I often solve mathematical problems in a few steps. I get some numerical values for example from numerical integration. Then I want to use them in the next steps of computing for example as a part of integrand, a part of derivative or to find roots for new functions which are built on the basis of earlier points.
I am interesting in finding the most effective application of this function. Maybe you know interesting links, tips or particular application of this function. It is important to obtain the maximal accuracy for this application.
Dear Qutaibeh,
Thank you for your clear example.
I would like to consult with you the following simple example. I calculate numerical values certain function f[x] in the range [0, 1]. They change monotonically from 0 to Infinity.
What step for x would you chose for calculating integral in this range.(for example step =0.1; 0.01 or 0.001) to obtain optimal InterpolatingFunction[] for the next computing.
Let’s assume that next we want to calculate integral from Exp[n f[x]] where n=10,100,..
Here is one suggestion. Let suppose you are interested in evaluating the following integral: \int_0^2(n \sin x)dx for several ranges of n. Where in our case f(x)=sin(x) in a set of data points:
Then we might try something like this in Mathematica ( I am using Version 10):
myfunc[n_,f_]:=Nintegrate[Exp[n f[x]],{x,0,2}]
f1[p_]:=Interpolation[Table{x,Sin[x]},{x,0,2,p}]
In[15]:= myfunc[10, f1[0.05]]
Out[15]= 16064.6
and a check on accuracy:
In[14]:= NIntegrate[Exp[10 Sin[x]], {x, 0, 2}]
Out[14]= 16064.6
Thus by vary p we can find an acceptable value of p. Of course you can program Mathematica to find the value of p so that the error difference between p_n and p_{n+1} meets your convergence requirement.
You can also specify the the options for Interpolation, e.g, for periodic functions you can use PeriodicInterpolation->True; you can also specify the order or method of interpolation.
Hope this helps.
Comment regrading Qutaibeh Katatbeh suggestion: I would be careful about using InterpolationPolynomial. As the number of data points increases the order of the polynomial increases and consequently the interpolating curve can have large oscillations. Best to use the function Interpolation which by default uses a cubic polynomial between successive data points. This ensure the function is continuous at data points. You can change the interpolation order between data points if needed, or use the option Method->"Spline". That ensures the function and first derivative at data points are continuous. Check the Help section in Mathematica for further details
Dear Brian,
Thank you for your very interesting and instructive suggestions. I have read them with pleasure. I thought about examples like yours.
Regarding to your second comment about InterpolationPolynomial could you explain main difference between InterpolatingFunction[] and InterpolationPolynomial[].
If you have n data points the InterpolatingPolynomial function will generate a Lagrange interpolation polynomial of degree n such that the polynomial goes precisely through each data point. If you use Interpolation on the n data points, the function will generate a piecewise polynomial through each successive data points. By default it is a cubic polynomial. This ensures that the interpolation goes through each data point, but the derivative at each data point may not be continuous. A spline method ensures the first derivative is smooth. You can get more details from a web site I maintain when I teach Mathematica in Hanoi, Vietnam : https://sites.google.com/site/ekayasolutionsvietnam2/home
Dear Brian,
Thank you for your explanation of the idea of Interpolating function in Mathematica. The website looks very interesting and I think can be useful for me in this and other problems connected with using (programming in) Mathematica
- Badges
- Science topic