Exercise · ZW-FIXED-POINT-ITERATION-009 · ★★☆ Intermediate
If \( g(x) = \frac{x}{2} + 1 \), what is the limit of \( x_n \) starting from any \( x_0 \)?
Choose one option.
Show answer
- D. \( 2 \)
Explanation
The fixed point satisfies \( p = \frac{p}{2} + 1 \), so \( p = 2 \). Since \( |g'| = \frac{1}{2} < 1 \), the sequence converges to \( 2 \). The option \( 1 \) is the constant term, not the fixed point.
Review the concept: Fixed-point iteration
Iterative method that approximates a solution of the equation \( x = g(x) \) by generating the sequence \( x_{n+1} = g(x_n) \) from an initial value \( x_0 \).
\[x_{n+1} = g(x_n), \quad |g'(p)| < 1 \Rightarrow \text{convergencia local}\]
More practice problems: fixed-point iteration
- ZW-FIXED-POINT-ITERATION-001 What is the general formula of fixed-point iteration?
- ZW-FIXED-POINT-ITERATION-002 What condition defines a fixed point p of g ?
- ZW-FIXED-POINT-ITERATION-003 If |g'(p)| < 1 , what type of convergence is expected near p ?
- ZW-FIXED-POINT-ITERATION-004 What does the fixed-point theorem state about the existence of a fixed point?
- ZW-FIXED-POINT-ITERATION-005 What is x_1 if g(x) = cos (x) and x_0 = 0 ?
- ZW-FIXED-POINT-ITERATION-006 If g(x) = (x^2 + 2)/(3) and x_0 = 1 , what is x_2 ?
- ZW-FIXED-POINT-ITERATION-007 What is the linear convergence constant when g'(p) = -0.5 ?
- ZW-FIXED-POINT-ITERATION-008 For g(x) = √x + 6 , what is the positive fixed point?
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