Fixed-Point Iteration
Repeat x = g(x) until it converges.
Key idea
Repeat x = g(x) until it converges.
In depth
Rewrite f(x) = 0 as x = g(x) and iterate xₙ₊₁ = g(xₙ). It converges if |g′| < 1 near the fixed point, by the contraction mapping principle. Convergence is linear.
Worked example
x = cos x: iterating from 1 gives 0.540, 0.858, 0.654, ..., → 0.739.
Real-world uses
- Repeating a simple formula to estimate values such as interest rates or loan payments
Common mistake
Choosing a g with |g′| > 1, which diverges.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Fixed-Point Iteration?
Repeat x = g(x) until it converges. It is a topic at the advanced level in Differential Equations, Abstract Algebra & Applied Math.
Where is Fixed-Point Iteration used in real life?
Real-world examples: Repeating a simple formula to estimate values such as interest rates or loan payments.
What should I learn before Fixed-Point Iteration?
Make sure you are comfortable with Bisection Method, the previous topic in Differential Equations, Abstract Algebra & Applied Math.
What is a common mistake with Fixed-Point Iteration?
Choosing a g with |g′| > 1, which diverges.
How can I practise Fixed-Point Iteration?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Fixed-Point Iteration?
Move on to Interpolation next.
Related advanced topics in Differential Equations, Abstract Algebra & Applied Math
- What Is a Differential Equation? – An equation relating a function and its derivatives.
- Order and Linearity of ODEs – Classify equations by highest derivative and linearity.
- Direction Fields – Slope fields show solution curves visually.
- Separable Equations – Separate variables and integrate both sides.
- First-Order Linear Equations – Use an integrating factor.
- Exponential Growth and Decay Models – dy/dt = ky gives y = y₀eᵏᵗ.