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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

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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Hi;

What expression are they iterating through Newton Raphson?

**In mathematics, you don't understand things. You just get used to them.**

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They? Who are they?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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Good question. Is there some function being iterated?

**In mathematics, you don't understand things. You just get used to them.**

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Actually, this is not something I am being taught at school.

I am just trying to learn it and know about its basics

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,883

Hi;

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

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

Have you read Gleicks book?

**In mathematics, you don't understand things. You just get used to them.**

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No! What's the name of the book?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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"Chaos Making a New Science" by James Gleick

**In mathematics, you don't understand things. You just get used to them.**

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What is the basic principle behind generation of fractals?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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The instability of Newton's iteration is the main thing!

**In mathematics, you don't understand things. You just get used to them.**

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Whats "instable" about it?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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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!

**In mathematics, you don't understand things. You just get used to them.**

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Chaos is nice!

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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Nope! Chaos is a big mess.

**In mathematics, you don't understand things. You just get used to them.**

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Why? Don't you enjoy randomness?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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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.

**In mathematics, you don't understand things. You just get used to them.**

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,979

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)*

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,883

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

**In mathematics, you don't understand things. You just get used to them.**

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,979

Hi Bobby,

I don't understand why that line needs changing.

Would this be better?

IF a-z=0 THEN

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,883

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

**In mathematics, you don't understand things. You just get used to them.**

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,979

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)

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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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

**In mathematics, you don't understand things. You just get used to them.**

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,979

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?

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,979

Maybe a FOR/NEXT loop?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,883

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,

**In mathematics, you don't understand things. You just get used to them.**

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