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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

It sure does!

I feel just like my teachers.

**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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**ElainaVW****Member**- Registered: 2013-04-29
- Posts: 580

You aren't going to finish with that awful Colonel story?

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

You can bet a whole lot of bananas I am!

**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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```
nestList :: (Double -> Double -> Double) -> Double -> Double -> [Double]
nestList f x n = (x : (nestList f (f x (n+1.0)) (n+1.0)) )
f :: Double -> Double -> Double
f x n = (1.0 - 8.0*x*n)/n
```

```
*Main> take 21 $ nestList f 0.11778303565638346 0
[0.11778303565638346,5.7735714748932354e-2,3.811428200854117e-2,2.8419077265003995e-2,2.2647381879968037e-2,1.8820944960255704e-2,1.6099106984621026e-2,1.4064286980174654e-2,1.248570415860277e-2,1.1225477842288951e-2,1.0196177261688389e-2,9.339672815583799e-3,8.615950808662945e-3,7.995470453773365e-3,7.464807798384503e-3,6.948204279590642e-3,6.914365763274866e-3,3.508603305565781e-3,2.748672911102931e-2,-0.16726225394086605,1.3880980315269285]
```

*Last edited by Agnishom (2016-06-19 14:21:59)*

'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: 109,606

So your answer for a[20] was?

**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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1.3880980315269285

The error amplifies with each step

'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: 109,606

Correct a mundo.

The correct answer to 30 digits is 0.00531798937699268677949682431548, notice that a[19] does not even have the right sign!

What do you think went wrong?

**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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That multiplication by 8 and the subtraction.

'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: 109,606

The effect is called smearing. The subtraction had little effect.

So is that the end? Nope, there is a nice workaround. It will seem like magic!

**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 does smearing happen?

'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: 109,606

Why does smearing happen?

It happens all the time.

Anytime a multiplication occurs 101 * .123456 = 12.4690, actual answer 12.469006. We lost 2 digits, they were smeared out.

Anytime you have a number n, unless you are dealing with integers you have a small error called little e. You do not really have n but instead n + e. Everytime you do a calculation the little e is either no trouble because it remains small or...

Here is what happened.

1st iteration) 8 (n + e) = 8n + 8e. For the sake of brevity I will just chart the e.

2nd iteration) 8 (n + 8 e) = 64e

3rd iteration) 8 (n + 64 e) = 512e

3th iteration) 8 (n + 512 e) = 4096e etc.

The small e is now 4000 times larger than it was and that is only the third iteration! You lost about 18 - 20 digits doing that calculation and since you only had about 16 to start your final answer does not have a single correct digit.

**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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Oh I see!

What can we do about this?

'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: 109,606

Run the recursion in a different way!

**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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There is a different way?

'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: 109,606

If multiplying by some constant is unstable when that constant is greater than 1 than dividing by it must be stable.

We can run the recurrence backwards, first you solve for a[n-1].

Now we have the 8 on the bottom as a divisor. The error is going to work for us! What does this difference equation say. Well it runs from the higher numbers downwards ( backwards direction ). We need to start from say a[50] then we would get a[49], a[48], a[47]... all the way down to a[20].

Only one problem, how do we know what a[50] is?

**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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Something like 1 or 0 or Infinity?

'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: 109,606

Right! Start with anything at all, try a[50]=1241. At each step the system which is now an error divider will pick up almost 1 digit per iteration. By the time you get to a[20] you should have about 25 correct digits! Try it.

**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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And if I work all the way down to a[0], would it converge to log(9/8)?

'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: 109,606

That I would doubt, get a[20], which is the point.

**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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That is strange. I have never seen anything like this before

'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: 109,606

Numerical analysis, it is fantaaastic!

What did you get?

**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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You'll have to wait. I have a computer issue.

Does the sequence of a converge as n goes towards infty?

'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: 109,606

I do not understand the question

I am getting .005317989376992687 with all digits correct.

**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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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,037

Hmm, out Intro to NA professor mentioned that trick with the example of the integral

It's quite brilliant.

Also, yes, if you ran it to a[0], you would get an approximation of ln(9/8). This is also, I think, the only value of a[0] for which the sequence a[n] converges.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Thanks, I will try that for a[0]. I think the idea is credited to Wilkinson.

**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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