Newton Fractal

Agnishom · 2012-11-23 14:47:13

Will anyone expain me the basic principle behind the generation of Newton Fractals?

I think I have a basic( and only a very basic) idea of The Newton-Raphson Method, Calculus and Complex Number

bobbym · 2012-11-23 17:17:00

Hi;

What expression are they iterating through Newton Raphson?

Agnishom · 2012-11-23 18:11:31

They? Who are they?

bobbym · 2012-11-23 18:18:34

Good question. Is there some function being iterated?

Agnishom · 2012-11-23 18:24:00

Actually, this is not something I am being taught at school.
I am just trying to learn it and know about its basics

bobbym · 2012-11-23 18:30:33

Hi;

newton's generates lots of fractals. here is one;

http://upload.wikimedia.org/wikipedia/c … nction.png

Have you read Gleicks book?

Agnishom · 2012-11-23 18:33:53

No! What's the name of the book?

bobbym · 2012-11-23 18:36:50

"Chaos Making a New Science" by James Gleick

Agnishom · 2012-11-23 19:36:37

What is the basic principle behind generation of fractals?

bobbym · 2012-11-23 19:39:46

The instability of Newton's iteration is the main thing!

Agnishom · 2012-11-23 19:45:01

Whats "instable" about it?

bobbym · 2012-11-23 19:50:22

We would have to go over it to really understand that. But I can say that everyone is taught to use it as model of efficiency. Oh, the way it zooms in on the roots of the canned problems that are given it in books.

Truth is unless you get a good initial guess, one that is close to the root in a special Newton's will go haywire.

In short it exhibits sensitivity to initial conditions, the mark of Chaos!

Agnishom · 2012-11-23 21:08:04

Chaos is nice!

bobbym · 2012-11-23 21:08:59

Nope! Chaos is a big mess.

Agnishom · 2012-11-23 21:16:29

Why? Don't you enjoy randomness?

bobbym · 2012-11-23 21:30:17

I am a numerical analyst. I consider that nothing but round off error, algorithm error. We try to eliminate it and Chaoticians are trying to show it, highlight it.

phrontister · 2012-11-23 23:02:52

Hi Bobby,

Truth is unless you get a good initial guess, one that is close to the root in a special Newton's will go haywire.

Yes, that's what happens with your towers problem if I use Newton's Method.

I wrote an LB program for it, and unless my initial guess is between 33.886745 and 39.999999, LB goes crosseyed.

    PROMPT "Enter approx. x (>33.886745 and <39.999999)";x$
    x=VAL(x$)
[loop]
    a=1/SQR(2500-x^2)+1/SQR(1600-x^2)-1/10
    IF ABS(a-z)<0.000000001 THEN PRINT "15-digit approximation of x: ";USING("##.#############",x):END
    z=a
    b=x/(2500-x^2)^(3/2)+x/(1600-x^2)^(3/2)
    x=x-a/b
    GOTO [loop]

Output (for input between 33.886745 and 39.999999):
    "15-digit approximation of x: 37.3550853341325"

Last edited by phrontister (2012-11-24 01:42:04)

bobbym · 2012-11-23 23:06:24

Hi;

Newton's has that problem and a few others.

In the case of that program you might try improving it by replacing this line:

IF a=z THEN

phrontister · 2012-11-24 00:05:21

Hi Bobby,

I don't understand why that line needs changing.

Would this be better?

IF a-z=0 THEN

bobbym · 2012-11-24 00:16:17

Remember a while back with the loop that counted down from 1 to 0 by .1?

phrontister · 2012-11-24 00:33:49

Hmmmm......thinking........still thinking.........and.......nope!

I think you mentioned that case not all that long ago, and I think that I recalled it then. But not now! Grumble.

What was it again? (sorry)

bobbym · 2012-11-24 00:40:38

The comparison of two floating point numbers for equality is always dangerous.

z = a is a possible problem

Take a look over here and see what I mean:

http://www.mathisfunforum.com/viewtopic … 56#p117156

post #18

phrontister · 2012-11-24 01:03:25

Ah...yes. I remember that now. *slaps forehead*

When I wrote "IF a=z THEN" I thought that a & z would never actually equal each other (except with an integer solution, maybe?), but I went for that option anyway because I couldn't come up with anything better then, and it happened to halt program execution at the right time.

I still haven't thought of a way to overcome that, and I couldn't glean anything from that past discussion. I probably don't understand it well enough now, although I think I did at the time.

Any hints for this one?

phrontister · 2012-11-24 01:05:08

Maybe a FOR/NEXT loop?

bobbym · 2012-11-24 01:29:09

Hi;

How about the solution of not testing for equality but test whether they are close

Abs[a-z]<.0005

You can adjust the .0005, it is just for illustration,

Math Is Fun Forum

#1 2012-11-23 14:47:13

Newton Fractal

#2 2012-11-23 17:17:00

Re: Newton Fractal

#3 2012-11-23 18:11:31

Re: Newton Fractal

#4 2012-11-23 18:18:34

Re: Newton Fractal

#5 2012-11-23 18:24:00

Re: Newton Fractal

#6 2012-11-23 18:30:33

Re: Newton Fractal

#7 2012-11-23 18:33:53

Re: Newton Fractal

#8 2012-11-23 18:36:50

Re: Newton Fractal

#9 2012-11-23 19:36:37

Re: Newton Fractal

#10 2012-11-23 19:39:46

Re: Newton Fractal

#11 2012-11-23 19:45:01

Re: Newton Fractal

#12 2012-11-23 19:50:22

Re: Newton Fractal

#13 2012-11-23 21:08:04

Re: Newton Fractal

#14 2012-11-23 21:08:59

Re: Newton Fractal

#15 2012-11-23 21:16:29

Re: Newton Fractal

#16 2012-11-23 21:30:17

Re: Newton Fractal

#17 2012-11-23 23:02:52

Re: Newton Fractal

#18 2012-11-23 23:06:24

Re: Newton Fractal

#19 2012-11-24 00:05:21

Re: Newton Fractal

#20 2012-11-24 00:16:17

Re: Newton Fractal

#21 2012-11-24 00:33:49

Re: Newton Fractal

#22 2012-11-24 00:40:38

Re: Newton Fractal

#23 2012-11-24 01:03:25

Re: Newton Fractal

#24 2012-11-24 01:05:08

Re: Newton Fractal

#25 2012-11-24 01:29:09

Re: Newton Fractal

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