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#1 2012-12-26 00:42:08
- Serj
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Calculation with a given accuracy
Hello, guys! Plzz help me out here..
The proplem:
Real numbers X, ε given (X not equal to 0, ε > 0). Calculate the sum of the series with the accuracy ε (ε = 10^-3, 10^-4, 10^-5, 10^-6) and specify the number of summands. Put results into columns ε, sum, N. Run the calculation only for the first 10 terms. Numerical part of the problem is attached.
This actually is a problem for my programming class, but I can't understand math behind the problem. What is X here? How can I perform these calculations with the given accuracy. Pls. advice if possible. Thanks!!
#2 2012-12-26 00:57:30
- bobbym
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Re: Calculation with a given accuracy
Hi Serj;
Usually you are given the function first and then you use a Taylor series to determine what the error will be. Here you are deprived of that. Do you know what x is?
But since it appears to be an alternating series I will play with it because there are other ways to bound this.
I latexed the series for you
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#3 2012-12-26 01:16:49
- Serj
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Re: Calculation with a given accuracy
bobbym, thank you for your reply!
Unfortunately I do not know what X is and this makes it more confusing. Two things I am currently confused about are what I should do with X here and how to determine the level of accuracy. Can you pls. formulate what I should search for to find some information on solving for this kind of problems. Thanks!
#4 2012-12-26 01:23:35
- bobbym
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Re: Calculation with a given accuracy
Okay I have identified the function and it is
1 - Jo which is a Bessel function of the first kind.
You might not need to know what x is because convergence for series like that are usually when -1 < x < 1.
As a programming problem it is too tough because it is much better formulated and solved in Numerical Analysis.
I will need some time to work on it.
Okay, we are in luck. The Bessel function converges for all values of x. This is not difficult to see.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#6 2012-12-26 01:41:40
- bobbym
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Re: Calculation with a given accuracy
I have the answer that you require it will take some time to post and to check. Please hold on.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#8 2012-12-26 01:56:29
- bobbym
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Re: Calculation with a given accuracy
Hi;
Now we come to the first problem.
and specify the number of summands. Put results into columns ε, sum, N. Run the calculation only for the first 10 terms.
They obviously want a table. And they only want you to run the series for the first 10 terms. The problem is as x gets larger you will need more terms to meet the error.
To give the answer I would like to have what restrictions are on x. Is there more to the problem?
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#10 2012-12-26 02:24:50
- bobbym
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Re: Calculation with a given accuracy
Ahh! The other variant is correct! That is the bound you need to do what is expected.
I will post with
Basically without this constraint you would have x, ε, and the number of summands exceeding 10.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#12 2012-12-26 02:32:54
- bobbym
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Re: Calculation with a given accuracy
Hi;
Okay, I will solve with that constraint. If you need more later come back.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#14 2012-12-26 02:53:29
- bobbym
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Re: Calculation with a given accuracy
Hi;
I am assuming N is the number of summands or terms. For Abs(x)<1, I have chosen x = 1. This way the error is even slightly less than in the table.
What this means is you need 2 terms for an error < .001 For .0001 and .00001, 3 terms will suffice. For .000001, 4 terms are needed.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#15 2013-02-11 19:57:39
- Serj
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Re: Calculation with a given accuracy
pls. don't judge me harshly, but the problem has not been completely solved yet 
(but the deadline is coming... uggg)
You can see what I got so far in the attached picture.
I am able to run summation only for its first 9 terms. Then the denominator apparently gets to large for Turbo Pascal and it returns me an error.
What I cannot understand is the epsilon column. bobbym, can u pls. tell me what was the algorithm to find those errors in the table you posted. Shall I now, having my table, somehow derive them from the sums that I got? Thanks!
PS
I cannot upload the image, so it can be found here - master-chinese.ru/screen_shot.jpg
Last edited by Serj (2013-02-11 20:08:50)
#16 2013-02-11 22:29:01
- bobbym
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Re: Calculation with a given accuracy
Hi;
There is no algorithm. I just summed the series.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#18 2013-02-11 22:44:03
- bobbym
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Re: Calculation with a given accuracy
Yes, epsilon is the estimate of the error between the truncated sum and the actual answer. Do you want to know how to calculate that?
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#20 2013-02-11 23:06:30
- bobbym
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Re: Calculation with a given accuracy
Hi;
First notice that the series you are working on
is an alternating series. There is a nifty way to get the error of those types of series.
Take a look at this table:
Last edited by bobbym (2013-02-12 00:11:05)
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#22 2013-02-12 01:21:36
- bobbym
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Re: Calculation with a given accuracy
but do not now how I can get the error for each sum.
Sometimes calculating the error can be quite difficult or perhaps impossible. In this case it is easy.
But what I need is the formula that will allow me to calculate those errors.
You do not need a formula for this particular problem we use the alternating series rule. If you look at the table you might see what it is. If not I will show you.
Last edited by bobbym (2013-02-12 02:22:16)
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#25 2013-02-12 03:42:29
- bobbym
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Re: Calculation with a given accuracy
That is right. The actual error is always less then the first neglected term of an alternating series.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.