NEED FULL DETAILED ANSWER AND PROCESS. 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.

Answered: If Newton's method is used to find a…

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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NEED FULL DETAILED ANSWER AND PROCESS.

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.

Expert Solution

Step 1

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,

$N (x) = x - \frac{f (x)}{f^{I} (x)} .$ We have $N (3) = 3$ , (N(x) is the Newton Iteration function)

$N (3) = 3 - \frac{{(x - 3)}^{7}}{7 {(x - 3)}^{6}} = 3 - \frac{(x - 3)}{7} .$

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 $|f^{I} (0)| < 1 .$ (With this condition the Newton's iteration steps attracted towards (converges) that fixed point).