposite signs.

What is the difference between a cobweb and a staircase diagram?

They are both graphical representations of the iteration. A staircase diagram occurs when 0 < g'(x) < 1, and the x-values approach the root from one side. A cobweb diagram occurs when -1 < g'(x) < 0, and the x-values oscillate around the root as they converge.