root r. If en is the error of the nth iterate, i.e., en = In – r, then obtain lim en+1 n00 en 5. Bonus question: What happens to the above if the root r has a multiplicity m?

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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4.
It is given that a sequence of Newton iterates converge to a rootr of the function f(x). Further,
it is given that the root r is a root of multiplicity 2, i.e., f(x) = (x – r) g(x), where g(r) # 0. It is also
given that the function f, its derivatives till the second order are continuous in the neighbourhood of the
root r. If en is the error of the nth iterate, i.e., en = xn – r, then obtain
En+1
lim
n00 en
5.
Bonus question: What happens to the above if the root r has a multiplicity m?
Transcribed Image Text:4. It is given that a sequence of Newton iterates converge to a rootr of the function f(x). Further, it is given that the root r is a root of multiplicity 2, i.e., f(x) = (x – r) g(x), where g(r) # 0. It is also given that the function f, its derivatives till the second order are continuous in the neighbourhood of the root r. If en is the error of the nth iterate, i.e., en = xn – r, then obtain En+1 lim n00 en 5. Bonus question: What happens to the above if the root r has a multiplicity m?
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