Solved examples: Fixed-Point Iteration for implicit equations in DE.

📚 Quick Study Guide

• 🔍 Fixed-Point Iteration is a method for approximating solutions to equations of the form $x = g(x)$.
• 🔢 In the context of differential equations, it's used to solve implicit equations where the unknown appears on both sides.
• 📝 The iteration formula is $x_{n+1} = g(x_n)$, where $x_0$ is an initial guess.
• 📈 Convergence depends on the choice of $g(x)$ and the initial guess $x_0$. Specifically, $|g'(x)| < 1$ near the fixed point is often a sufficient condition for convergence.
• 💡 To apply fixed-point iteration to an implicit equation $f(x, y) = 0$, rewrite it in the form $x = g(x, y)$ or $y = g(x, y)$.
• 🧪 The choice of rewriting can affect the convergence. Sometimes, rearranging the equation differently can lead to a convergent iteration.
• 🎯 The method continues until the difference between successive approximations, $|x_{n+1} - x_n|$, is smaller than a predefined tolerance.

Practice Quiz

1. Which of the following equations is suitable for fixed-point iteration?
   1. A. $x^2 - 3x + 2 = 0$
   2. B. $x = \cos(x)$
   3. C. $e^x = 0$
   4. D. $x + 5 = 0$
2. What is the general form of the fixed-point iteration formula?
   1. A. $x_{n+1} = f(x_n)$
   2. B. $x_{n+1} = g(x_n)$
   3. C. $x_{n+1} = h(x_n)$
   4. D. $x_{n+1} = x_n$
3. What is a sufficient condition for the convergence of fixed-point iteration?
   1. A. $|g'(x)| > 1$
   2. B. $|g'(x)| < 1$
   3. C. $g'(x) = 0$
   4. D. $g'(x) = 1$
4. Consider the equation $x = x^2 - 1$. What is the iteration formula?
   1. A. $x_{n+1} = x_n - 1$
   2. B. $x_{n+1} = x_n^2 - 1$
   3. C. $x_{n+1} = x_n^2 + 1$
   4. D. $x_{n+1} = 2x_n$
5. How do you typically start a fixed-point iteration?
   1. A. By setting $x_0 = 0$
   2. B. By choosing an arbitrary initial guess $x_0$
   3. C. By solving the equation analytically
   4. D. By setting $x_0 = 1$
6. What is the stopping criterion for fixed-point iteration?
   1. A. $|x_{n+1} - x_n| > \epsilon$
   2. B. $|x_{n+1} - x_n| < \epsilon$
   3. C. $x_{n+1} = x_n$
   4. D. $n \rightarrow \infty$
7. Consider the equation $f(x) = x^3 + x - 1 = 0$. Which rearrangement is suitable for fixed-point iteration?
   1. A. $x = 1 - x^3$
   2. B. $x = \frac{1}{x^2 + 1}$
   3. C. $x = x^3 - 1$
   4. D. $x = x^3 + 1$