newton methode with square root Use Newton iteration to find:√91, start with Xo = 10Xn+1Xn+ (A)Χη2

Let us assume that we need to find the square root of n∈N.Consider,x=nx2=nx2−n=0Assume f(x)=x2−n.…

Answered: Use Newton iteration to find: √91,…

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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newton methode with square root

Use Newton iteration to find:
√91, start with Xo = 10
Xn+1
Xn+ (A)
Χη
2

Expert Solution

Step 1: Explanation to Newton's Method

Let us assume that we need to find the square root of $n \in N$ .

Consider,

$x = \sqrt{n}$

$x^{2} = n$

$x^{2} - n = 0$

Assume $f (x) = x^{2} - n$ . Then $f^{'} (x) = 2 x$ .

By Newton's Method, we know that

$x_{n + 1} = x_{n} - \frac{f (x_{n})}{f^{'} (x_{n})}$

$x_{n + 1} = x_{n} - \frac{x_{n}^{2} - n}{2 x_{n}}$

after solving we get,

$x_{n + 1} = \frac{x_{n} + \frac{n}{x_{n}}}{2}$

In our problem, $n = 91$ . Therefore, formula reduces to

$x_{n + 1} = \frac{x_{n} + \frac{91}{x_{n}}}{2}$

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Use Newton iteration to find: √91, start with xo 10 Xn+1 A Xn+ ( = `xn' 2