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Then why are we not using them instead?

'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'

I'm not crazy, my mother had me tested.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,407

You are learning and this one is famous. Remember Newton discovered this in the 1600's when there were no computers or packages.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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What a sad time! He must have been very bored

'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'

I'm not crazy, my mother had me tested.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,407

Newton was a prodigious calculator too.He got about 35 digits of pi just for fun.

Heard of the Fibonacci cubic?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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Nope.

The next will tell me about it.

'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'

I'm not crazy, my mother had me tested.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,407

x^3 + 2x^2 + 10x = 20 he managed to get 10 correct digits or so for one of the roots. No why that is amazing?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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Because he had no computers and no bobbym?

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

I'm not crazy, my mother had me tested.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,407

He was lucky, my ancestors probably would have eaten him.

It is amazing because it was 500 years before Newton! No one knows how he got such an answer way back then.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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Why did he not write that down?

Mind I talk to you later?

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

I'm not crazy, my mother had me tested.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,407

Okay, I will see you then.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**thickhead****Member**- Registered: 2016-04-16
- Posts: 1,059

bobbym suggested Newton Raphson meyhod on excel sheet

```
iteration x -f(x)/f'(x) iteration -f(x)/f'(x)
0 0.333333333 0.166666667
1 0.333333333 -0.092227696 1 1.166666667 -0.029937788
2 0.241105638 -3.22768E-05 2 1.136728879 -0.000599103
3 0.241073361 -3.27892E-10 3 1.136129776 -2.82214E-07
4 0.241073361 0 4 1.136129493 -6.2817E-14
5 1.136129493 -1.04521E-16
iteration x -f(x)/f'(x)
-1 0.5 iteration x -f(x)/f'(x)
1 -0.5 4537037 -2 0.047619048
2 -0.74537037 0.036455252 1 -1.952380952 0.010069362
3 -0.708915119 -0.000294648 2 -1.94231159 0.000427204
4 -0.709209767 -7.12133E-09 3 -1.941884386 7.51194E-07
5 -0.709209774 2.10294E-16 4 -1.941883635 2.31898E-12
6 -0.709209774 -1.05147E-16 5 -1.941883635 -8.44021E-16
iteratin x -f(x)/f'(x) iteration x -f(x)/f'(x)
-1.5 0.00297619 2 -0.142857143
1 -1.49702381 2.31318E-06 1 1.857142857 -0.068711871
2 -1.497021496 1.51145E-12 2 1.788430986 -0.016602135
3 -1.497021496 8.47227E-16 3 1.771828851 -0.000914114
4 -1.497021496 0 4 1.770914737 -2.68561E-06
5 -1.497021496 0 5 1.770912051 -2.31345E-11
```

I tried various values of x as starting point to avoid long division.

**{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha{Gods rejoice at those places where ladies are respected.}**

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,407

Hi thickhead;

You got it the first time? That is very good.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**thickhead****Member**- Registered: 2016-04-16
- Posts: 1,059

But I had made this table to see the change in sign

```
x f(x)
-3 233
-2 1
-1 1
0 -1
1 1
2 13
```

**{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha{Gods rejoice at those places where ladies are respected.}**

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I got it the first time too!

So, what's the next thing?

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

I'm not crazy, my mother had me tested.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,407

The Fibonacci polynomial.

x^3 + 2x^2 + 10x - 20 = 0

1) We will not use Newton but will make our own iteration.

2) We will learn when an iteration will converge to a value.

3) We will learn how to cobweb.

4) We will see some Neumannian error analysis in action.

5) Some old guy with no teeth will rant about teakettles, computers and seipper.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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Let's start with 1

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

I'm not crazy, my mother had me tested.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,407

1) it is!

Supposing we pull an x out of that poly:

Now we have another iteration.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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How do you know it converges?

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

I'm not crazy, my mother had me tested.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,407

There are tests but first use your command and enter 1. as the initial guess.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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It converges to 1.368808107821373

EM guys always check their work.

```
In[63]:= f[x_] := x^3 + 2 x^2 + 10 x - 20
In[64]:= f[1.36880810782137255657266599931866062649`16.]
Out[64]= 0.*10^-15
```

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

I'm not crazy, my mother had me tested.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,407

You can probably see that there are other iterations possible.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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Not really. The only other ones I know are bisection and Newton's

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

I'm not crazy, my mother had me tested.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,407

Hmmm:

Here are more:

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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Basically, any expression of x that could be obtained from the equation?

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

I'm not crazy, my mother had me tested.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,407

Here is another:

Basically, any expression of x that could be obtained from the equation?

Yep, as many ways as you can solve for x.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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