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Yes. Would you change the title of this thread to 'An Integral and the Computer' or something more interesting

'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;

I will change it to exactly that.

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

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**anonimnystefy****Real Member**- From: The Foundation
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Hm, what about Romberg?

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**bobbym****Administrator**- From: Bumpkinland
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Romberg uses the trapezoidal rule and then generates an array of values. Hopefully with each column being more accurate because each column is a higher and higher Newton Cotes.

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

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**anonimnystefy****Real Member**- From: The Foundation
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So, how is each value generated, exactly?

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**bobbym****Administrator**- From: Bumpkinland
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It uses something called Richardson's Extrapolation to do that. I generally do not use it for anything but sequence acceleration and I am not sure how it does that either.

As I said the first column is as trapezoidal rule the successive columns are computed using this formula:

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

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**anonimnystefy****Real Member**- From: The Foundation
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The original integral to a 100 places (with trap):

Requires n=215004709762716014868701639070092447035983684173824 according to the error estimate.

*Last edited by anonimnystefy (2013-08-04 02:55:36)*

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**bobbym****Administrator**- From: Bumpkinland
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I would not recommend the trap rule for that many digits of precision.

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

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With what did you compute that?

'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
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He used the big boy on the block, M!

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

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**anonimnystefy****Real Member**- From: The Foundation
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I do not recommend it either. Just look at how large n it took!

*Last edited by anonimnystefy (2013-08-04 03:09:01)*

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**bobbym****Administrator**- From: Bumpkinland
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You do not believe that M actually did anything that many times?!

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

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**anonimnystefy****Real Member**- From: The Foundation
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Of course not! Nothing can do something like that that many times. What I do not know is what it did.

*Last edited by anonimnystefy (2013-08-04 03:22:58)*

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**bobbym****Administrator**- From: Bumpkinland
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Used acceleration techniques. What command did you use?

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

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**anonimnystefy****Real Member**- From: The Foundation
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Sum.

*Last edited by anonimnystefy (2013-08-04 03:31:28)*

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**bobbym****Administrator**- From: Bumpkinland
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Okay, that uses lots of acceleration making that command fast.

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

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**anonimnystefy****Real Member**- From: The Foundation
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Yes, thought it was something like that, but that number is still very large. Maybe it has to do with the fact that it all of them are very very small.

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**bobbym****Administrator**- From: Bumpkinland
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What is very small?

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

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**anonimnystefy****Real Member**- From: The Foundation
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(Cos[x])^100.

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**bobbym****Administrator**- From: Bumpkinland
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If you used Sum he probably was able to do the definite summation and get a closed form.

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

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**anonimnystefy****Real Member**- From: The Foundation
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It seems it really does have a closed form for the sum!

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**bobbym****Administrator**- From: Bumpkinland
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That would account for the speed and accuracy.

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

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**anonimnystefy****Real Member**- From: The Foundation
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I'm now trying to implement the Simpson's rule in M.

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**bobbym****Administrator**- From: Bumpkinland
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Should not be difficult. Nice clean formula for math style.

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

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**anonimnystefy****Real Member**- From: The Foundation
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Yes, it is a pretty one. Unlike Romberg...

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