Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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4. If Newton's method is used to find a root of f(x) = (x – 3)" = 0,
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a. Will Newton's method converge for xo close to the root r = 3?
Explain.
b. What is the order of convergence, if it converges?
c. Will Newton's method converge for all xo? Explain.
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Transcribed Image Text:4. If Newton's method is used to find a root of f(x) = (x – 3)" = 0, - a. Will Newton's method converge for xo close to the root r = 3? Explain. b. What is the order of convergence, if it converges? c. Will Newton's method converge for all xo? Explain.
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Given f(x)=(x-3)7=0 . By just seeing the function we can realise that x = 3 is a root of the function with multiplicity 7. (Which means that 3 is a root "7 times!" ).

Now let us consider the Newton's formula for convergence,

N(x)=x-f(x)fI(x). We have N(3)=3, (N(x) is the Newton Iteration function)

N(3)=3-(x-3)77(x-3)6=3-(x-3)7.

Hence 3 is a fixed point of N.

Attracting point-Definition

A point x0 is a fixed point of a function f(x) if and only if f(x0) = x0. Such a  point x0 is called an attracting fixed point if fI(0)<1.(With this condition the Newton's iteration steps attracted towards (converges) that fixed point).

 

 

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