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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

RRA: Riemann Rearrangement Algorithm

First shown to me by the courtesy of Thomas E. Gantner

This was supposedly used by Riemann himself to show the folly of rearranging series irresponsibly. Few people are aware of its real usefulness.

For any convergent alternating series

we can usually generate a faster converging series with the transformation

Notice that the series inside the curly brackets is itself an alternating series and the algorithm can be reapplied on it!

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

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,603

Hi bobbym

Good morning from me!

Thanks for creating a new thread! I assume you are going to expand your post so that we can later talk about it.

*Last edited by anonimnystefy (2012-04-10 18:17:18)*

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

Hi;

This thread will be devoted to it entirely. Examples and theory.

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

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,603

Cool! Again,thank you!

I figured out immideately why it works,but I am not sure how it applies to finite series or how many terms to take...

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

Hi;

I think the purpose is geared more to an infinite series which is being estimated by a finite one to some desired accuracy.

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

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,603

But how many terms do we take for a 10 digits approximation?

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

You can usually get the general term of the new alternating series

that is produced. From that you use the Leibniz rule.

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

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,603

Aha...Just what I thought. I just thought you had something better.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

It is enough to do the job, why do we need something else?

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

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,603

Dunno,but it seems that every time I try to reuse a method,you have a better one.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

There are always other methods. In numerical analysis there are thousands of methods. Experience helps in the choice of the correct one. Someday, you might read "Numerical Methods that Usually Work," by the great Forman S. Acton, my teacher. You will then learn that the greatest numerical mathematician is a chemist!

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

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,603

But,for now we should use Leibniz?

Can you give me a series to apply Leibniz to?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

Any alternating series that converges. What kind of example do you want?

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

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
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An easy one,for starters.Then maybe we can gradually go to some unusual series.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

I am not following you. All that Leibniz does is bound the tail of an alternating series. Is that what you want to do?

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

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,603

No.Give me a simple alternating function to estimate to a certain number of digits and then lower the number of terms needed to get that estimate.I want to do the process you showed me.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

How many terms are needed to estimate the above sum with 11 correct digits.

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

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
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Well,our first approximation using Leibniz rules says we need 38 terms.But we can do better than that!

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

Hi;

38 is correct! That is approximately what is going to be needed with a straight summation.

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

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
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I applied the RRA I got

When I used Leibniz again it told me I needed 37 terms.

*Last edited by anonimnystefy (2012-04-11 00:02:13)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

That is essentially what I got. You know what happened?

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

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,603

Hi bobbym

No I don't.Did it make a faster converging series (obviously)?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

From post #1

we can usually generate a faster converging series with the transformation

In this case it does not accelerate the convergence.

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

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,603

Then what?

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

Numerical techniques have conditions on them. When you violate those conditions strange things can occur. Take the simplest and most familiar numerical technique, Newtons iteration. People think they understand it but they do not. It is a very temperamental beast like the RRA.

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

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