Question

can someone explain the Newton-Raphson to me please???
i dont get it :S
what does it work out and how???

thanks
its for maths coursework so please help me
*desperate*

Answer 1

Well Newton-Raphson, also known as Newton's Method uses the slope of the function to approximate the value of the actual function.. I will have to draw this out and explain..

Answer 2

http://www.twiddla.com/569411

Answer 3

it would be great if you had an example to work with

Answer 4

the idea is this: you want a zero of some function, say
\[f(x)=x^3-4\] which is another way of saying you want the cube root of 4

Answer 5

now make a guess

Answer 6

use this x4+x-9=0
cause thats the equation i need to use do do the newton raphson method on

Answer 7

ok fine we do that one

Answer 8

\[f(x)=x^4+x-9\] make a guess

Answer 9

it doesn''t even have to be a particularly good guess, just something that gets you relatively close to 0

Answer 10

would you like me to make a guess?

Answer 11

??
this is what ive done so far
f(x)=x4+x-9
f’(x)=4x3+1


xr+1=xr - x4+x-9
4x3+1

the attachment is what ive got so far

Answer 12

ok you have the idea. now just make a guess. i guess 1

Answer 13

basically you look at an x value close to the solution , then t that x value, you go to the graph and draw a tangent line, and you use the x intercept of that tangent line as your new approximation ( which isnt always better )

Answer 14

but guess what???
:S

Answer 15

guess at some number that gives you something close to zero. i guess 1

Answer 16

i did the tangent thing on autograph :)

Answer 17

\[f(1)=-7\]

Answer 18

a lousy guess but who cares

Answer 19

\[f'(1)=5\]

Answer 20

so my new guess is
\[x_2=1-\frac{-7}{5}=1+\frac{7}{5}=2.2\]

Answer 21

now i repeat the process with 2.2. in other words you make a guess at what you think the zero is

Answer 22

call it g for guess. then you adjust by taking
\[g-\frac{f(g)}{f'(g)}\] and that is your new guess.

Answer 23

repeat until you think you are close enough, which you can tell because you will get the same decimals repeatedly

Answer 24

its 2.4
its in the table of mydoc.
but what does the newton raphson work out???
:S

Answer 25

its repeating tangent lines, and it doesnt always work

Answer 26

it gets you closer and closer to the zero. i can explain the geometry if you like

Answer 27

imagine you have a point close to the zero you are looking for. say
\[(x_1,y_1)\] where
\[y_1\] is near zero, but not exactly 0 (otherwise \[x_1\] would be your answer.

Answer 28

then find the equation of the line tangent to the graph. you do this by finding the slope, and using the point slope formula. you already have the point
\[(x_1,y_1)\] and the slope is
\[f'(x_1)\]

Answer 29

the equation for the tangent line is
\[y-y_1=f'(x_1)(x-x_1)\] now you want to know where this line crosses the x - axis (because you are trying to get to the zero of the function0 so set y = 0 and solve you get
\[-y_1=f'(x)(x-x_1)\]
\[x-x_1=-\frac{y_1}{f'(x_1)}\]
\[x-x_1-\frac{f(x_1)}{f'(x_1)}\]

Answer 30

im really sorry but i haven't got a clue what you are talking about :(
i get it when your talking about tangents and a repeated root
and getting closer to zero
but i dont understand what the x1, y1) stuff??
im lost
and what does the newton raphson actually work out???
roots???

Answer 31

ok lets start at the very beginning and forget about tangents etc

Answer 32

its just a linear approximation , dnt worry bout it too much

Answer 33

what it works out is a numerical method to find the zeros of some function

Answer 34

lets repeat your example because maybe it will make it clear. your equation is
\[x^4+x-9=0\] which is the same as saying we want the zeros of
\[f(x)=x^4+x-9\] yes?

Answer 35

yes

Answer 36

ok now here is the idea in a nutshell. it is a numerical method so you have to start somewhere. that is what i mean by "make a guess" guess some number \[x_1\] so that
\[f(x_1)=0\] approximately

Answer 37

so you approximate by a straight line ( which is easy to solve )

Answer 38

it doesn't have to be a very good guess. i would guess
\[x_1=1\] because i have no imagination and like integers

Answer 39

with me so far?

Answer 40

a linear eqn is alot easier to solve than a cubic

Answer 41

as a matter of fact now that i look that was the first guess on the sheet you sent

Answer 42

so far we have done nothing other than to make a guess at some number that gets us close to zero.

Answer 43

doesnt matter what you first guess is , as long as its not a stationary point, or near one.

Answer 44

unless the function has multiple turning points

Answer 45

now the idea is this. see you close you are by finding
\[f(x_1)\] in our case we get
\[f(1)=1^4+1-9=-7\] so it was a rather lousy guess because -7 is not very close to zero. but whatever, we will adjust

Answer 46

and this is how you adjust. find
\[f'(x)=4x^3+1\] and now find
\[f'(1)=5\]

Answer 47

"adjust" :| lol

Answer 48

so our first guess was 1
our adjusted guess will be
\[1-\frac{f(1)}{f'(1)}\]

Answer 49

in other words
\[1-\frac{-7}{5}=2.2\] and that number is on your table in the sheet you have. that is the second guess

Answer 50

well actually it is 2.4

Answer 51

so now what? well now you do it again, this time with 2.4 instead of 1
\[2.4-\frac{f(2.4)}{f'(2.4)}\]

Answer 52

now is where a calculator or spreadsheet would be handy because the numbers are annoying

Answer 53

but the table on the left is the sequence of numbers you get by repeating the process.

Answer 54

1
2.4
1.927895
1.700591
1.649107
1.646727
1.646722
1.646722

Answer 55

and the numbers in the next column show that when you evaluate the function at these numbers you get closer and closer to 0

Answer 56

-7
26.5776
6.74235427
1.064316798
0.045073275
9.23294E-05
3.89861E-10
0

Answer 57

i am fairly sure that this did not answer your question. but i am not sure if the question is a mechanical one "how do we use this method?" or a conceptual one "how does it work""

Answer 58

http://en.wikipedia.org/wiki/File:NewtonIteration_Ani.gif

Answer 59

so what does the newton raphson work out?
the roots?
and if you use this method does the equation have to be ^4?

Answer 60

copy paste that into the browser, it seemed to have broken the link :\

Answer 61

Maybe a pic would help a bit.

Answer 62

oh no it does not have to be 4th degree

Answer 63

yes, it works out the roots

Answer 64

you can use it to find square roots for example

Answer 65

in fact it is more or less how your calculator finds them

Answer 66

can be anything , e^(3x) + tan(x) +x^5 = 2

Answer 67

if you want to find
\[\sqrt{5}\] it is like saying find the roots of
\[f(x)=x^2-5\] and you can use this method for that as well

Answer 68

what elecengineer said. anything more or less

Answer 69

@estudier nice picture!

Answer 70

Thanks, did it help though?

Answer 71

any mix of functions , inverse functions etc, they can all be linearly approximated.

Thats the one thing you should understand/ appreciate , this method takes any messy equation and breaks it down to solving a linear equation

Answer 72

i like the little balloon.

problem is from the sheet this looks like a numerical methods class and not a calc one. there is precious little explanation on this paper

Answer 73

i mean on the paper missmystery sent

Answer 74

Yeah, methods, no background\theory, usual story...

Answer 75

well yes, i now know how to work it out cause my friends did it for me on the doc
and satellite you explained it very detailed :) thanks
but i dunno what else to put for my coursework tho
soz but my pc is really slow so it always takes a long while to reply :@

Answer 76

Which bit of the coursework is still causing a problem?

Answer 77

can you help me finish this sentence please???
But a disadvantage of using the decimal search is .....
and change in sign means.....
thanks
sorry pc is really really slow :(

Answer 78

can you hepl me complete the two sentences pleasE???
change in signs means???
and a disadvantage of the decimal search please

thanks

sorry my pc is very slow at the moment and a while ago it didnt even let me go onto openstudy
some message keeps poppin up :S

Answer 79

A disadvantage of NR is that the sequence may not converge to a zero of the function.

I'm not sure what u mean by change in signs (normally a sign change in relation to a gradient would mean that the curve had "turned").

Answer 80

whats nr???
is that the same as decimal serach???

thanks

and yes, i kinda remember now the change in sign is a turning point :)

Answer 81

nr is Newton Raphson method, were you meaning the decimal search algorithm?

Main problem with that is that the number of candidate solutions grows very fast and you need some kind of heuristics to keep things managable.

Answer 82

thats for the decimal search right

THANKS!!! :)