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Q5. Consider the function f(x) = (x – v2)² e². a) What is the multiplicity k of the root r = v2? Explain why we cannot apply Newton's method theorem here. b) Indeed Newton's method applied to f will only converge linearly (you don't need to prove this). Show that by using the modified iteration k f(xn) f'(rn) Xn+1 = Xn - (1) with k from (a) and ro 2 v2, xn converges to the root r = /2 quadratically.

Q5. Consider the function f(x) = (x – v2)² e². a) What is the multiplicity k of the root r = v2? Explain why we cannot apply Newton's method theorem here. b) Indeed Newton's method applied to f will only converge linearly (you don't need to prove this). Show that by using the modified iteration k f(xn) f'(rn) Xn+1 = Xn - (1) with k from (a) and ro 2 v2, xn converges to the root r = /2 quadratically.

Advanced Engineering Mathematics
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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Q5. Consider the function f(x) = (x – v2)? e". V2? Explain why we cannot apply a) What is the multiplicity k of the root r = Newton's method theorem here. b) Indeed Newton's method applied to f will only converge linearly (you don't need to prove this). Show that by using the modified iteration k f(xn) f'(x,) Xn+1 = Xn - (1) with k from (a) and ro > v2, xn converges to the root r = v2 quadratically. Note: This procedure can be applied in general for roots of multiplicity k > 1, i.e. run the modified iteration (1) instead of the classical Newton's method. There is a problem however: in applications, we don't know what the k is.
Transcribed Image Text:Q5. Consider the function f(x) = (x – v2)? e". V2? Explain why we cannot apply a) What is the multiplicity k of the root r = Newton's method theorem here. b) Indeed Newton's method applied to f will only converge linearly (you don't need to prove this). Show that by using the modified iteration k f(xn) f'(x,) Xn+1 = Xn - (1) with k from (a) and ro > v2, xn converges to the root r = v2 quadratically. Note: This procedure can be applied in general for roots of multiplicity k > 1, i.e. run the modified iteration (1) instead of the classical Newton's method. There is a problem however: in applications, we don't know what the k is.
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