Looking for a particular algorithm for numerical integration

Tag: algorithm,numerical-methods,numerical,numerical-integration

Consider the following differential equation
f(x) = g'(x)
I have a build a code that spits out values of the function f(x) for the variable x, where x goes from 0 to very large.

Now, I'm looking for a scheme that will analyse these values of f(x) in order to determine g(x). Does anybody have any suggestions? The main problem is that if I would calculate g(x) = Integral (f(x) * dx), then I'll end up with just a number (i.e. the area under the graph), but I need to know the actual function of g(x).

I've cross-posted this question here: http://math.stackexchange.com/questions/1326854/looking-for-a-particular-algorithm-for-numerical-integration

Best How To :

numerical integration always return just a number
- if you do not want the number but function instead
- then you can not use numerical integration for this task directly
Polynomial approach
- you can use any approximation/interpolation technique to obtain a polynomial representing f(x)
- then integrate as standard polynomial (just change in exponent and multiplication constant)
- this is not suited for transcendent, periodical or complex shaped functions
- most common approaches is use of L'Grange or Taylor series
- for both you need a parser capable of returning value of f(x) for any given x
algebraic integration
- this is not solvable for any f(x) because we do not know how to integrate everything
- so you would need to program all the rules for integration
- like per-partes,substitutions,Z or L'Place transforms
- and write a solver within string/symbol paradigm
- that is huge amount of work
- may be there are libs or dlls that can do that
- from programs like Derive or Matlab ...

[edit1] As the function f(x) is just a table in form

double f[][2]={ x1,f(x1),x2,f(x2),...xn,f(xn) };
you can create the same table for g(x)=Integral(f(x)) at interval <0,x>

so:

g(x1)=f(x1)*(x1-0)
g(x2)=f(x1)*(x1-0)+f(x2)*(x2-x1)
g(x3)=f(x1)*(x1-0)+f(x2)*(x2-x1)+f(x3)*(x3-x2)
...

this is just a table so if you want actual function you need to convert this to polynomial via L'Grange or any other interpolation...
you can also use DFFT and for the function as set of sin-waves

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