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Hi
Supposing we have:
Supposing we wished to know what a was in:
We could form a recurrence, we could expand out g(x) with a computer,
we could get the Taylor series of a few terms of g(x) and hope to spot a pattern.
Or we could try to get an asymptotic expression for the coefficients of a. Here is
how to do it with the help of a CAS.
Start by getting the Laurent expansion for g(x):
Now we expand L(x) around 0.
Substitute x = 1 to form a sequence.
Fit a function to the above values. In this case a 4th degree polynomial was tried and found to be okay.
Now we have an asymptotic form for the coefficients of g(x)!
Plug in 1000000
That should be a very good approximation to the exact answer
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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Hi bobbym,
Good one!
Have you tried to find the exact value?
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
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Hi gAr;
No, do you need 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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Hi,
No, I was trying to compute, just to compare.
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
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Hi gAr;
I will try to get at least an accurate answer using another method.
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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Hi bobbym,
Found that to be:
Close to your approximation, so I hope it's correct.
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
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Hi gAr;
You used the Knuth idea to get a smaller recurrence, very good. I am working
on a method that uses residues, it will take a while.
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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Hi,
Yes, I followed what the wise man said.
Okay, I'll wait.
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
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Hi gAr;
Both residue methods failed. I do not have any other methods other than the two
up there.
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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Hi bobbym,
Okay.
But what is the residue method? Will that yield exact value?
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
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Hi gAr;
Yes! But it is slowwwwww! If you want the coefficient
of x^100, you find the residue of g(x) / 101 at x = 0.
It is fast for small numbers but slows down for 1000001
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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Hi,
Hmmm, finding residues also involves expansion, isn't it? It's just as bad as expansion!
Knuth's method is the best bet for now!
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
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Hi gAr;
I did not mean to expand them. The residues are computed by an integral
and a limit. Also Sage should have a residue command.
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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Hi bobbym,
I thought that residues are calculated by CAS by Laurent's series, I don't know!
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
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Hi gAr;
Maybe, they have a definite integral over here but I can not get that
to work either.
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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Hi bobbym,
Can you put that integral here?
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
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Hi gAr;
I will try, they have another idea that I am not following at all.
Please give me some time.
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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Okay, no problem.
Take your time!
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
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Hi gAr;
I see what went wrong, the method is not for that rational type of generating function
it is for another type. In other words g(x) is not the correct form.
It also requires a numerical integration?
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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Hi bobbym,
Okay, you mean the other idea or the one you were trying earlier?
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
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Hi;
The residue idea should work for g(x) and I tested up to say x^10000
where it got exact answers. I just used Mathematica's built in
residue command. Residue[ g(x) / x^10001, {x, 0}]
The other idea uses an integral it too gets exact answers, I have tried on small problems
but not for rational forms like g(x). At least I do not know to use 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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Hi bobbym,
Okay.
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
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Hi gAr;
Here is an example of it working:
You need the coefficient of x^10
Which is correct without the i.
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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Hi bobbym,
I think a CAS would still need to expand it in order to integrate?
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
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Hi;
Not if you numerically integrate 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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