Let p = [a, b] be the root of ƒ € C¹([a, b]), and assume f'(p) = f'(po) for some po € [a, b]. Consider an iteration scheme that is similar to, but different from Newton's method: given po, define Pn+1 = Pn f (Pn) f'(po)' n ≥ 0. Assuming that the iterative scheme converges, i.e. that pn → p as n → ∞, show that this method has order of convergence a = = 1.
Let p = [a, b] be the root of ƒ € C¹([a, b]), and assume f'(p) = f'(po) for some po € [a, b]. Consider an iteration scheme that is similar to, but different from Newton's method: given po, define Pn+1 = Pn f (Pn) f'(po)' n ≥ 0. Assuming that the iterative scheme converges, i.e. that pn → p as n → ∞, show that this method has order of convergence a = = 1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Newton's method
![Let \( p \in [a, b] \) be the root of \( f \in C^1([a, b]) \), and assume \( f'(p) \neq f'(p_0) \) for some \( p_0 \in [a, b] \). Consider an iteration scheme that is similar to, but different from Newton’s method: given \( p_0 \), define
\[
p_{n+1} = p_n - \frac{f(p_n)}{f'(p_0)}, \quad n \geq 0.
\]
Assuming that the iterative scheme converges, i.e. that \( p_n \rightarrow p \) as \( n \rightarrow \infty \), show that this method has order of convergence \( \alpha = 1 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fd47556-f3ce-4f39-818d-be563d9523c8%2Fb2824e8f-9fb7-4d72-a46e-c43f6942658f%2Fthfch09_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( p \in [a, b] \) be the root of \( f \in C^1([a, b]) \), and assume \( f'(p) \neq f'(p_0) \) for some \( p_0 \in [a, b] \). Consider an iteration scheme that is similar to, but different from Newton’s method: given \( p_0 \), define
\[
p_{n+1} = p_n - \frac{f(p_n)}{f'(p_0)}, \quad n \geq 0.
\]
Assuming that the iterative scheme converges, i.e. that \( p_n \rightarrow p \) as \( n \rightarrow \infty \), show that this method has order of convergence \( \alpha = 1 \).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
