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#1 2012-04-11 16:08:58
- bobbym
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The RRA
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.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#2 2012-04-11 16:14:35
- anonimnystefy
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Re: The RRA
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 (2012-04-11 16:17:18)
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#3 2012-04-11 16:21:51
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Re: The RRA
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.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#4 2012-04-11 16:31:20
- anonimnystefy
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Re: The RRA
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...
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#5 2012-04-11 16:44:38
- bobbym
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Re: The RRA
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.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#6 2012-04-11 17:08:37
- anonimnystefy
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Re: The RRA
But how many terms do we take for a 10 digits approximation?
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#7 2012-04-11 17:10:49
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Re: The RRA
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.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#8 2012-04-11 17:18:01
- anonimnystefy
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Re: The RRA
Aha...Just what I thought. I just thought you had something better.
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#9 2012-04-11 17:25:55
- bobbym
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Re: The RRA
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.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#10 2012-04-11 17:30:03
- anonimnystefy
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Re: The RRA
Dunno,but it seems that every time I try to reuse a method,you have a better one.
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#11 2012-04-11 17:37:03
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Re: The RRA
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.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#12 2012-04-11 19:06:16
- anonimnystefy
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Re: The RRA
But,for now we should use Leibniz?
Can you give me a series to apply Leibniz to?
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#13 2012-04-11 20:52:35
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Re: The RRA
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.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#14 2012-04-11 21:32:14
- anonimnystefy
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Re: The RRA
An easy one,for starters.Then maybe we can gradually go to some unusual series.
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#15 2012-04-11 21:34:21
- bobbym
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Re: The RRA
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.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#16 2012-04-11 21:38:00
- anonimnystefy
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Re: The RRA
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.
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#17 2012-04-11 21:41:52
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Re: The RRA
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.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#18 2012-04-11 21:49:04
- anonimnystefy
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Re: The RRA
Well,our first approximation using Leibniz rules says we need 38 terms.But we can do better than that!
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#19 2012-04-11 21:53:28
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Re: The RRA
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.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#20 2012-04-11 22:01:47
- anonimnystefy
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Re: The RRA
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 22:02:13)
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#21 2012-04-11 22:12:35
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Re: The RRA
That is essentially what I got. You know what happened?
In mathematics, you don't understand things. You just get used to them.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#22 2012-04-11 22:16:00
- anonimnystefy
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Re: The RRA
Hi bobbym
No I don't.Did it make a faster converging series (obviously)?
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#23 2012-04-11 22:19:46
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Re: The RRA
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.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#24 2012-04-11 22:20:23
- anonimnystefy
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Re: The RRA
Then what?
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#25 2012-04-11 22:23:06
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Re: The RRA
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.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.