General Iterative Processes [H]

General Iterative Processes

An iterative process repeatedly applies the same rule — each output becomes the next input. To solve \( g(x)=0 \) iteratively: rearrange to \( x=f(x) \), write \( x_{n+1}=f(x_n) \), choose a starting value, and compute successive terms until convergence.

Convergence

Verified numerically: if successive values stabilise, the iteration converges. Different rearrangements of the same equation may converge or diverge — not every form works.

Growth and Decay as Iteration

\[ A_{n+1} = A_n \times m \qquad (m = 1+r/100) \implies A_n = P \cdot m^{n-1} \]

Compound growth/decay is the linear special case.

Worked Examples

\( x_{n+1}=\frac{1}{2}\left(x_n+\frac{6}{x_n}\right) \), \( x_1=3 \). Find \( x_2,x_3,x_4 \). What root?
\[ x_2=2.5 \qquad x_3=2.45 \qquad x_4\approx 2.449 \to \sqrt{6} \]
Show \( x^3-2x-5=0 \) gives \( x_{n+1}=\sqrt[3]{2x_n+5} \).
\[ x^3=2x+5 \implies x=\sqrt[3]{2x+5} \checkmark \]
From \( x_1=1.5 \) with \( x_{n+1}=\sqrt[3]{2x_n+5} \), find \( x_2,x_3 \) to 3 d.p.
\[ x_2=\sqrt[3]{8}=2.000 \qquad x_3=\sqrt[3]{9}\approx 2.080 \]

Converging toward root \( \approx 2.094 \).

 Key Takeaways

  • Rearrange \( g(x)=0 \) to \( x=f(x) \), write \( x_{n+1}=f(x_n) \). Fixed point = solution.
  • Verify convergence numerically. Try a different rearrangement if it diverges.
  • Confirm accuracy with a change-of-sign test at the rounding boundary.