Computation Contest #5 [2 SBI]

in #programming2 years ago (edited)

Here you can solve interesting problems using whatever programming language you like. Also you will earn SBI and sometimes STEM by doing so.
Also you might learn new things by doing so.
The tasks will be rather hard to solve without a programmable computer and some programming skills, but if you want to add a few million numbers by hand or similar, I would still give you the reward.
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Rules

No upvote, No resteem, No follow required!

I will give the prize randomly to those who solved the problem.

If two pieces of code are to closely related I might consider the later of them as copied which results in no prize for that person.

You have 4 days to solve it.

Even though this is about computation I will also accept algebraic solutions if you find one.

In order to get accepted you need to somehow attach your code.

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Problem

The problem of this contest is similar to what you did last week:
Screenshot from 2019-11-10 23-20-06.png

Integrate that function using a computer, but use a different technique:

  1. Generate the taylor polynomial at x=1(using your computer) for about 100 terms of the function you should integrate.
  2. Integrate that taylor polynomial using the power rule.
  3. Use that integral to calculate the integral.

Use that procedure to integrate said function from 0.9 to 1.1.
Compare the result to what the method from #4 gets.

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To everyone who already participated in a past contest, come back today and try a new problem(tell me if you don't want to be tagged):
@crokkon @kaeserotor @tonimontana

In case no one gets a result(which I doubt), I will give away the prize to the person who makes the most constructive description why the problem is too hard in your opinion.

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@contrabourdon sponsors my contests with 2 STEEM weekly.
You can support him by using a witness vote on untersatz, so he can further support this and other contests.

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I have a hard time modelling the taylor expansion already :l Just can't figure out the right expressions right now and have other work to do.

Should be doable with two loops, the outer one gives the order of the derivative, the inner one creates all the elements f'(a), f''(a)/2 and so on. The elements are put into a list since they are just constant factors. Integrating is easy and can be done by using the index of the list i. Simply multiply a factor (x-a)^(i+1)/(i+1)to every list element, put borders in there and calculate upper border minus lower border. Put those values into a list and take the sum over all of them to get the result.

I will give you a hint regarding the taylor expansion:

You can first find the taylor expansion of e^x and then divide this taylor expansion by x. Then you already have a part of the taylor series and only need to add the taylor series of 1/x.
This is easy, because deriving 1/x can be done easily and exact in the computer in comparison to deriving e^x/x.

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