(a) Verify that √a is a fixed point of the function 1 a g(x) 9) =j (+4). x 2 (b) Assume that, for some po, the fixed point iteration Pn+1 converges to p = √a as n → ∞. Determine the order of convergence. = g(Pn)
(a) Verify that √a is a fixed point of the function 1 a g(x) 9) =j (+4). x 2 (b) Assume that, for some po, the fixed point iteration Pn+1 converges to p = √a as n → ∞. Determine the order of convergence. = g(Pn)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Fixed Points and Convergence
**(a)** Verify that \( \sqrt{a} \) is a fixed point of the function
\[
g(x) = \frac{1}{2} \left( x + \frac{a}{x} \right).
\]
**(b)** Assume that, for some \( p_0 \), the fixed point iteration \( p_{n+1} = g(p_n) \) converges to \( p = \sqrt{a} \) as \( n \to \infty \). Determine the order of convergence.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fd47556-f3ce-4f39-818d-be563d9523c8%2F0d6203a4-892a-4c3d-85f9-2e28ea6c027f%2Frn3de5_processed.png&w=3840&q=75)
Transcribed Image Text:### Fixed Points and Convergence
**(a)** Verify that \( \sqrt{a} \) is a fixed point of the function
\[
g(x) = \frac{1}{2} \left( x + \frac{a}{x} \right).
\]
**(b)** Assume that, for some \( p_0 \), the fixed point iteration \( p_{n+1} = g(p_n) \) converges to \( p = \sqrt{a} \) as \( n \to \infty \). Determine the order of convergence.
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