Question

A small doubt regarding Newton Raphson method.

Answer 1

I have to find a real root to the equation x = e^(-x) , using Newton Raphson method correct to 3 decimal places.
I know that the relation is :
\[\Large x_{n+1} = x_{n} - \frac{f(x_n)}{f'(x_n)}\]
One thing I notice here is that
\[\Large \frac{f(x_n)}{f'(x_n)} = -1\] always.
So our relation boils down to
\[\Large x_{n+1} = x_n + 1\].

Answer 2

Now my doubt lies in the fact that, I need to make an initial approximation here. Say I take x0 = 0. I evaluate f(0) and it turns out to be -1. Then my x1 would be 1 ..then 2..3 and so on.
This is a never ending sequence..because answer is nearly 0.60 (by manual calculations) but I don't ever come across it.

Answer 3

that's clever! very useful observation !

Answer 4

hehe thanks :P

Answer 5

So that relation never allows me to visit 0.60 because minimum difference is always 1. So how do I do this via NR?

Answer 6

it seems your relation may not work

Answer 7

what expression are you taking as f(x) ?

Answer 8

e^(-x)

Answer 9

why ?

Answer 10

f(x)=x-e^-x

Answer 11

Don't we represent the given function as x = (something) ?

Answer 12

f'(x)=1+e^-x

Answer 13

x= something in fixed point iteration

Answer 14

you're given an "equation" to solve
not function

@DLS

Answer 15

ohh..I see.. I confused both the methods XD
thanks for clarification^^ so it should be :
f(x) = e^-x - x right?

Answer 16

yeah or x-e^-x

Answer 17

any of them is correct

Answer 18

I see, thanks for clearing :) @ganeshie8 @jango_IN_DTOWN

Answer 19

Yeah N-R method will fail only if f'(x) is zero in the nbd of the point you took

Answer 20

So this question can't be done with fixed point iteration right ?

Answer 21

https://www.quora.com/When-does-Newton-Raphson-fail

Answer 22

why not?

Answer 23

you check if it satisfies the condition

Answer 24

okay yes, the condition of convergence holds true :D

Answer 25

then it will work

Answer 26

Any tips on how to take the initial approximation btw?

Answer 27

Currently I try to graph the given function if possible and find an approximate root..then take a value nearby.

Answer 28

use graphical method or tabulation metod

Answer 29

if you know where the root will be, say the root lies in between 0 and 1

Answer 30

f= e^(-x)
f'=-e^(-x)

y= c
y= f'(c) (x-c) + f(c)

0= f'(c) (x-c) + f(c) - c

0= xf'(c) -c f'(c) + f(c) - c

xf'(c) = c f'(c) - f(c) + c

xf'(c) = c (f'(c)+1) - f(c)

x = c (1 + 1/f'(c)) - f(c)/f'(c)

Answer 31

then you divide the inteval [0,1] into 10 equal subintervals and check in which interval the sign changes