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#176 2013-08-03 03:45:15

Agnishom
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From: The Complex Plane
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Re: An Integral and the Computer

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


'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'
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#177 2013-08-03 04:21:37

bobbym
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From: Bumpkinland
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Re: An Integral and the Computer

Hi;

I will change it to exactly that.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#178 2013-08-03 05:32:36

anonimnystefy
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Re: An Integral and the Computer

Hm, what about Romberg?


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#179 2013-08-03 05:40:17

bobbym
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Re: An Integral and the Computer

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#180 2013-08-03 05:42:36

anonimnystefy
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Re: An Integral and the Computer

So, how is each value generated, exactly?


“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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#181 2013-08-03 05:47:43

bobbym
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Re: An Integral and the Computer

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#182 2013-08-04 02:54:31

anonimnystefy
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Re: An Integral and the Computer

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)


“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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#183 2013-08-04 02:57:01

bobbym
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Re: An Integral and the Computer

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#184 2013-08-04 02:58:14

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 16,876
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Re: An Integral and the Computer

With what did you compute that?


'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'
'Humanity is still kept intact. It remains within.' -Alokananda

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#185 2013-08-04 02:59:15

bobbym
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From: Bumpkinland
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Re: An Integral and the Computer

He used the big boy on the block, M!


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#186 2013-08-04 03:08:38

anonimnystefy
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Re: An Integral and the Computer

I do not recommend it either. Just look at how large n it took!

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


“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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#187 2013-08-04 03:10:26

bobbym
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From: Bumpkinland
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Re: An Integral and the Computer

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#188 2013-08-04 03:22:32

anonimnystefy
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Posts: 15,532

Re: An Integral and the Computer

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)


“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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#189 2013-08-04 03:26:24

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,776

Re: An Integral and the Computer

Used acceleration techniques. What command did you use?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#190 2013-08-04 03:31:07

anonimnystefy
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Registered: 2011-05-23
Posts: 15,532

Re: An Integral and the Computer

Sum.

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


“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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#191 2013-08-04 03:37:49

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,776

Re: An Integral and the Computer

Okay, that uses lots of acceleration making that command fast.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#192 2013-08-04 04:08:55

anonimnystefy
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Registered: 2011-05-23
Posts: 15,532

Re: An Integral and the Computer

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.


“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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#193 2013-08-04 04:14:35

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,776

Re: An Integral and the Computer

What is very small?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#194 2013-08-04 04:16:28

anonimnystefy
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Registered: 2011-05-23
Posts: 15,532

Re: An Integral and the Computer

(Cos[x])^100.


“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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#195 2013-08-04 04:19:46

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,776

Re: An Integral and the Computer

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#196 2013-08-04 04:25:13

anonimnystefy
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From: The Foundation
Registered: 2011-05-23
Posts: 15,532

Re: An Integral and the Computer

It seems it really does have a closed form for the sum!


“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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#197 2013-08-04 04:26:05

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,776

Re: An Integral and the Computer

That would account for the speed and accuracy.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#198 2013-08-04 04:33:20

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,532

Re: An Integral and the Computer

I'm now trying to implement the Simpson's rule in M.


“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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#199 2013-08-04 04:37:32

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,776

Re: An Integral and the Computer

Should not be difficult. Nice clean formula for math style.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#200 2013-08-04 06:49:53

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,532

Re: An Integral and the Computer

Yes, it is a pretty one. Unlike Romberg...


“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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